Franc ¸ois Flory
email: francois.flory@ec-marseille.fr

Ludovic Escoubas

Gérard Berginc

Optical properties of nanostructured materials: a review

Keywords: nanostructured thin films, surface scattering, photonic crystals, quantum dots, nanoplasmonics, solar cells

Depending on the size of the smallest feature, the interaction of light with structured materials can be very different. This fundamental problem is treated by different theories. If first order theories are sufficient to describe the scattering from low roughness surfaces, second order or even higher order theories must be used for high roughness surfaces. Random surface structures can then be designed to distribute the light in different propagation directions. For complex structures such as black silicon, which reflects very little light, the theory needs further development. When the material is periodically structured, we speak about photonic crystals or metamaterials. Different theoretical approaches have been developed and experimental techniques are rapidly progressing. However, some work still remains to understand the full potential of this field. When the material is structured in dimension much smaller than the wavelength, the notion of complex refractive index must be revisited. Plasmon resonance can be excited by a progressing wave on metallic nanoparticles inducing a shaping of the absorption band and of the dispersion of the extinction coefficient. This addresses the problem of the permittivity of such metallic nanoparticles. The coupling between several metallic nanoparticles induces a field enhancement in the surrounding media, which can increase phenomena like scattering, absorption, luminescence, or Raman scattering. For semiconductor nanoparticles, electron confinement also induces a modulated absorption spectra. The refractive index is then modified.

The bandgap of the material is changed because of the discretization of the electron energy, which can be controlled by the nanometers size particles. Such quantum dots behave like atoms and become luminescent. The lifetime of the electron in the excited states are much larger than in continuous energy bands. Electrons in coupled quantum dots behave as they do in molecules.

Many applications should be forthcoming in the near future in this field of research.

Introduction

From atoms and molecules to crystals and bulk components, optical materials are naturally structured at different scales. Thanks to tremendous progresses in nanotechnologies, the optical materials can also be artificially structured at different scales.

The interaction of materials with optical waves and photons is strongly dependent on the structure, which can then be used to control light field distribution and light propagation. This 1934-2608/2011/$25.00 C 2011 SPIE allows the development of a large range of key components for optical systems and it is now a major field of photonics.

Refraction, interferences, diffraction, scattering, anisotropy, absorption, light emission, and nonlinear effects are all widely used to develop photonic components. Here, we will focus our attention only on a small part of this wide field. Our purpose is to try to have a general look at linear and passive interaction of light with structured transparent materials and to give a personal view on the related phenomena. The applications are numerous and generally belong to what are called information and communication technologies in its wider meaning and green photonics, in particular photovoltaic solar cells.

The electromagnetic theory based on Maxwell equations with a rigorous approach or with more or less simplifying considerations, allows the description of light propagation in complex media when the complex refraction indices of the materials are known. We will see that the notion of refractive index must be revisited in some cases of nanometer structures. With the increase of computer power Maxwell equations can be solved, whatever the structure is, but one has to keep in mind that the sampling of spatial and temporal variables, which is necessarily finite, is still an approximation.

The general aim of photonic components is to control the way light is distributed and propagates, and its polarization and frequency.

We will consider at first surface structures. The roughness of the surface induces light scattering even with a high-quality polishing process. The surface can also be artificially structured with a periodic pattern or a random surface to have antireflecting properties or to control the light direction by diffraction. A brief review of major results in the field is discussed.

Stacking thin films of different materials and different thicknesses is the important field of optical interference coatings. Depending on deposition process and the material, the natural micro/nanostructure of the films can be more or less complex. This has different consequences on their optical properties, which are reviewed. When the films are periodically structured in one-(1D), two-(2D), or three dimensions (3D), one speaks about photonic crystals because light waves behave like the wave function of electrons in crystals. Optical interference coatings can be considered as 1D photonic crystals. Metamaterials also belongs to the photonic crystal field. The name photonic crystal is now used in a wider meaning as it can also concern nonperiodic structures.

