Zeinab Chehadi

Mohammed Bouabdellaoui

Mehrnaz Modaresialam

Thomas Bottein

Marco Salvalaglio

Monica Bollani

David Grosso

Marco Abbarchi

Scalable Disordered Hyperuniform Architectures Via Nano-Imprint Lithography Of Metal Oxides

Keywords: Disordered hyperuniform materials, Sol-gel dip coating, Nano-imprint lithography, titania, silica

Fabrication and scaling of disordered hyperuniform materials remain hampered by the difficulties in controlling the spontaneous phenomena leading to this novel kind of exotic arrangement of objects. Here we demonstrate a hybrid top-down/bottom-up approach based on sol-gel dip-coating and nano-imprint lithography for the faithful reproduction of disordered hyperuniform metasurfaces in metal oxides. Nano-to micro-structures made of silica and titania can be directly printed over several cm 2 on glass and on silicon substrates. First, we describe the polymer mold fabrication starting from a hard master obtained via spontaneous solid-state dewetting of SiGe and Ge thin layers on SiO2. Then we assess the effective disordered hyperuniform character of master and replica and the role of the thickness of the sol-gel layer on the metal oxide replicas and on the presence of a residual layer underneath. Finally, as a potential application, we show the antireflective character of titania structures on silicon. Our results are relevant for the realistic implementation over large scales of disordered hyperuniform nano-and micro-architectures made of metal oxides thus opening their exploitation in the framework of wet chemical assembly.

INTRODUCTION

Disordered hyperuniform (dHU) materials are an emerging class of amorphous systems at the edge between ordered crystals (as they do not display long-range density fluctuations) and liquids (as they do not have Bragg peaks in diffraction). This idea was theoretically introduced about twenty years ago by S. Torquato and F. H Stillinger [START_REF] Torquato | Local Density Fluctuations, Hyperuniformity, and Order Metrics[END_REF] and its comprehension and exploitation in practical applications are still at their beginning. In the last years, dHU features have been highlighted in a variety of natural phenomena spanning from classical [START_REF] Donev | Unexpected Density Fluctuations in Jammed Disordered Sphere Packings[END_REF][START_REF] Kurita | Incompressibility of Polydisperse Random-Close-Packed Colloidal Particles[END_REF] to quantum systems [START_REF] Lesanovsky | Out-of-Equilibrium Structures in Strongly Interacting Rydberg Gases with Dissipation[END_REF][START_REF] Mitchell | Amorphous Topological Insulators Constructed from Random Point Sets[END_REF][START_REF] Gerasimenko | Quantum Jamming Transition to a Correlated Electron Glass in 1T-TaS2[END_REF] , from out-of-equilibrium at room temperature [START_REF] Hexner | Hyperuniformity of Critical Absorbing States[END_REF] to in equilibrium [START_REF] Zhang | Ground States of Stealthy Hyperuniform Potentials: I. Entropically Favored Configurations[END_REF][START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] including biological frameworks [START_REF] Jiao | Avian Photoreceptor Patterns Represent a Disordered Hyperuniform Solution to a Multiscale Packing Problem[END_REF] . Beyond fundamental understanding of natural phenomena, the interest in this peculiar class of correlated disorder springs from the potential applications it may have in anomalous light and matter transport. Generally speaking, disordered materials are less prone to fabrication imperfections with respect to the ordered counterparts while displaying spatial steadiness of the photonic modes [START_REF] Caselli | Spatial Steadiness of Individual Disorder Modes upon Controlled Spectral Tuning[END_REF] . As such, if obtained via self-assembly over large scales, they may be much less expensive. In optics (e.g. for anti-reflection coatings) another advantage of disordered structures with respect to ordered arrays of objects (e.g. such as moth-eye structures for anti-reflection coatings), is the lack of diffraction from which neat colorization and angular dependencies can spring [START_REF] Bochet-Modaresialam | Methylated Silica Surfaces Having Tapered Nipple-Dimple Nanopillar Morphologies as Robust Broad-Angle and Broadband Antireflection Coatings[END_REF] .

Artificial structures, such as waveguide polarizers [START_REF] Zhou | Hyperuniform Disordered Network Polarizers[END_REF] , lasers [START_REF] Degl'innocenti | Hyperuniform Disordered Terahertz Quantum Cascade Laser[END_REF] , metalenses [START_REF] Zhang | Experimental Demonstration of Luneburg Lens Based on Hyperuniform Disordered Media[END_REF] , photonic band gaps for light propagation [START_REF] Florescu | Designer Disordered Materials with Large, Complete Photonic Band Gaps[END_REF][START_REF] Man | Isotropic Band Gaps and Freeform Waveguides Observed in Hyperuniform Disordered Photonic Solids[END_REF][START_REF] Muller | Silicon Hyperuniform Disordered Photonic Materials with a Pronounced Gap in the Shortwave Infrared[END_REF] plasmonic metasurfaces [START_REF] Froufe-Pérez | Role of Short-Range Order and Hyperuniformity in the Formation of Band Gaps in Disordered Photonic Materials[END_REF] , and efficient light trapping in thin Si films [START_REF] Castro-Lopez | Reciprocal Space Engineering with Hyperuniform Gold Disordered Surfaces[END_REF] are recent examples of dHU systems, typically obtained via top-down fabrication methods or hybrid self-assembly and top-down [START_REF] Piechulla | Tailored Light Scattering through Hyperuniform Disorder in Self-Organized Arrays of High-Index Nanodisks[END_REF] . Nevertheless, dHU systems can be, in principle, obtained by self-assembly, exploiting advection [START_REF] Tavakoli | Over 65% Sunlight Absorption in a 1 Μm Si Slab with Hyperuniform Texture[END_REF] , emulsions [START_REF] Weijs | Emergent Hyperuniformity in Periodically Driven Emulsions[END_REF] , block copolymers assembly [START_REF] Weijs | Emergent Hyperuniformity in Periodically Driven Emulsions[END_REF] , and sedimentation [START_REF] Zito | Nanoscale Engineering of Two-Dimensional Disordered Hyperuniform Block-Copolymer Assemblies[END_REF][START_REF] Goldfriend | Hyperuniformity, and Instability in the Sedimentation of Irregular Objects[END_REF] . Other remarkable examples of dHU materials are those obtained via spontaneous pattern formation in systems following Swift-Hohenberg [START_REF] Wang | Hyperuniformity with No Fine Tuning in Sheared Sedimenting Suspensions[END_REF][START_REF] Becker | Complex Dewetting Scenarios Captured by Thin-Film Models[END_REF] and Cahn-Hilliard [START_REF] Stoop | Curvature-Induced Symmetry Breaking Determines Elastic Surface Patterns[END_REF] dynamics (respectively describing wrinkling and phase separation via spinodal decomposition) as recently pointed out by Ma and coworkers [START_REF] Cahn | On Spinodal Decomposition[END_REF][START_REF] Ma | Random Scalar Fields and Hyperuniformity[END_REF] . These spontaneous phenomena are mainly observed in liquid and soft materials.

Very recently, spinodal-like solid-state dewetting of SiGe alloys on SiO2 was demonstrated as an alternative, self-assembly method for the fabrication of high-dielectric constant, effectively dHU metasurfaces [START_REF] Ma | Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions[END_REF] . The advantage of this approach is the lack of constraints in the extension of the dewetted wafers [START_REF] Benali | Flexible Photonic Devices Based on Dielectric Antennas[END_REF] that, in principle, can be extended up to 8 inches using state-of-the-art molecular beam epitaxy reactors. Nevertheless, the implementation of dHU architectures made of such hard materials and having a typical size that can be tuned at will from tens of nm up to tens of m can be conveniently exploited as masters for nano-imprint lithography [START_REF] Naffouti | Complex Dewetting Scenarios of Ultrathin Silicon Films for Large-Scale Nanoarchitectures[END_REF] .