Another field of nanophotonics that is strongly developing is the field of nanoplasmonics. It concerns the use of plasmon resonance in small metallic structures of different shapes. A lot of work is performed in this field for different applications like plasmonic circuits, plasmonic solar cells, surface enhanced Raman scattering, or tip enhanced Raman scattering. Another domain that emerged a decade ago and which is another hot topic, concerns the optical properties of quantum structures to make biological tags, efficient light emitting diodes (LEDs), efficient solar cells, or low-consumption flat panel displays. The modeling of the constitutive properties of films including quantum dots cannot be performed by the usual harmonic oscillator model. New approaches taking quantum confinement of electrons into account are discussed.

Surface Structures

Random Surfaces

Electromagnetic wave scattering from randomly rough surfaces and films has emerged as a discipline of considerable research in areas including surfaces optics and plasmonics. A randomly rough surface is described in terms of statistical parameters. We usually consider that there are two parts that define the surface roughness: its deviation from a plane reference surface and the variation of these heights along the surface. These surface properties are described by statistical distributions as surface height distribution functions and correlation functions that appear in the theory of wave scattering from randomly rough surfaces. In nanoscale roughness region, surfaces with large roughness height and small correlation length will have the steepest slopes. Some synthetic surfaces have much larger slopes than natural surfaces. A main tool for studying these surfaces is wave scattering simulation.

Because of its roughness, an optical surface scatters light. In 1956, Giacomo already gave the results of measurements of the angular distribution of light scattered by Fabry-Perot filters showing scattering peaks corresponding to angle of resonances of the filter. [START_REF] Giacomo | Les couches réfléchissantes multidiélectriques appliquées à l'interférométrie de Fabry-Perot. Étude théorique et expérimentale des couches réelles[END_REF] Since then, many works have been performed on scattering theory. For slightly rough surfaces and thin films, Kröger et al. [START_REF] Kröger | Scattering of light by slightly rough surfaces on thin films including plasma resonance emission[END_REF] proposed a first order theory in 1970. The relation between the angular dependence of scattering and the statistical properties of optical surfaces was later described by Elson et al. in 1976. [START_REF] Elson | Relation between the angular dependence of scattering and the statistical properties of optical surfaces[END_REF] Each scattering angle is related to a spatial frequency of the autocorrelation function of surface roughness. In other words, the roughness of the surface is decomposed in a sum of gratings of different steps. A spatial frequency corresponds to a step of a grating and then a scattering direction. The depth of each grating determines the scattering amplitude.

When concerned with multilayers made of homogeneous thin films, the roughness of each layer boundary must be taken into account. The roughness of one boundry is linked to the next with an intercorrelation function. Depending on the spatial frequency, the deposition technique, and on the material considered, this intercorrelation function can be close to one. Both a first order theory and measurement of scattering from multilayer thin films have been given by Bousquet et al. in 1981. 4 In this theory, a slightly rough surface is replaced by a plane surface holding a surface distribution of currents. These secondary sources send light in different directions for which the multilayer has different optical properties. The scattering diagram results from both the amplitude of these secondary sources and the transmission of the stack. Different theoretical and numerical approaches have been developed to model the scattering of an electromagnetic field by random rough surfaces. The small-perturbation method [START_REF] Rice | Reflection of electromagnetic waves from slightly rough surfaces[END_REF] (SPM) is used for surfaces with small roughness. An overview of the method for one-dimensionally rough surfaces is given by O'Donnel in chapter 5 of Ref. 6. The Kirchhoff approximation method [START_REF] Beckmann | The Scattering of Electromagnetic Waves from Rough Surfaces[END_REF] is used for long correlation length surfaces, the method is described by Voronovich in Ref. 8 and in chapter 2 of Ref. 7. We can also cite the full-wave method analysis developed by Bahar, 9 the surface-field phase-perturbation technique developed by Ishimaru and Maradudin, [START_REF] Shen | Multiple scattering of waves from random rough surfaces[END_REF][START_REF] Winebrenner | Investigation of a surface field phase-perturbation technique for scattering from rough surfaces[END_REF] and the quasi-slope approximation developed by Tatarskii. [START_REF] Tatarskii | The expansion of the solution of the rough surface scattering problem in powers of quasi-slopes[END_REF] Research on numerically exact methods for randomly rough surfaces has increased over the past quarter century. Modeling rough surface scattering by exact methods often involves the choice of a one-dimension surface model due to the reduced computational complexity for these types of surfaces. But one-dimensional models are inadequate for a set of scattering effects such as cross-polarized scattering and scattering outside the plane of incidence. In the last decade, results have also been obtained by rigorous computational approaches for the scattering of polarized light from 2D randomly rough surfaces (see chapter 7 in Ref. 7 for an overview). For a randomly rough surface (i.e., stochastic problems), one can compute averages of scattered fields obtained by an ensemble of surface realizations and direct examinations of convergence are often appropriate. Problems with very large surface and Monte Carlo averaging require supercomputing resources to obtain results within a reasonable amount of time. Comparison with results obtained by analytical predictions of randomly rough surface scattering can give some insight about the accuracy of numerical computation, for example, in the case of small magnitudes of scattered intensity. We can notice that in studies of scattering from nanoscale rough surfaces, the goal is to understand the physical scattering process, subtle phenomenon whose magnitude and angle range are not very large, therefore, new analytical techniques must be developed rather than extensive use of numerical exact methods.

In the mid-1980s, a new popular method, called the small-slope approximation (SSA), [START_REF] Voronovich | Wave Scattering from Rough Surfaces[END_REF][START_REF] Voronovich | Small-slope approximation in wave scattering by rough surfaces[END_REF][START_REF] Voronovich | Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric halfspace[END_REF][START_REF] Berginc | Small-slope approximation method: A further study of vector wave scattering from two-dimensional surfaces and comparison with experimental data[END_REF][START_REF] Berginc | The small slope approximation method applied to a three dimensional slab with rough boundaries[END_REF] was proposed by Voronovich. The SSA is an appropriate candidate to bridge the gap between the Kirchhoff approximation and the small-perturbation method. It is valid for arbitrary roughness, provided that the slopes of the surface are smaller than the angles of incidence and scattering. This method has been extended up to the third-order to 3D vector cases for Gaussian and non-Gaussian randomly rough surfaces. The coherent and incoherent component formulations of the electromagnetic intensity for cross-and co-polarization have been developed in bistatic and monostatic cases. [START_REF] Berginc | Small-slope approximation method: A further study of vector wave scattering from two-dimensional surfaces and comparison with experimental data[END_REF] The SSA method has also been used to study different rough structures like a 3D slab or a film, considering the effects of higher orders in a perturbative expansion. The small slope approximation has been extended to the fourth-order terms of the perturbative development, and the expression of the cross-sections for the different polarization states can be found in Ref. 16. New formulations of the SSA are described in Ref. 16. In the development of the SSA series, the third-order SPM kernel has been taken into account. An expression of this kernel is developed in chapter 6 of Ref. 7 for 2D surfaces and a 3D slab, the development up to third order of the SPM is mandatory to analyze the phenomenon of backscattering enhancement, a well-defined peak in the retro-reflection direction of incidence for the case of nanoscale surface roughness. The Voronovich SSA was generalized to a layer bounded with two randomly rough surfaces. New terms in the SSA were introduced to consider the coupling between the two rough surfaces. Complete expressions of the scattering matrices and the expression of the needed crosssection for the different polarization states have been developed. [START_REF] Berginc | The small slope approximation method applied to a three dimensional slab with rough boundaries[END_REF] With this new formulation of the SSA, the backscattering enhancement for a slightly rough layer can be calculated and new analytical theories have been developed to understand the backscattering enhancement effects. [START_REF] Bourlier | Multiple scattering in the high-frequency limit with secondorder shadowing function from 2-D anisotropic rough dielectric surfaces: I Theoretical study[END_REF] The last problem is described by random volume and surface scattering. Useful phenomena in the optical range can be produced by random media with randomly rough surfaces. Designing these disordered slabs with rough surfaces can produce new optical components, which can transmit or scatter optical field with specified angular, spatial, or spectral properties. Understanding how light interacts with disordered matter is a fundamental issue in optoelectronics and photonics and has huge consequences in communications, imaging, and sensing. New formulations have been developed for a three-dimensional disordered medium with randomly rough interfaces. [START_REF] Berginc | Light scattering from 3D nanoscale disordered media[END_REF] This structure describes a device based on metallic nanoparticles embedded in insulators or dielectric media. A theory of transport based on the Bethe-Salpeter equation is presented in Ref. 18. For a three-dimensional system composed of a random medium bounded by two randomly rough surfaces, the Bethe-Salpeter equation is constructed in order that the medium and the boundaries are treated on the same footing. With this unified Bethe-Salpeter, a general expression is obtained, whatever the choice of the scattering operators used at the boundaries.