Here we show that SiGe-based micro-and nano-architectures can be faithfully reproduced in high-and low-dielectric constant metal oxides such as titania and silica, on silicon and glass wafers combining sol-gel dip-coating and nano-imprint lithography (soft-NIL). First, we describe shortly the SiGe master preparation via spinodal-like solid-state dewetting. Then, we discuss the differences in the polymeric mold preparation starting from a SiGe master with respect to conventional Si-based ones. We provide a detailed description of the sol-gel formulations and of the nano-imprint lithography process. We address the effective dHU character of the masters and of the metal oxide replicas by measuring their bidirectional reflectance distribution function under laser and white light illumination thus assessing the properties of the corresponding structure factor S(k). Note that, differently with previous investigations on these systems, here we directly measure the dHU character through light scattering rather than inferring it from the analysis of the morphology emerging patterns, thus addressing distinctive features of this class of disordered materials. Finally, study the effect of the initial sol-gel layer thickness on the quality of the metal oxide replicas and show its influence on the properties of our dHU metasurfaces as anti-reflection coatings on silicon. The hard masters used in this work were elaborated by a self-assembly method based on solid state dewetting of ultra-thin silicon films on insulator (UT-SOI) and SiGe (or Ge) deposition via molecular beam epitaxy (Figure 1a) [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF][START_REF] Abbarchi | Solid-State Dewetting of Single-Crystal Silicon on Insulator: Effect of Annealing Temperature and Patch Size[END_REF] . The UT-SOI substrates used here were commercial monocrystalline Si films, 14 nm thick, (001)-oriented on 25 nm thick buried oxide on a Si (001) substrate. All the samples in use were diced in squares of 2 cm × 2 cm in size. The surface preparation for epitaxy consists of ex-situ and in-situ cleaning. Ex-situ chemical cleaning consisted of: 1) 10' in acetone in ultrasonic bath; 2) 10' Ethanol in ultrasonic bath; 3) 10' in NO2 plasma; 4) 30'' in a solution of 10% HF-deionized H2O in N2 atmosphere in order to leave the surface passivated by H + . The samples are immediately transferred in the ultra-high vacuum of a molecular beam epitaxy growth chamber (RIBER MBE32) with a base pressure of ∼10 -10 Torr. In-situ cleaning consisted of a flash annealing at 700 °C for 30' in the molecular beam epitaxy chamber, in order to remove any residual native oxide on the UT-SOI surface.

EXPERIMENTAL SECTION MASTER PREPARATION

The molecular beam epitaxy reactor is equipped with solid sources (silicon and germanium). We use an electron gun for the deposition of silicon, which consists of a charge containing a silicon ingot, which is heated by electron bombardment vaporizing the Si atoms. The germanium is source instead, is placed in an effusion Knudsen cell. In this case the evaporation is activated via Joule effect dissipating an electric current in the cell. The growth rates of silicon and germanium for these samples are ∼1.2 nm per minute and ∼0.51 nm per minute respectively. Ge composition from 30 to 100% and thickness from 10 to 1500 nm are deposited depending on the final size of the 3D structures obtained via solid state dewetting.

Solid state dewetting was obtained by high temperature annealing. Depending on the initial SiGe layer thickness, Ge content and target morphology (e.g. separated islands, connected structures), the process can be done at temperature ranging between 400 to 800°C and time from 30 minutes to 6 hours. Examples of the final morphologies of dewetted structures used as hard masters for nano-imprint lithography are provided in Figure 5 [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] .

MOLD PREPARATION

SiGe is an unusual material for masters that more conventionally are made of Si, metal or plastic.

A specific development has been necessary to cast a polymer e.g. (polydimethylsiloxane, PDMS) mold on SiGe for its faithful replication (Figure 1 b). Owing to the surface properties of dewetted islands of SiGe or Ge, an improved hydrophobicity is necessary for demolding. This is obtained only with a much longer aging in 1 H, 1 H, 2 H, 2 H-perfluorooctyltriethoxysilane (POTS) vapor with respect to other types of masters used for soft-NIL [START_REF] Bochet-Modaresialam | Methylated Silica Surfaces Having Tapered Nipple-Dimple Nanopillar Morphologies as Robust Broad-Angle and Broadband Antireflection Coatings[END_REF][START_REF] Bottein | Environment-Controlled Sol-Gel Soft-NIL Processing for Optimized Titania, Alumina, Silica and Yttria-Zirconia Imprinting at Sub-Micron Dimensions[END_REF] . The same process on a flat Ge surface was obtained just after 1h, pointing to peculiar properties of Ge after dewetting. Indeed, in its crystalline state, germanium, similar to silicon and carbon (diamond), is a covalent solid that crystallizes into a diamond cubic lattice structure. However, when prepared under vacuum conditions, both (100) and (111) surfaces of Ge undergo extensive reconstructions in their surface atomic geometry exhibiting a zwitterionic property [START_REF] Loscutoff | REACTIVITY OF THE GERMANIUM SURFACE: Chemical Passivation and Functionalization[END_REF][START_REF] Mui | Proton Transfer Reactions on Semiconductor Surfaces[END_REF] : the upper Ge atoms are more nucleophilic, whereas the lower ones are more electrophilic. During dewetting many different crystallographic faces form (according to the equilibrium shape of the material in use). These reconstructions involve the variation of the underlying nucleophilic properties. Thus, in order to enable the silane groups of POTS vapor to react with all nucleophilic sites on the islands surface, the functionalization requires a long time (e.g., about 12 hours in our case, against 1 hour or less, for conventional masters). PDMS molds were prepared as follows (Figure 1b): firstly, the surface of the master was functionalized with silane group by using 1 H, 1 H, 2 H, 2 H-perfluorooctyltriethoxysilane (POTS) in vapor phase for about 12 hours to decrease the surface energy, reducing the interactions with the PDMS chains. A bi-layer hard-PDMS/soft-PDMS stamp was then obtained in two steps. Hard-PDMS: 1) PP2-RG07 from GELEST kit, part A (base) and part B (for crosslinking) mixed with a molar ratio of 1:10. 2) Pouring PP2-RG07 on the functionalized hard master surface (depending on the master size and morphology the quantity is different) and spin coated: i) Spin speed 300 round-per-minute for 3''; ii) Spin speed 4000 round-per-minute for 30''; iii) Curing for 15 min at 65-70 o C providing a final thickness of ~ 2000 nm. Soft-PDMS is prepared as follows: i) Mixing the precursors reactants 90 wt% RTV141A; 10 wt % RTV141B from BLUESIL; ii) Pouring them atop the hard PDMS. The master with hard and soft-PDMS is cured at 70 °C for about 12 hours before demolding.

SOL-GEL DIP-COTING AND NANO-IMPRINT LITHOGRAPHY

The final metal oxide structures are obtained as follows (Figure 1c). For dense titania (TiO2) replicas, the precursor solution was prepared by mixing 1 TiCl4 (TiCl4 99% Sigma-Aldrich): 40 EtOH: 12 H2O: 5× 10 -4 F127 (Pluronic F127: polyethylene oxide-polypropylene oxide triblock copolymer (PEO-PPO-PEO, from Aldrich) molar ratios. Dense methylated silica (Si4O7Me2) was obtained from a solution containing 0.5 TEOS (tetraethylorthosilicate): 0.5 MTEOS (methyltriethoxysilane): 14 EtOH: 5 H2O: 0.2 HCl (1M):1x10 -4 F127 molar ratios. After 24 hours of aging, the solutions were dip-coated on glass or on Si substrates at a constant relative humidity (ambient humidity for SiO2 and relative humidity RH < 10% for TiO2) and with controlled withdrawal speeds between 0.5 and 8 mm/s in order to adjust the final thickness [START_REF] Grosso | How to Exploit the Full Potential of the Dip-Coating Process to Better Control Film Formation[END_REF] . In some cases, the final sol-gel solutions were diluted to obtain the appropriate film thickness.

The PDMS mold was pumped for 10 minutes under primary vacuum and then readily applied (in less than one minute) on the as-deposited sol-gel thin film into a nanoimprint chamber with temperature and relative humidity control [START_REF] Bottein | Environment-Controlled Sol-Gel Soft-NIL Processing for Optimized Titania, Alumina, Silica and Yttria-Zirconia Imprinting at Sub-Micron Dimensions[END_REF] . The relative humidity was maintained in the NIL chamber at 70% for SiO2 and 55% for TiO2 xerogel films. Before unmolding, the system was cured at 70 °C for 20 min. The imprinted samples were then annealed at 450 °C for 10 min.

SPECTROSCOPIC CHARACTERIZATION

Spectroscopic investigation of the optical properties of the samples in study was performed using 3 different setups. A conventional spectrophotometer (from Perkin Elmer) is used to assess total reflection and diffusion from 250 to 1500 nm. A gonio-reflectometer is used to measure the bidirectional-reflectance distribution function [START_REF] Germer | Spectrophotometry: Accurate Measurement of Optical Properties of Materials[END_REF] (BRDF) assessing the structure factor S(k) of the structures under investigation. We set independently the angle of the incident excitation light beam (exc, exc) and that of detection (det, det). exc is kept constant during a measurement, whereas the excitation azimuthal angle exc is always kept fix at 0 degrees in all the measurements. The detection angle det is scanned from -80 to +80 degrees during a measurement, whereas det is always kept constant at 0 degrees. For these measurements, the light sources used for excitation are a 630 nm CW laser diode and a broad-band tungsten lamp. They are fed in a multimode optical fiber, collimated and shined on the sample as a collimated beam of about 2 mm in diameter. The scattered light is detected with a multimode optical fiber and detected by a Si-based photodiode (integrating the signal in wavelength, used for the laser excitation) or by a Si-based array (3648 pixels from Oceans Optics, used for measuring the scattering spectra under white light illumination). All the measurements are normalized by a Lambertian reference. We thus measure the BRDF which units are ster. -1 . An optical microscope is used to excite the sample and collect the reflected light with a high numerical aperture objective lens (NA = 0.9) allowing to integrate over a broad angle of about ±64 degrees with respect to the sample normal (see Figure 12a for a sketch of the setup). For these measurements we used a white LED lamp (emitting from ~400 to 750 nm) to illuminate the sample and collecting the light with a multi-mode optical fiber having a diameter of 200 nm in a confocal geometry. This provides a collection spot on the sample surface of about 5 m in diameter. The light is then fed into a spectrometer and detected by a Si-based array.