This type of simulation computation can give some experimental conditions and specifications to realize highly integrated optical devices that use metallic or metallo-dielectric nanoscale structures. For 3D slabs with nanoscale roughness and nanoparticles, the enhanced backscattering is produced by different mechanisms, the wave scattering by the same boundary, the wave coupling by the two boundaries, or the wave scattering by the random volume. All of these models and simulations can demonstrate that plasmon-polariton resonances are responsible for remarkable optical phenomena.

Controlled Random Surfaces

The control of surface roughness with tuned statistical properties is a mean to choose the scattered light distribution. [START_REF] Maradudin | Structured Surfaces as Optical Metamaterials[END_REF] The method based on photofabrication using the speckle of a diffused laser beam has been proposed by Gray. [START_REF] Gray | Method of forming optical diffusers of simple known statistical properties[END_REF] It allows tuning the statistical properties of the generated surface by modifying parameters of the exposure. Several other results based on this method have been derived after Refs. 21-23, in particular for extending depth of focus or light uniformization. [START_REF] Garcia-Guerrero | Design and fabrication of random phase diffusers for extending the depth of focus[END_REF] More recently, a method based on a spatial light modulator to control the laser beam shape appeared to be flexible. [START_REF] Brissonneau | Laser assisted fabrication of random rough surfaces for optoelectronics[END_REF] Controlling the stochastic parameters of randomly rough surfaces can be a method to "design" the scattering of an electromagnetic wave. For example, the problem is how to design a 2D randomly rough surface that scatters, in a specified manner, an optical wave incident on it and the surface realized is not a deterministic one. We notice that this kind of process is an inverse one; we must determine the structure of the surface considering the pattern of the scattered wave and the mean intensity of the scattered field, which is regarded as a spatial average of the scattered intensity over an area containing many realizations of the surface roughness. A subtle question arises when we compare the definition of wave scattering from a randomly rough surface. In the case of a theoretical computation, the scattered field is averaged over the ensemble of realizations of the surface profile function. The measurement is considered on a single realization of the surface profile function. The assumption that the spatial average is equivalent to the ensemble average, can only be proven if the generated surfaces are stationary and it is not always the case. The validity of this assumption is common to all experimental work.