ENVIRONMENTAL ELLIPSOMETRY POROSIMETRY

A characterization of the 2D flat layers was performed in order to determine their properties in terms of refractive index, porosity and roughness. This is done by Environmental Ellipsometry Porosimetry (Figure 2 and3). The measurements were carried out with a Wollam M200V ellipsometer (350nm-1000nm) measuring in real time the changes of refractive index of a metal oxide thin layer kept in a chamber with controlled atmosphere where air and water were progressively mixed. We thus measured the refractive index evolution upon adsorption/desorption of water, from which isotherms can be extracted and used to obtain the mesoporosity of the layers (pore volume, pore size distribution, pore interconnection through adapted Kelvin models [START_REF] Loizillon | In-Depth Study of Coating Multimodal Porosity Using Ellipsometry Porosimetry in Desorption Scanning Mode[END_REF][START_REF] Boissiere | Porosity and Mechanical Properties of Mesoporous Thin Films Assessed by Environmental Ellipsometric Porosimetry[END_REF] .

EXPERIMENTAL RESULTS

CHARACTERIZATION OF FLAT METAL-OXIDES LAYERS

In the case of porous layers (Figure 2 a andd, respectively for porous TiO2 and porous SiO2), when the relative solvent pressure increases, adsorption and capillary condensation occur in the pores.

The corresponding adsorbed volume fraction of adsorbate is then deduced from the Bruggeman Effective Medium Approximation [START_REF] Loizillon | In-Depth Study of Coating Multimodal Porosity Using Ellipsometry Porosimetry in Desorption Scanning Mode[END_REF] (Figure 2 b and e, respectively for porous TiO2 and porous SiO2). The pore size distribution is determined from the isotherm via a modified Kelvin equation to consider the non-spherical geometry of the pores and the thickness of the water layer adsorbed in the pores before capillary condensation (Figure 2 c andf, respectively for porous TiO2 and porous SiO2) [START_REF] Loizillon | In-Depth Study of Coating Multimodal Porosity Using Ellipsometry Porosimetry in Desorption Scanning Mode[END_REF] . It can be extracted for nanoporous films (up to pores of ~50 nm in diameter). Note that the adsorption and desorption curves do not coincide owing to a well-known poreblocking effect induced by interconnections between pores smaller than the pores themselves (bottleneck pores). Thus, the desorption branch is associated with the interconnections and only the adsorption branch can be exploited to estimate the pore size (even if it occurs out of equilibrium, leading to an overestimation of their dimension). As a first approximation, the adsorption curve provides the size of the smallest dimension of the pores, while the desorption curve provides the size of the interconnections (pore blocking effect) [START_REF] Loizillon | In-Depth Study of Coating Multimodal Porosity Using Ellipsometry Porosimetry in Desorption Scanning Mode[END_REF][START_REF] Boissiere | Porosity and Mechanical Properties of Mesoporous Thin Films Assessed by Environmental Ellipsometric Porosimetry[END_REF] In the case of thin, porous TiO2 layer (Figure 2 a, b andc) pore and connections size are of the order of 9 nm and 6 nm respectively with an overall porosity of about 50% determining n ~ 1.6.

Thus, the case presented here is representative of a rather porous titania layer with a low refractive index. For the thin, porous SiO2 layer (Figure 2 c, d ande) the porosity is about 25% of the total volume, the pore diameter and connections are below 1-2 nm. The corresponding refractive index is about 1.32, thus smaller than the typical value of 1.5 found for dense silica.

A second example of metal oxides layers is provided for dense materials (Figure 3). In these cases, adsorption/desorption isotherms obtained by ellipsometry porosimetry do not provide a precise estimation of the pore size distribution owing to the low variation of refractive index and low pressure at which the pores load with water, preventing the use of Kelvin's law for capillary condensation. However, a measured refractive index of ~2.34 for dense TiO2 allows to roughly estimate the layer porosity in 10-15% (Figure 3 Further examples of titania layers with different pore size distributions can be found in [START_REF] Bottein | Black" Titania Coatings Composed of Sol-Gel Imprinted Mie Resonators Arrays[END_REF][START_REF] Checcucci | Multifunctional Metasurfaces Based on Direct Nanoimprint of Titania Sol-Gel Coatings[END_REF] (D~2-3 nm). For applications in optics, the possibility to adjust the porosity of the sol-gel layers provides a fine tuning of the optical constant of the thin layers and 3D printed structures. It is thus a powerful tool to adjust n to a desired value (e.g. for anti-reflection coatings [START_REF] Bochet-Modaresialam | Methylated Silica Surfaces Having Tapered Nipple-Dimple Nanopillar Morphologies as Robust Broad-Angle and Broadband Antireflection Coatings[END_REF] and Bragg mirrors).

As for the case of titania, other examples of silica layers featuring a different porosity can be found in [START_REF] Checcucci | Multifunctional Metasurfaces Based on Direct Nanoimprint of Titania Sol-Gel Coatings[END_REF] . In this latter case, an average pore size of about 7 nm is found with a porosity of 37% determining a refractive index of 1.28. More generally, a plethora of combinations of porosity, pore size distribution, interconnections between pores can be created at will, providing a full control over the micro-and meso-porosity of metal oxides [START_REF] Loizillon | In-Depth Study of Coating Multimodal Porosity Using Ellipsometry Porosimetry in Desorption Scanning Mode[END_REF] and of their refractive index from low (~1.1 for porous MgF2 [START_REF] Grosso | Ultralow-Dielectric-Constant Optical Thin Films Built from Magnesium Oxyfluoride Vesicle-like Hollow Nanoparticles[END_REF] ) to high (~2.4 for dense titania). This is for instance possible by playing with the initial concentration of templating agents (e.g. cetyltrimethylammonium tetrachloride, Pluronics F127, and poly(butadiene-b-ethyleneoxide). Beyond the intrinsic porosity of the sol-gel materials, an important feature is their roughness. In order to address this feature, we measured via atomic force microscopy the height profile of dense TiO2 and SiO2 flat layers (Figure 4). The case of porous layers is not shown. From this characterization we deduce a roughness of about 0.1 nm for TiO2 and 0.24 nm for SiO2. In this latter case we can observe the presence of a few pits, up to ~10 nm dip and having a density of about ~5*10 6 cm -2 . Thus, we conclude that overall, the sol-gel layers are of optical quality. To complement this information, we also mention that from optical spectroscopy measurements, we can address the light scattering of flat layers revealing a diffusion of about 0.4%, as low as that one measured on a polished silicon wafer, which further accounts for the flatness of the sol-gel deposition and lack of scattering centers (see Supplementary Information Starting from plain, as-prepared, sol-gel coatings we can replicate all the different morphologies (e.g. islands, connected structures, Figure 5). At a glance, we show the hard SiGe masters obtained via solid state dewetting and the corresponding TiO2 and SiO2 replicas of small disconnected islands for dense materials (Figure 5 a) and connected structures for dense materials (Figure 5 b andc). These metal oxides-based nano-and micro-architectures can be printed on glass and on silicon [START_REF] Bochet-Modaresialam | Methylated Silica Surfaces Having Tapered Nipple-Dimple Nanopillar Morphologies as Robust Broad-Angle and Broadband Antireflection Coatings[END_REF][START_REF] Bottein | Black" Titania Coatings Composed of Sol-Gel Imprinted Mie Resonators Arrays[END_REF][START_REF] Checcucci | Multifunctional Metasurfaces Based on Direct Nanoimprint of Titania Sol-Gel Coatings[END_REF] and thus, all combinations of morphology, material, size and substrate are in principle possible.

ROUGHNESS OF FLAT SOL-GEL LAYERS

INVERTED STRUCTURES: NEGATIVE REPLICAS

As a further example of the versatility of our method, we showcase the possibility to invert the original structures by printing their negative replica. Inverting a mold allows to limit the use (and deterioration) of the original hard master, increasing its lifetime. Moreover, some designs might be very complex or time-consuming to obtain by lithography, whereas their inverse is not.

A simple way to invert a mold is summarized in Supplementary Information Figure S.I. 2.

If liquid PDMS before (reticulation) is directly poured on a PDMS mold and cured, cross-linking occurs at the interface and demolding is impossible. In fact, PDMS is never fully crosslinked and even after long annealing time some functions can still be reactive. In order to easily separate negative from positive PDMS, different techniques were developed to change the surface functionalization of PDMS [START_REF] Gitlin | Rapid Replication of Master Structures by Double Casting with PDMS[END_REF][START_REF] Mutreja | Positive and Negative Bioimprinted Polymeric Substrates: New Platforms for Cell Culture[END_REF][START_REF] He | Artificial Honeycomb-Inspired TiO2 Nanorod Arrays with Tunable Nano/Micro Interfaces for Improving Poly(Dimethylsiloxane) Surface Hydrophobicity[END_REF] These techniques are hardly exploitable for small structures (e.g. of sub-micron dimension), in particular when densely packed. Thus, we added a thin conformal SiO2 layer of few tens of nanometers deposited via plasma enhanced chemical vapor deposition.

Positive and negative molds, made using 15 nm of SiO2 as buffer layer, were then used for imprinting metal oxides.

To confirm the good quality of the inverted molds, the replication of two different kinds of ordered patterns were evaluated: a network of densely packed dots (depth 100 nm, diameter 270 nm, pitch 700 nm) and large rings (depth 400 nm, outer diameter Dext = 6 µm, inner diameter Dint This technique is used here to replicate and invert the complex features framed by spontaneous dewetting to obtain more diverse structures from the same master. Positive and negative molds were fabricated by following the protocol detailed above on various sol-gel formulations (Figure 5 c) such as dense TiO2, porous TiO2 and porous SiO2 (not shown) producing positive and negative replicas of the dewetted structures.

DISORDERED HYPERUNIFORMITY OF METAL OXIDE STRUCTURES

In what follows we will address the features of characterizing the disorder of the master and replicas. We will focus only on dense TiO2 as it is a more interesting material for applications in photonics (e.g. provided its larger refractive index with respect to SiO2). However, as shown in Figure 5, the replication of the same structures in SiO2 and in porous materials is straightforward. We performed a spectroscopic investigation of the dHU character of a SiGe master and its metal oxide replica, addressing their structure factor S(k//) via light scattering in the far field (where k// = 2*sindet is the in-plane wavevector,  is the wavelength of the incident light and det is the detection polar angle with respect to the sample normal).

We first characterize the BRDF of a SiGe-based master obtained by spinodal solid-state dewetting of 150 nm of Si0.7Ge0.3 on ultra-thin silicon on insulator [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] (Figure 2). The far-field scattering of the laser shined at normal incidence (exc = 0 degrees) at the center of the sample (where the morphology of the dewetted structures displays disconnected islands, Figure 6 a, d andf), shows an intense peak at k// = 0 m -1 (det = 0 degrees), a sharp decrease and two symmetric lobes at about k// ~ ±3.4 m -1 . For the sake of thoroughness, we mention that the deep in the specular reflection at k// = 0 m -1 degrees is due to a shadow effect of the detecting arm passing in front of the excitation arm. It is just an artificial effect. Measuring S(k//) in two orthogonal directions ([110] and [010], in-plane crystallographic directions of the (001) surface of the original SOI, by rotating the sample of 90 degrees), provides identical results within experimental error accounting for the uniformity of the scattering that is configured as a ring in the far-field (Figure 6 d andf).

Generally speaking the degree of hyperuniformity of a disordered material can be quantified by extrapolating the value of S (k = 0). This quantity is strictly zero for ideal dHU materials whereas in all experimental cases it always features a finite value. The smaller the value, the higher the degree of hyperuniformity [START_REF] Xie | Hyperuniformity in Amorphous Silicon Based on the Measurement of the Infinite-Wavelength Limit of the Structure Factor[END_REF] . Very often, accessing the real value of S (k = 0) can be challenging and depending on the experimental tools and method analysis overestimations are likely possible.

The laser scattering method used here does not allows to explore very small k, owing to a limited angular resolution (~1 degree) and the presence of a very intense specular reflection peak at k = 0. Thus, we can extrapolate an S(k = 0) ~ 0.15 which is not far from typical values found for spontaneous dHU patterns [START_REF] Xie | Hyperuniformity in Amorphous Silicon Based on the Measurement of the Infinite-Wavelength Limit of the Structure Factor[END_REF][START_REF] Wilken | Hyperuniform Structures Formed by Shearing Colloidal Suspensions[END_REF][START_REF] De Graff | The Long-Wavelength Limit of the Structure Factor of Amorphous Silicon and Vitreous Silica[END_REF] . From this analysis it emerges an effective hyperuniform character of the masters in use as expected for these kinds of patterns [START_REF] Ma | Random Scalar Fields and Hyperuniformity[END_REF] and from the analysis of the spectral density measured from large atomic force microscopy images in similar samples [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] .

It is interesting to note that S(k//) is sensitive to the different morphologies of the underlying structures. Owing to gradients in the annealing temperature of the sample in the molecular beam epitaxy machine, the dewetting at the center and at the edges evolves at a slightly different speed, resulting in different morphologies, ranging from holes (Figure 6 c), to connected structures (Figure 6 b), to islands (Figure 6 a). Measuring S(k//) in these three areas is possible to follow its evolution (Figure 6 e and g) with the maximum moving from about 3 to 3.4 m -1 . The deep at small wavevectors is more pronounced for connected structures with respect to the islands, in agreement with the previous assessment of the dHU character of this kind of samples [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] .

A similar measurement is repeated by changing the angle of incidence of the excitation from exc = 0 to -40 degrees in steps of 10 degrees (Supplementary Information Figure S.I. 4). Also, in this case the scattering is redistributed in a ring centered around the specular reflection peak with a maximal intensity at about k// ~ ± 3.4 µm -1 with respect to the specular reflection peak.

Thus, the scattering ring in the far-field shifts according to the incidence angle of the excitation laser, as expected for this kind of disordered structures (Iring(k//) = S(k// -kin) where kin is the projection on the plane of the sample of the wavevector of the incident radiation [START_REF] Tavakoli | Over 65% Sunlight Absorption in a 1 Μm Si Slab with Hyperuniform Texture[END_REF] . The same measurement performed on a titania-based replica of the master shows exactly the same behavior with the ring at k// ~ ±3.4 µm -1 degrees and an overall lower scattering intensity (Figure 7 a). This latter feature is explained by the lower refractive index of the titania used for soft-NIL (n~2.2) with respect to the SiGe master (n~4). Thus, beyond the structural characterization performed with the AFM that can provide only local information [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] , the use of the BRDF allows us to average the properties over a larger scale (e.g., in our case several mm 2 ) accounting for the similar dHU features of master and replicas. A final assessment of the dHU character is provided by measuring the BRDF by using white light and spectrally resolving the measurement (Figure 7 b). As expected from the presence of a ring at a defined in-plane wavevector in S(k//), the far-field scattering measured by the BRDF as a function of the incident wavelength shows a linear trend of the angle at which this ring is found (k//ring = 2/*sinmax). This latter result mimics what was found in reference [START_REF] Castro-Lopez | Reciprocal Space Engineering with Hyperuniform Gold Disordered Surfaces[END_REF] by M. Castro-Lopez and coworkers for a dHU metasurface obtained via top-down lithography of a gold layer.

We also note that similar measurements performed on dewetted samples made of SiGe, but obtained in completely different growth conditions did not provide the same kind of results in terms of BRDF [START_REF] Bouabdellaoui | Self-Assembled Antireflection Coatings for Light Trapping Based on SiGe Random Metasurfaces[END_REF] (no ring was observed) confirming that the dHU features shown here are specific of spinodal-like dewetting. Although to a first look, the metal oxide replicas correspond well to the master morphologies, a more precise assessment of their features shows that for dense TiO2 a relevant size shrinking occurs (Figure 8). We use atomic force microscopy to measure the overall volume of the 3D, dewetted SiGe structures (obtained from a 150 nm thick SiGe layer) resulting in ~15 m 3 on a surface of 100 m 2 and maximal height of about 400 nm (Figure 8 a and b). By using an equivalent volume of dense titania (evaluated by ellipsometry after calcination) for the replica, we find a total volume of ~6 m 3 on a surface of 100 m 2 and a maximal height of ~200 nm (Figure 8 a andb). Optical microscope image (dark field, 100X magnification, numerical aperture NA = 0.9) (f) for a 20 nm (g) for a 38 nm (h) for a 65 nm thick, dense TiO2 replica. This observation clearly points out that the printed structures are shrink with respect to those on the original master and that they sit atop a flat residual layer (see the inset of Figure 8 d).