Black Silicon

Black silicon has been studied for more than 10 years. Black silicon texture has been produced by techniques including laser-chemical, [START_REF] Her | Microstructuring of silicon with femtosecond laser pulses[END_REF] electrochemical, [START_REF] Ma | Wide-band 'black silicon' based on porous silicon[END_REF] and reactive-ion etching. [START_REF] Chow | Reactive Ion Etching of Silicon in CCl4 and HCl Plasmas[END_REF] Whatever the production technique, a black silicon surface generally exhibits quasiperiodical conical structures of height and a period of a few microns (Fig. 1). In some cases, irregular nanometer Q1 structures on the top of the cones make the surface more complex. Such a surface has been obtained by Mazur et al. [START_REF] Her | Femtosecond laser-induced for-mation of spikes on silicon[END_REF] by sending femtosecond UV laser pulses in a SF 6 gas on a silicon surface. Sulphur is embedded into the silicon surface. Black silicon presents remarkable wide band antireflection properties and makes the silicon highly absorbing both in the visible and in the near infrared, below silicon band gap. [START_REF] Wu | Near-unity below-band gap absorption by microstructured silicon[END_REF] Wet etching of silicon using a chloroauric acid bath is also an inexpensive way to make black silicon.

Reactive ion etching has also been used to replicate a photoresist pattern on a silicon Q2 substrate. Troconical pyramidal shapes made by photolithography with one or two periods also gives a very wide band antireflection silicon surfaces. [START_REF] Bouffaron | Enhanced antireflecting properties of micro-structured top-flat pyramids[END_REF][START_REF] Escoubas | Sand-castle bi-periodic pattern for spectral and angular broadening of antireflective properties[END_REF] Other techniques to make black silicon do not need photolithography. The masks are made of silica or polystyrene [START_REF] Sun | Broadband moth-eye antireflection coatings on silicon[END_REF][START_REF] Zhang | Application of two-dimensional polystyrene arrays in the fabrication of ordered silicon pillars[END_REF][START_REF] Zhang | Ordered nanostructures array fabricated by nanosphere lithography[END_REF] particles directly deposited on silicon.

If the remarkable properties of black silicon are not yet well explained, it seems that both the structure and a possible high doping of the material may play a role. A question that is still opened concerns the smallest structures, as nanometer structures may induce quantum confinement effects.

A major issue arises with black silicon: can black silicon be the inexpensive way toward visible-short IR wavelength detectors? Besides HgCdTe and InGaAs, black silicon is a new way to capture visible, near infrared, and small wave infrared light.

Thin Film Structures and Optical Coatings

The use of interferences in multilayer coatings has been developed for many years to reduce the reflection of an optical surface, to make wavelength filters of different types like mirrors, band pass, high pass, low pass, etc. Optical coatings are now widely used in optical systems and very often they are key elements for the implementation of very important applications. Between others, low-loss mirrors are in production for gyrolasers used in automatic navigation systems. Very narrow-band pass filters are key components for dense wavelength division multiplexing optical communications or for astronomy. A very rich literature is available on this subject. [START_REF] Macleod | Thin-Film Optical Filters[END_REF] Depending on the deposition technique used, the micro/nanostructure of thin films can be more or less complex. It is well known that low energy evaporation condensation techniques based on e-beam gun or on heated crucible give columnar thin films for most of the materials. [START_REF] Dirks | Columnar microstructure of optical thin films[END_REF] The typical dimension of a column is in the order of 20 nm and in some cases substructures of smaller dimension can be detected. [START_REF] Flory | Optical properties of nanostructured thin films[END_REF] The refractive index is generally smaller than the refractive index of the bulk material. This is attributed to the lacunar structure of these films. In particular columnar thin films have a packing density that can be much smaller than one. The packing density is given by the ratio of the volume occupied by bulk material over the total volume of the layer.