SHRINKING AND RESIDUAL LAYER OF PRINTED DISORDERED HYPERUNIFORM

A further indirect evidence of these features can be observed by optical inspection of the printed structures (Figure 9). When changing the thickness of the dense TiO2 used for printing, the reflected colors from both 3D structures and from the flat areas nearby, abruptly changes as revealed by bright-filed, optical microscope images (Figure 9 b-d). This can be explained by constructive and destructive interferences in the residual layer as well as light scattering from the 3D structures acting ad Mie resonators as visible in dark-field optical microscope images (Figure 9 f-h). As shown later, these morphological features have a deep influence on the optical properties of the replicas. A more quantitative and systematic investigation of shrinking and residual layer thickness can be performed for samples bearing separate islands (Figure 10). By measuring the island height (Figure 10 increase of the residual layer as a function of the deposited layer thickness. Moreover, the 3D structures display a relevant shrinking both in height (55-40%) and in volume (75-60%). Although the shrinking is reduced for thicker layers, the tendency is very weak.

OPTICAL PROPERTIES OF DISORDERED HYPERUNIFORM PRINTED

STRUCTURES

As a possible application we provide a characterization of the optical response under white light illumination of dense TiO2-based structures, showing an anti-reflective coating (ARC) character when printed on bulk Si. Part of the samples used as masters have been already characterized in reference [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] . For optical applications we produced series of ad-hoc SiGe and Ge islands in order to optimize their size and shape for as anti-reflection coatings at visible and near-infrared frequencies.

Light reflection and diffusion are measured from near-UV up to near-IR at quasi-normal incidence (11 degrees with respect to the sample normal) with a spectrophotometer mounting an integrating sphere. We showcase four different examples of TiO2 samples having an initial sol-gel layer thickness of 70, 95, 120 and 147 nm deposited on bulk Si (Figure 11), all obtained starting from a 100 nm dewetted Ge master. All the samples show a remarkable reduction of the reflected intensity with respect to the bare bulk Si in a broad spectral range, extending from visible to near-infrared. Note that the presence of a sharp increase in the reflected light intensity at about 1150 nm is due to the transparency of Si below the energy bandgap. Thus, at near-infrared frequency, the reflection spectrum is determined also by the light reflected at the bottom surface of the wafer. Overall, the ARC effect shifts to longer wavelengths when increasing the thickness of the printed TiO2 layers. This behavior can be explained as the combination of two effects: 1) the larger size of the 3D printed objects that act as Mie resonators [START_REF] Checcucci | Multifunctional Metasurfaces Based on Direct Nanoimprint of Titania Sol-Gel Coatings[END_REF] shifting the resonant light coupling into the substrate to longer wavelength [START_REF] Bouabdellaoui | Self-Assembled Antireflection Coatings for Light Trapping Based on SiGe Random Metasurfaces[END_REF][START_REF] Spinelli | Broadband Omnidirectional Antireflection Coating Based on Subwavelength Surface Mie Resonators[END_REF] and 2) the increase of the 2D residual layer thickness underneath the 3D objects that produces etaloning with a first reflection minimum at progressively longer wavelength.

At some wavelength the reflected intensity can be as small as 2% (70 nm TiO2 printed structures, Figure 11a) and below 5% in broad interval of wavelength (e.g. about a 400 nm for 120 nm TiO2 printed structures, Figure 11a). The 2D flat layers counterpart can be simulated by transfer matrix formalism based on the Fresnel equation (Figure 11b). If compared to this latter case, the 3D printed structures show better performances as ARC. In fact, although the 2D layers can exhibit minima of reflection well below 0.05% (e.g. at see the case of flat TiO2 at ~440 nm, Figure 11b), these low values are limited to a very narrow spectral band-width.

Further insight on the optical properties of 3D printed structures can be obtained by detecting only on the scattered light (removing the specular reflection from the spectra). Diffusion [START_REF] Bottein | Black" Titania Coatings Composed of Sol-Gel Imprinted Mie Resonators Arrays[END_REF][START_REF] Checcucci | Multifunctional Metasurfaces Based on Direct Nanoimprint of Titania Sol-Gel Coatings[END_REF][START_REF] Checcucci | Titania-Based Spherical Mie Resonators Elaborated by High-Throughput Aerosol Spray: Single Object Investigation[END_REF] plays an important role in limiting the ARC performances of the titania-based soft-NIL structures (not shown). Thus, to limit the scattering and improve the ARC effect, we further processed the samples by adding via dip-coating a SiO2 layer with a variable thickness ranging from 50 to 150 nm atop the 70 nm TiO2 structures (Figure 11c). We observe a further reduction in the reflected intensity explained by two effects: 1) a smoother change of refractive index from air (n = 1), to SiO2 (n ~ 1.35), to TiO2 (n ~ 2.2), and to Si (n ~ 4), and 2) a neat reduction of light scattering from the TiO2 structures as confirmed by diffusion measurements (not shown). As found in the previous case, overall, the performances of 3D structures topped by SiO2 are superior to the flat 2D layers counterpart (Figure 11c). 3D ARC can couple the light in the underlying substrate also when changing the incident angle of light, a feature not matched by conventional stacks of 2D layers. In order to assess this property, we compare the reflection of 3D structures with the flat 2D layers by exciting the sample and collecting the reflected light with a high numerical aperture objective lens (NA = 0.9) allowing to integrate over an angle of about ±64 degrees with respect to the sample normal (Figure 12a). These spectra were normalized to that one detected on a bare Si substrate (thus a reflection intensity of 1 corresponds to Si reflection).

Exploiting the lateral resolution of the confocal microscope, we can test the homogeneity of the reflection spectra in different parts of the sample (Figure 12.b). Fluctuation below 5% are found form one point to the other, accounting for the uniform ARC properties across the sample. Furthermore, comparing the reflection spectra from the printed areas with those collected in the flat parts (Figure 12c andd), we can observe again a superior performance of the former case with respect to the latter as expected for 3D ARC. Thus, in spite of the good properties of stacks of 2D layers at certain wavelength, the 3D counterpart can overperform this classic approach being both broad-band and broad angle. It is in fact well known that 2D layers can be optimized for a limited set of incidence angles (see also the Supplementary Information Figure S.I. 8 for simulations of stacks of 2D layers as a function of incidence angle).

DISCUSSION

The advantage of our fabrication protocol over previous reports [START_REF] Zhou | Hyperuniform Disordered Network Polarizers[END_REF][START_REF] Degl'innocenti | Hyperuniform Disordered Terahertz Quantum Cascade Laser[END_REF][START_REF] Florescu | Designer Disordered Materials with Large, Complete Photonic Band Gaps[END_REF][START_REF] Man | Isotropic Band Gaps and Freeform Waveguides Observed in Hyperuniform Disordered Photonic Solids[END_REF][START_REF] Muller | Silicon Hyperuniform Disordered Photonic Materials with a Pronounced Gap in the Shortwave Infrared[END_REF][START_REF] Froufe-Pérez | Role of Short-Range Order and Hyperuniformity in the Formation of Band Gaps in Disordered Photonic Materials[END_REF][START_REF] Castro-Lopez | Reciprocal Space Engineering with Hyperuniform Gold Disordered Surfaces[END_REF][START_REF] Tavakoli | Over 65% Sunlight Absorption in a 1 Μm Si Slab with Hyperuniform Texture[END_REF] of dHU devices is manifold. Most of the techniques used for photonic applications at visible and near-infrared frequencies exploit top-down methods (e.g., e-beam lithography and reactive ion etching). As such, they are hardly scalable, limiting the range of applicability of dHU materials. Self-assembly for dHU system [START_REF] Weijs | Emergent Hyperuniformity in Periodically Driven Emulsions[END_REF][START_REF] Zito | Nanoscale Engineering of Two-Dimensional Disordered Hyperuniform Block-Copolymer Assemblies[END_REF][START_REF] Goldfriend | Hyperuniformity, and Instability in the Sedimentation of Irregular Objects[END_REF][START_REF] Wang | Hyperuniformity with No Fine Tuning in Sheared Sedimenting Suspensions[END_REF][START_REF] Becker | Complex Dewetting Scenarios Captured by Thin-Film Models[END_REF][START_REF] Stoop | Curvature-Induced Symmetry Breaking Determines Elastic Surface Patterns[END_REF][START_REF] Cahn | On Spinodal Decomposition[END_REF][START_REF] Ma | Random Scalar Fields and Hyperuniformity[END_REF][START_REF] Ma | Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions[END_REF][START_REF] Weijs | Mixing by Unstirring: Hyperuniform Dispersion of Interacting Particles upon Chaotic Advection[END_REF] can be typically implemented in liquid and soft materials, featuring a low dielectric constant (low refractive index) less appropriate for light management, and often results in out-of-equilibrium systems. Thus, in spite of the appeal of self-assembly, these features constitute a limit for the exploitation of these methods for practical applications.

Solid-state dewetting is a valid alternative to build dHU architectures as it provides hard materials with large dielectric constant with size tuneability spanning more than 3 orders of magnitude [START_REF] Salvalaglio | Hyperuniform Monocrystalline Structures by Spinodal Solid-State Dewetting[END_REF] . However, it is limited to few materials and requires epitaxial growth and expensive silicon on insulator wafers that for instance are not transparent at visible frequencies and are provided in a limited set of specifications (e.g. SOI, BOX and handle thickness). In addition to this, SiGe compounds feature strong absorption losses at near-UV frequencies, rendering them less adapted to applications at short wavelength. All these issues limit the range of exploitability of this self-assembly approach.

The method proposed here enjoys all the flexibility of sol-gel chemistry, allowing to print a plethora of metal oxides [START_REF] Bottein | Environment-Controlled Sol-Gel Soft-NIL Processing for Optimized Titania, Alumina, Silica and Yttria-Zirconia Imprinting at Sub-Micron Dimensions[END_REF] . We showed two prototypical cases of materials relevant to applications such as Silica and Titania, having a low and high refractive index respectively, and sharing reduced absorption at near-UV frequency. Nonetheless, it is possible to tune at will the refractive index (e.g., by mixing them [START_REF] Louis | Understanding Crystallization of Anatase into Binary SiO 2 /TiO 2 Sol-Gel Optical Thin Films: An in Situ Thermal Ellipsometry Analysis[END_REF] or by changing their porosity [START_REF] Li | Suppressing Structural Colors of Photocatalytic Optical Coatings on Glass: The Critical Role of SiO 2[END_REF] ), and introduce nanoparticles or organic moieties [START_REF] Sanchez | Synthesis, and Properties of Inorganic and Hybrid Thin Films Having Periodically Organized Nanoporosity[END_REF] . It is possible to print on large surfaces up to 8 inches [START_REF] Bochet-Modaresialam | Methylated Silica Surfaces Having Tapered Nipple-Dimple Nanopillar Morphologies as Robust Broad-Angle and Broadband Antireflection Coatings[END_REF][START_REF] Zhang | Preparation of SiO 2 Anti-Reflection Coatings by Sol-Gel Method[END_REF] , on metals, plastic, semiconductors and glasses. Thus, provided an appropriate master with ad-hoc morphologies, it is possible to reproduce it hundreds of times by framing several polymeric mastermolds from it. Note that, provided the sol-gel solutions and the mold, the overall process of soft-NIL can be done in less than 30 minutes in a research laboratory, independently from the sample size. All these features render our solution for dHU metasurfaces scalable to large surfaces and appealing for realistic applications.

A full assessment of the optical properties of these metal oxides-based dHU nanoarchitectures goes beyond the aim of this work. However, our results support the anti-reflective effect obtained with this kind of materials, as also recently reported by other groups [START_REF] Zhang | Experimental Demonstration of Luneburg Lens Based on Hyperuniform Disordered Media[END_REF][START_REF] Tavakoli | Over 65% Sunlight Absorption in a 1 Μm Si Slab with Hyperuniform Texture[END_REF] . Thus, the ARC effects shown here are potentially important for devices obtained via sol-NIL, especially considering the scalability of the dewetting approach to produce hard masters and the ease of implementation of the overall process. Note that there was no optimization of the ARC properties.

It is also worth stressing that, differently from similar 3D ARCs based on dielectric structures [START_REF] Spinelli | Broadband Omnidirectional Antireflection Coating Based on Subwavelength Surface Mie Resonators[END_REF] , here we avoid any top-down etching of the Si crystal as we directly print the structures in a single NIL step atop the flat Si surface. This avoids any introduction of surface defects that can be detrimental for an efficient light-to-electricity conversion in photovoltaic cells. At the same time, differently from ARCs on Si based on SiGe deposition and dewetting at high temperature [START_REF] Bouabdellaoui | Self-Assembled Antireflection Coatings for Light Trapping Based on SiGe Random Metasurfaces[END_REF] , the temperature budget for soft-NIL can be as low as 450 °C [START_REF] Bochet-Modaresialam | Methylated Silica Surfaces Having Tapered Nipple-Dimple Nanopillar Morphologies as Robust Broad-Angle and Broadband Antireflection Coatings[END_REF][START_REF] Bottein | Environment-Controlled Sol-Gel Soft-NIL Processing for Optimized Titania, Alumina, Silica and Yttria-Zirconia Imprinting at Sub-Micron Dimensions[END_REF][START_REF] Dalstein | Submicrometric, MOF-Based 2D Photonic Structures: Toward Easy Selective Vapors Sensing by a Smartphone Camera[END_REF] rendering our process potentially back-end C-MOS compatible [START_REF] Lee | Back-End Deposited Silicon Photonics for Monolithic Integration on CMOS[END_REF] . Sol-gel deposition of 2D layers of metal-oxides is a common approach to produce anti reflection coatings see for instance [START_REF] Prado | Development of Multifunctional Sol-Gel Coatings: Anti-Reflection Coatings with Enhanced Self-Cleaning Capacity[END_REF] for the cases of glass and 63 for the case of silicon. Here we provided a direct quantitative comparison of dHU ARC with the flat counterpart made with the same materials and nominal thickness showing an overall better performance of 3D structures with respect to the 2D counterpart.

CONCLUSION

In conclusion, we showed a simple and versatile approach based on sol-gel coatings and nano-imprint lithography of metal oxides to replicate disordered hyperuniform patterns on glass and silicon wafers. The strength of our approach is manifold: i) it is scalable as it is based on masters obtained via spontaneous dewetting and is compatible with industrial processing (such as roll-to-roll), ii) it can be transferred in a plethora of materials which density and refractive index can be tuned at will and iii) can be further functionalized adding other particles, iv) it can reproduce fine features ranging from ~10 nm to few nm. Thus, the importance of our results relies in the new possibility to exploit sol-gel coatings and nano-imprint lithography of metal oxides in the framework of disordered hyperuniform coatings, providing a solid ground to exploit this emerging class of materials to build functional metasurfaces.

Inverted structures, negative replicas

Figure 1 .

 1 Figure 1. (a) Elaboration of hard masters by solid state dewetting. (b) Preparation of PDMS mold. (c) Dip coating of metal oxide sol gel film and transfer of dewetted structures by nanoimprint lithography on sol-gel film.

Figure 2 .

 2 Figure 2. Adsorption/desorption isotherms obtained by ellipsometry porosimetry on a porous TiO2 layer, 100 nm thick on silicon and on a porous SiO2 layer, 174 nm thick on silicon. (a) Evolution of the refractive index n of a porous TiO2 layer a function of the relative humidity (RH%) in the atmosphere. (b) Adsorbed water volume (%) as a function of the water vapor partial pressure for a porous TiO2 layer. (c) Pore size distribution deduced from the adsorption (black) and desorption (red) curves for a porous TiO2 layer. (d) Evolution of the refractive index n of a porous SiO2 layer a function of the relative humidity (RH%) in the atmosphere. (e) Adsorbed water volume (%) as a function of the water vapor partial pressure for a porous SiO2 layer. (f) Pore size distribution deduced from the adsorption (black) and desorption (red) curves for a porous SiO2 layer.

  a). For dense SiO2 instead the measured refractive index is about 1.43, suggesting a porosity < 10% vol (Figure3 b).

Figure 3 .

 3 Figure 3. Adsorption/desorption isotherms obtained by ellipsometry porosimetry (a) for a dense TiO2 layer, 62 nm thick on silicon and (b) for a dense SiO2 layer, 137 nm thick on silicon.

Figure 4 .

 4 Figure 4. (a) Atomic force microscopy image of a dense TiO2 layer. (b) Height profile deduced from a. (c) Statistical distribution of height deduced from a. (d) Atomic force microscopy image of a dense SiO2 layer. (e) Height profile deduced from d. (f) Statistical distribution of height deduced from d.

  Figure 5. (a) SiGe master with isolated islands (left panel) and positive replicas in dense TiO2 and dense SiO2 (respectively central and right panels). (b) SiGe master of connected structures (left panel) and positive replica in dense TiO2 (right panel). (c) SiGe master of connected structures (left panel) and positive replica in dense SiO2 (right panel). (d) SiGe master of connected structures (top left panel) and positive replica in porous TiO2 (top right panel), negative replica in dense TiO2 (bottom left panel) and negative replica in porous TiO2 (bottom right panel).

= 3

 3 µm and pitch 10 µm). The masters and resulting positive/negative replicas were thus analyzed by AFM in order to compare their dimensions. AFM images are shown in the Supplementary Information Figure S.I. 2, 3 and summarized in the corresponding Supplementary Information Table S.I. 1 for the dense TiO2 case.

Figure 6 .

 6 Figure 6. (a) Scanning electron micrograph (SEM) of a master (150 nm SiGe after dewetting) at its center. (b) SEM of the master at the edge. (c) SEM of the master at the corner. (d) Bidirectional reflectance distribution function (BRDF) of the master measured at its center for fixed excitation at normal incidence for two orthogonal crystallographic directions. (e) Blow-up of (d). (f) Bidirectional reflectance distribution function (BRDF) of the master measured at its center, its side and its corner for fixed excitation at normal incidence. (g) Blow-up of (f).