In air, even for low moisture, water is adsorbed in the pore of the material, leading to an increase of the refractive index and then to a red shift of the optical properties of interference filters. [START_REF] Ogura | Water sorption phenomena in optical thin films[END_REF] Under such conditions, thin film materials are made of a mixture of bulk material and vacuum or water. The problem of the refractive index of these mixed materials induced a lot of work some years ago and different models [START_REF] Kinosita | Effects of vacuum exposure on stress and spectral shift of high reflective coatings[END_REF][START_REF] Maxwell-Garnett | Colors in metal glasses and in metal films[END_REF][START_REF] Bragg | The form birefringence of macromolecules[END_REF] taking into account the mixing have been proposed without giving a very satisfactory solution. In particular, the change induced by water adsorption cannot be fully explained. Porous films can be seen as a mixture of two materials and, following the homogenization theory, the permittivity ε, i.e., the square of the refractive index, of the mixture is the weighted meaning value of the ε 1 and ε 2 of the two materials. The weights are the proportions of the volume occupied by each material. This homogenization theory was first proposed by Rytov. [START_REF] Rytov | Electromagnetic properties of a finely stratified medium[END_REF] In particular, he has shown that a periodic stack of films, very thin relative to the wavelength, with alternating high and low refractive indices behaves like a single-axis homogeneous material with the optical axis perpendicular to the surface.

Columnar thin films also exhibit a form birefringence. When the deposition is performed with an oblique incidence of vapor (angle α in front of the normal to the surface), the columns are inclined (angle of β) (Fig. 2). The relation between α and β is approximately given by tanα = 2.tanβ for α < 45 • . The anisotropy increases with an increasing angle [START_REF] Jänchen | Determination of the refractive indices of highly biaxial anisotropic coatings using guided modes[END_REF] and the material has a biaxial behavior. [START_REF] Flory | Anisotropy in thin films. Modelization and measurement of guided and non guided optical properties. Application to TiO 2 films[END_REF][START_REF] Horowitz | Form birefringence in thin films[END_REF] When the angle α slowly changes during deposition, the material can have a zig-zag, an S, or a helicoidal nanostructure with a nematic or chiral behavior. [START_REF] Robbie | Chiral sculptured thin films[END_REF][START_REF] Lakhtakia | Sculptured thin films[END_REF][START_REF] Lakhtakia | Sculptured Thin Films: Nanoengineered Morphology and Optics[END_REF] Such "sculptured thin films" can be classified as nanoengineered metamaterials.

The optical properties of stacks made of anisotropic layers can be calculated by the use of a 4 × 4 matrix formalism. [START_REF] Berreman | Optics in stratified and anisotropic media: 4×4 matrix formulation[END_REF][START_REF] Hodgkinson | Birefringent Thin Films and Polarizing Elements[END_REF] Polarizing filters can be made with these materials like halfwave plates [START_REF] Motohiro | Thin film retardation plate by oblique deposition[END_REF] or polarization rotator in reflection. [START_REF] Flory | Artificial anisotropy and polarizing filters[END_REF] With ion assisted deposition or ion beam sputtering, the deposition processes are more energetic and the materials do not exhibit columns any longer. [START_REF] Flory | Optical properties of nanostructured thin films[END_REF] As they are denser, they are also insensitive to moisture.

Light can also propagate in the thickness of transparent multilayers, being trapped by total internal reflection. [START_REF] Flory | Characterization: Guided wave techniques[END_REF] The modes are then guided and the propagation constants can take only a finite number of values. A particular electric field distribution corresponds to each guided mode with the guided energy more important in one layer, several layers, or in the whole stack. A guided mode can be seen as a resonance of the optical field in the direction perpendicular to the layer boundaries. The guided mode propagation conditions correspond to constructive interferences in the direction parallel to the layer boundaries. They also correspond to poles of the reflection coefficient. The guided modes can be excited with a prism and this has been widely used as characterization techniques as well for refractive index measurements [START_REF] Monneret | m-Lines technique: Prism coupling measurement and discussion of accuracy for homogeneous waveguides[END_REF]56 as for loss measurement. [START_REF] Flory | Characterization: Guided wave techniques[END_REF] The m-line technique is now developed for gas sensing. [START_REF] Mazingue | Nanostructured ZnO coatings grown by pulsed laser deposition for optical gas sensing of butane[END_REF]

Photonic Crystals

For optical coatings, the light is reflected or transmitted through the sample. On the contrary, diffraction structures are generally used to distribute the light in a controlled way in different directions of the space. The association of diffractive and interference structures in 3D com-Q3 ponents allows control, as well the directions, of propagation as the spectral distribution of the optical waves in the space. In particular, the local field and the propagation in the structure can be adjusted with the photon lifetime. The typical dimensions of the structure are in the order of a part of the wavelength. These periodically structured materials are also named photonic crystals because light waves similarly behave to the wave function of electrons in crystals. [START_REF] Kossel | Analogies between thin-film optics and electron band theory of solids[END_REF][START_REF] Sakoda | Optical Properties of Photonic Crystals[END_REF] Another way to describe the light behavior in such structures is to use the density of optical modes as Purcell defined the density of modes for electrons in solid. [START_REF] Purcell | Spontaneous emission probabilities at radio frequencies[END_REF] Photonic crystals can also be considered as metamaterials with optical properties impossible to achieve with nonstructured materials. As an example, the structure can behave like a material having a refractive index of zero or even negative. Many laboratories and companies pay a lot of attention to the study of metamaterials as important applications may come soon with cloaking properties and stealthiness or perfectly stigmatic optics. It is interesting to consider multilayer mirrors not only as an interference system but also as a diffractive component. A mirror is a periodic stack made of alternating high and low refractive indices, each layer having an optical thickness of a quarter wave for the centering wavelength. It can be considered as a Bragg grating. This concept can be extended to Bragg grating in waveguides. The main difference is that each half period must be quarter wave but for an optical thickness defined with the effective index of the guided mode. This principle can be used to design waveguide filters and well known optical interference coating design methods can be helpful. [START_REF] Escoubas | Designing waveguide filters with optical thin-film computational tools[END_REF] As a mirror does not transmit a part of the spectrum, it has a forbidden frequency band for photons and it is a 1D photonic crystal. Thin film Fabry-Perot filters can also be seen as photonic bandgap structures with a defect.

Optical Properties of Materials Including Quantum Structures

Generally, models used to study the optical properties of nanostructures are based on the electromagnetic theory, but when the dimension of a semiconductor nanocrystal is smaller than the de Broglie wavelength, quantum phenomena must be considered. The de Broglie wavelength can be seen as the coherence length of the electron wavefunction.

Because of their interesting electronic and luminescent properties, plenty of research concerning quantum structures is performed in different laboratories. [START_REF] Lakhtakia | Nanometer Structures: Theory, Modeling, and Simulation[END_REF] A structure limited to a few nm in the three dimensions of space is called quantum dot (QD). QDs have an atom-like behavior, as the energies of the electron or hole can only take discrete sets of values. Their absorption and luminescent spectra depend both on the properties of the semiconductor used and on the size of the QDs. They can be very efficient light emitters with photoluminescence quantum efficiency as high as 80%. Their emission can be continuously tuned through a large spectrum by changing their size. As an example, CdSe QDs emit anywhere from 470 to 630 nm by varying their size from 3 to 6.5 nm.

Apart from applications already implemented such as multiple quantum wells [START_REF] Zory | The origin of quantum wells and the quantum well laser[END_REF] for lasers and detectors, [START_REF] Schneider | Quantum Well Infrared Photodetectors: Physics and Applications[END_REF] other important applications are forthcoming with the study of quantum structures.

Among others, QDs can be used to make biological tags. [START_REF] Bruchez | Semiconductor nanocrystals as fluorescent biological labels[END_REF] More recently, they have been used to develop high efficiency white LEDs. [START_REF] Nizamoglu | Color-converting combinations of nanocrystal emitters for warm-white light generation with high color rendering index[END_REF] They are also under study to make more efficient OLEDs (QLED). [START_REF] Coe | Electroluminescence from single monolayers of nanocrystals in molecular organic devices[END_REF] Quantum structures are also studied to increase the photovoltaic solar cell efficiency and to go beyond the Shockley-Queisser limit of 32% for Si based solar cells. 68 There are two ways for enhancing the conversion efficiency. The photovoltage or the photocurrent can be increased. To increase the photovoltage, the hot carriers must be extracted from the photoconverter before they cool. The energetic hot carriers must produce two or more electron-hole pairs through impact ionization to increase the photocurrent.