Figure 7 .

 7 Figure 7. (a) BRDF for the SiGe master (dewetted 150 nm SiGe) and for the titania replica. Both measurements are taken at the sample center. (b) Color-intensity map of spectrally-resolved BRDF for the TiO2 replica on silicon under white light illumination.

Figure 8 .

 8 Figure 8. (a) Atomic force microscopy image of a SiGe master bearing connected structures. (b) Height profile extracted from (a). (c) Atomic force microscopy image of a dense TiO2 replica. (d) Height profile extracted from (c).The inset shows a sketch of the printed structures sitting atop of flat residual layer.

Figure 9 .

 9 Figure 9. (a) Scanning electron microscope image of the hard SiGe master bearing connected structures. Optical microscope image (bright field, 100X magnification, numerical aperture NA = 0.9) for a (b) 60 nm (c) 90 nm (d) 110 nm thick, dense TiO2 replica. (e) Scanning electron microscope image of the hard SiGe master bearing isolated islands.

Figure 10 .

 10 Figure 10. (a) From the top to the bottom panel: AFM images of a SiGe master bearing isolated islands, dense TiO2 replica printed on a 60 nm, 90, 110 and 130 nm thick layers. (b) Statistical distribution of islands height as deduced from the atomic force microscope images in a. On each panel the average height <h> and corresponding standard

  b), their total volume (Figure 10 c, top panel) and knowing the thickness of the deposited material, we estimate the thickness of the residual layer (Figure 10 c, central panel), height and volume shrinking with respect to the original structures on the master (Figure 10 c, bottom panel). From this investigation (and from other reported in the Supplementary Information

Figure

  Figure S.I. 5 and 6) we can see a sub-linear increase of the islands height and a faster (linear)

Figure 11 .

 11 Figure 11. (a) Reflectance measurements for bare bulk Si, and 3D disordered hyperuniform replicas made of dense TiO2 on Si printed on 70, 95, 120 and 147 nm thick layers. Atomic force microscope images of 70, 95 120 and 147 nm samples are provided in the Supplementary Information Figure S.I. 5 (a). (b) Theoretical simulations of reflectance of Si bulk and flat layers of TiO2 on Si having a refractive index of 2.2 (similar to that of the 3D printed structures) and thickness of 70, 95, 120 and 147 nm (as those used for nanoimprint in a). (c) Reflectance measurements for bare bulk Si and 3D disordered hyperuniform replicas made of dense TiO2 on Si printed on 70 nm of TiO2 on Si and covered

Figure 12 .

 12 Figure 12. (a) Sketch of the sample and the optical setup for light illumination and detection. We use a LED light source coupled into a high numerical aperture objective lens (NA = 0.9), determining an excitation/collection angle of ±64 degrees with respect to the sample normal. The light is dispersed by a grating and detected by a Si based CCD array. (b) Normalized reflection of a 3D replica on 70 nm thick TiO2 detected in several parts of the sample spaced by hundreds of m one from the other. (c) Normalized reflection of 3D replicas on 70 nm thick TiO2 for the bare case and for the SiO2 topped cases. (d) Normalized reflection of 2D layers for 70 nm thick TiO2 and for the SiO2 topped cases.

3 : 4 :

 34 Figure S.I. 1. Light scattering at near UV and visible frequency for a Si wafer and for a TiO2 layer deposited via sol gel.

Figure S.I. 2 .

 2 Figure S.I. 2. Scheme representing the principle of inverted PDMS molds. First, a mould is made from an initialmaster by pouring the PDMS reactants and curing it at 70°C for 1h before releasing from the master. This negative mould displays the inverse of the master features. Secondly, after passivation of its surface, the negative mould is used as a master, yielding a positive mould with the same features than the initial master.

Figure S.I. 3 .

 3 Figure S.I. 3. (a) Starting hard masters obtained via focused on beam etching of a bulk Si wafer. Top panel, holes. Bottom panels: rings. (b) Positive TiO2 replicas. (c) Negative TiO2 replicas.

Figure SI. 6 .Figure SI. 7 .

 67 Figure SI. 6. (a) From the top to the bottom panel: AFM images of a SiGe master (dewetted from a 100 nm thick layer) bearing isolated islands, dense TiO2 replica printed on a 20, 38, and 60 nm thick layers. (b) Statistical distribution of islands height as deduced from the atomic force microscope images in a. On each panel the average height <h> and corresponding standard deviation σ are reported. (c) Top panel: deposited volume (squares), total volume of the printed structures (circles) and residual layer volume (triangles) as a function of the deposited TiO2 thickness. Central panel: average islands height <h> (circles) and residual layer thickness (squares). Bottom panel: height (circles) and volume shrinking of the replicas with respect to the master as a function of the deposited layer thickness.

Figure SI. 8 .

 8 Figure SI. 8. Simulations of light reflection from a bilayer on Si composed of 70 nm of TiO2 and 150 nm SiO2. The simulations show different cases relative to different incidence angles with respect to the sample normal

Table S .

 S I. 2: NIL of dHU with different morphology, material, size and substrate.

	HOLES
	UNITS: nm	Depth (±5 nm)		Diamter (±5	Pitch (±5 nm)
				nm)
	master	100		270	700
	Positive replica 28		285	700
	Negative replica 30		285	700
	RINGS
	UNITS: µm	Height (±10	Dext (±0.2	Dint (±0.2µm)	Pitch (±0.1µm)
		µm)	µm)
	master	400	5.5	3	10
	Positive replica 150	5.1	2.8	10
	Negative replica 160	5.6	2.7	10

France ; https://orcid.org/0000-0001-8568-8323; Thomas Bottein-Aix Marseille Univ, Université de Toulon, CNRS, IM2NP Marseille, France ; https://orcid.org/0000-0003-0312-2057; Marco Salvalaglio-Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany, Dresden Center for Computational Materials Science (DCMS), TU Dresden, 01062 Dresden, Germany; https://orcid.org/0000-0002-4217-0951; Monica Bollani -Istituto di Fotonica e Nanotecnologie-Consiglio Nazionale delle Ricerche, Laboratory for Nanostructure Epitaxy and Spintronics on Silicon, Via Anzani 42, 22100 Como, Italy; https://orcid.org/0000-0002-0078-5085; David Grosso-Aix Marseille Univ, Université de Toulon, CNRS, IM2NP Marseille, France ; https://orcid.org/0000-0002-9156-6848; Present Addresses † Thomas Bottein's present address: Carbon Waters 14 avenue Pey Berland 33607 Pessac France Funding Sources We acknowledge funding provided by CNRS, UPMC and IUF (Institute University of France), the PRCI network ULYSSES (no. ANR-15-CE24-0027-01) funded by the French ANR agency, the A*MIDEX foundation (ANR-11-IDEX-0001-02), the A*MIDEX project TITANIDE (no. A-M-AAP-EI-17-58-170228-16.21-ABBARCHI-SAT), and the FET-OPEN project NARCISO (no. 828890).

ACKNOWLEDGMENT

We acknowledge the facilities of the NANOTECMAT platform at the IM2NP Institute. We thank J. B. Claude and J. Wenger of Fresnel Institute for support in electronic imaging.

AUTHOR INFORMATION Corresponding Author

*Marco Abbarci -Aix Marseille Univ, Université de Toulon, CNRS, IM2NP Marseille, France ; https://orcid.org/0000-0002-2760-4766; Email: marco.abbarchi@im2np.fr

Sol-gel thickness tuning


Hicham Allag

Jean-Paul Yonnet

3D Analytical Calculation of the Interactions between Permanent Magnets

Keywords: Analytical Calculation, Permanent Magnets, 3D, Interaction Energy, Force, Torque

Up to now, the analytical calculation has been made only when the magnets own parallel magnetization directions. We have succeeded in two new results of first importance for the analytical calculation: the torque between two magnets, and the force components and torque when the magnetization directions are perpendicular. The last result allows the analytical calculation of the interactions when the magnetizations are in all the directions.

The 3D analytical expressions are difficult to obtain, but the torque and force expressions are very simple to use. As example the analytical expression can be included in optimization software allowing to directly obtaining the shape optimization by a fast way.

1-Introduction

The analytical expressions are a very powerful and a very fast method to calculate magnetic interactions. It is why the analytical expressions of all the interactions, energy, forces and torques between two cuboïdal magnets are very important results. Many problems can be solved by the addition of element interactions. The simpler shape of elementary volume is the parallelepiped, with its cuboïdal volume. It is why many 3D calculations can be made by the way of 3D interactions between two elementary magnets of cuboïdal shape.

Up to now, only the force components between two magnets with their magnetization direction parallel to one edge of the parallelepipeds have been analytically solved. We have succeeded in two new results of first importance for the analytical calculation: -the torque between two magnets, -the force components and torque when the magnetization directions are perpendicular.

Consequently, by combining parallel and perpendicular magnetization directions, the interaction energy and all the components of the forces and the torque can be calculated by fully analytical expressions, for any magnetization direction, and for any relative position between the two magnets. The only two hypotheses are that the magnets own a cuboïdal shape, and they are uniformly magnetized.

2-3D analytical calculation background

Since the discovery and the development of Samarium Cobalt magnets in the 70 years, the designers can use magnets owning a really rigid magnetization. They are the first magnets which can be used in repulsion without any risk of demagnetization. Their magnetization can be easily modelled by magnetic charges on the poles, or by equivalent currents.

One of the first applications of Samarium Cobalt magnets were the magnetic bearings. Due to the circular symmetry, the calculation can be made in 2D. The first 2D analytical expressions of the forces between magnets were given by Marina Marinescu [1] and Jean-Paul Yonnet [2]. The stiffness of a magnetic bearing can be easily calculated by analytical expressions [START_REF] Yonnet | 5 th Int Workshop on Rare Earth Cobalt Permanent Magnets and their Applications (5 th REPM)[END_REF].

The magnetic couplings are another application working by interaction forces between permanent magnets. When their length is long in comparison with the airgap dimensions, the 2D analytical calculation can be used [START_REF] Yonnet | [END_REF]. Otherwise it is necessary to use 3D calculation.

The 3D analytical calculation is obviously more difficult than the 2D. It is not 2 but 4 successive analytical integrations which must be calculated, and the difficulty fastly increases with the number of integrations. Many persons thought that the last analytical integration was not possible, and must be made by a numerical way. We had worked on this problem, and we have succeeded in solving the calculation. At that time all the integrations had been made by brain-work, with a pen and a piece of paper.The first 3D forces analytical expressions were published in 1984 by Gilles Akoun and Jean-Paul Yonnet [5].

Figure 1: Magnet configuration

The forces were analytically calculated for two cuboïdal magnets with parallel magnetization directions (Figure 1). The calculus was made by the way of interaction energy determination. The forces expressions were obtained by derivation of the energy expression.

A new contribution to the 3D calculation has been presented by Edward P. Furlani [6] in 1993 and Yeong-Der Yao in 95 [7], with the analytical calculation of multipolar magnetic couplings with discoïdal shape. But the last integration is made by a numerical way, or by a series calculation.

In 1998, Fréderic Bancel, a mathematics teacher, reaches to obtain the analytical expressions of forces when the two magnets are in angled position. The force calculus was made by direct integration of magnetic induction. The result has never been published [START_REF] Bancel | Fy(θ) and Fz(θ) forces between magnets[END_REF]. Several colleagues of his laboratory, around Guy Lemarquand, have diffused the Bancel results by using them to calculate magnetic couplings. They have published a series of articles on the subject [for example [START_REF] Eliès | [END_REF]10].

An original approach of the 3D analytical calculation has been presented by Fréderic Bancel in 1999, by proposing the notion of magnetic nodes [11]. It comes from the fact that the forces expressions are only function of the distance between the corners of the magnets.

3-New results in 3D analytical calculation

Until now, all the analytical force calculation has been made for cuboïdal magnets which magnetization is parallel to one of the edge of the magnet. It means that the magnetic poles are only on two rectangular faces of the magnet [START_REF] Yonnet | Chap. 9 of "Rare Earth Iron Permanent Magnet[END_REF]. Recently, we have worked again on the 1984 formulations, and we succeed in the analytical interaction energy calculation when the magnetization directions of the two cuboïdal magnets are perpendicular (Figure 2). From the expression of energy, the forces can be easily calculated by derivation. It is an important step, because its opens the way to the calculation of the interactions when the magnetization directions are in every direction. When the two magnetization directions are not parallel to the magnet edges, it can be decomposed in its three projections on the axes. As it is a fully linear problem, the whole energy can be calculated by addition of the interactions between the magnetization components. From the energy expression, the force can be obtained by derivation. For the first time, the fully analytical force expressions can be written whatever the magnetization directions are.

Figure 2: Magnet configuration with perpendicular magnetization directions

Another important step has been made when we succeed in the analytical calculation of the torque component. The use the equivalence between charged surfaces and magnetic nodes allows obtaining an energy repartition on the magnet corners. The torque on the magnets can be analytically obtained by angular derivation of interaction energy. Another method of torque calculation is based on the force components exerted on the parallelogram corners. It leads to the same results. The three torque components between two magnets can be expressed by fully analytical expressions. By combining the magnetization components, the torque can be obtained for all the geometrical and magnetic cases.

In summary, all the interactions (energy, forces and torques) between two cuboïdal magnets can be analytically calculated, whatever the magnetization directions are and whatever the parallelepiped orientations are.

These original results represent a -the full analytical expressions of the torque calculation in the case of magnets with parallel magnetization directions [START_REF] Yonnet | Conference MmdE[END_REF],

-the full analytical expressions of the interaction energy and the force components for any magnetization directions [START_REF] Allag | Conference EMSA[END_REF].

The size allowed for this paper is not sufficient to give full analytical expressions. These original results and their applications will be shown at the time of the oral presentation in the conference.

4-The basic mathematical model

The interactions between two parallelepiped magnets are studied. Their edges are respectively parallel (see Figure 1). The magnetizations J and J' are supposed to be rigid and uniform in each magnet. The dimensions of the first magnet are 2a x 2b x 2c, and its polarization is J. Its centre is O, the origin of the axes Oxyz. For the second magnet, the dimensions are 2A x 2B x 2C, its polarization is J', and the coordinates of its centre O' are (α, β, γ). The side 2a is parallel to the side 2A, and so on.

The magnet dimensions are given on Table 1.

Axis Ox Oy Oz First Magnet (J) 2a 2b 2c Second Magnet (J') 2A 2B 2C Second Magnet Position O' α β γ

Table 1: Magnet dimensions and position

The magnetization directions shown on Figure 1 correspond to the case when the polarizations J and J' have the same direction, parallel to the side 2c. Note that the calculation stay valid when they are in opposite direction; only the expression sign is reversed.

The polarizations J and J' are supposed to be rigid and uniform. They can be replaced by distributions of magnetic charges on the poles. It is the coulombian representation of the magnetization. Their density σ is defined by: n J r r ⋅ = σ On the example of Figure 1, since J is perpendicular to the surfaces 2a x 2b and oriented to the top, these polar faces wear the density σ = +J on the upper face (North Pole), and σ = -J on the lower face (South Pole).

All the analytical calculations have been made by successive integrals, to calculate the interaction energy for the two magnets system. The forces and the torques can be obtained by linear and angular derivation. The analytical calculation of the interaction energy in 3D is made by four successive integrations. The first one gives a logarithm function. In the second one, you have two logarithm and two arc-tangent functions… The last one owns many complex functions based on logarithm and arc-tangent functions. The interaction energy expression has 256 terms. It seems to be complicated, but it can be used very easily. A simple pocket calculator for students is sufficient to obtain the results.

5-Examples of application

The analytical expressions are fully useable for the calculation of many systems working by permanent magnet interactions, like bearings or couplings.

But it is not the only type of application which can be calculated. Very often, permanent magnets are used with iron yokes for flux closing. These iron yokes can be taken into account by image effect. The magnets and the yokes can be replaced by the magnets and the images of the magnets.

Another family of problems are the electromagnetic applications. Some of them can be directly solved by analytical expressions. For example, to calculate the force and torque between coils and magnets, the coils can be seen as Amperian currents around magnets, and the problem can be directly solved by magnet interactions (Figure 3). The analytical expression can be used in software making direct shape optimization. CADES is an example of such software [START_REF] Allag | Conference EMSA[END_REF]. In a first step, the problem is described by a linear model, including analytical expressions. In a second step, the problem is reversed, and the optimized shape is obtained. The Figure 4 shows an example of such optimization for a magnetic coupling.

10-Conclusion

All the analytical expressions of the interaction energy, the force and torque components between two cuboïdal magnets have been calculated for any magnetization directions. The only hypotheses are that the magnets own a parallelepiped shape, and that are uniformly magnetized. These results allow the analytical calculation of the interactions when the magnetization directions are in any direction.

Many problems can be solved by these results. The simpler shape of elementary volume is the parallelepiped, with its cuboïdal volume. By the superposition of 3D interactions between elementary magnets, many 3D calculations can be made. These results allow calculating not only the direct interaction of permanent magnets, but also many other devices including soft magnetic materials, and currents.

  They have never been published for the moment, except two recent publications in conferences more oriented towards the applications of permanent magnets:

Figure 3 :

 3 Figure 3: Interaction between a coil and two magnets

Figure 4 :

 4 Figure 4: Geometry of the initial (a) and optimised (b) magnetic coupling