In principle, QDs can be used in these two ways, which exclude one to the other in different QD solar cell configurations. [START_REF] Kamat | Quantum dot solar cells. Semiconductor nanocrystals as light harvesters[END_REF] Another way to collect UV photons is to change their frequency to convert them into visible photons. QDs can also be used to make quantum dot-sensitized nanocrystalline TiO 2 solar cells. They can also be organized in a thin layer with an absorption band complementary to the absorption band of another material composing the solar cell. [START_REF] Vassiltsova | Ordered fabrication of luminescent multilayered thin films of CdSe quantum dots[END_REF] In most photonics components, quantum dots are embedded in thin films. So, to model the optical properties of such components, the refractive index n of thin films including quantum dots (TFIQD) must be known. The classical model used to describe n cannot apply to quantum structures and the quantization of the electron energy must be taken into account.

Frequency Conversion

If the radiative recombination rates of electrons generated by energetic photons are faster than cooling rates, frequency conversion can occur and can increase cell efficiency.

As an example, when considering wide band gap semiconductor nanocrystals like ZnO, the UV can be absorbed and changed into longer wavelengths. Green and UV are emitted. The luminescence in the green is not completely explained but it can be due to oxygen vacancies. [START_REF] Studenikin | Fabrication of green and orange photoluminescent, undoped ZnO films using spray pyrolysis[END_REF] The luminescence in the UV is coming from quantum confinement with an energy level close to the edge of the bandgap. The UV shift of the bandgap is well explained by calculation (Fig. 3). [START_REF] Flory | Optical properties of dielectric thin films including quantum dots[END_REF] In this case of luminescence, the charges generated by the UV photons must radiatively relax before being transferred to the surrounding media, being recombined at the nanocrystal surfaces, or being coupled to phonon nonradiative decays.

Charge Carrier Generation

Depending of their size, CdSe nanoparticles produce electrons and holes from different photon energies of the visible spectrum. The charges can be transferred to the surrounding medium if resonant transfers are possible. These particles can replace dye molecules in TiO 2 sensitized dye. [START_REF] Robel | Quantum dot solar cells. Harvesting light energy with CdSe nanocrystals molecularly linked to mesoscopic TiO 2 films[END_REF] It is interesting to notice that, thanks to the relative band offset, CdSe quantum dots have a resonant electron energy level with the conduction band of TiO 2 . "Rainbow" solar cells using CdSe of different-sized quantum dots assembled in an orderly fashion are under study. 74 Hybrid organic/inorganic cells can also be made with P3HT and CdSe nanorods or P3HT and PbS nanoparticles. With size-quantized PbS nanoparticles, 3% incident photon to current efficiencies (IPCE) under 550-nm monochromatic irradiation is obtained. [START_REF] Günesa | Hybrid solar cells using PbS nanoparticles[END_REF] A 6.9% IPCE under 0.1 milliwatt per square illumination at 515 nm is obtained with 7-nanometer by 60-nanometer CdSe nanorods. [START_REF] Jiu | Hybrid solar cells based on blends of CdSe nanorods and poly(3-alkylthiophene) nanofibers[END_REF]

Multiple Exciton Generation

When photoexcited carriers elastically collide with one another, no energy is lost. On the contrary, collisions with the atoms of the cell material are inelastic. They induce an energy loss with phonon induced decays. In principle, if atomic collisions can be avoided during the time it takes a photogenerated carrier to cross the cell, such a thermalization loss can be avoided.

Carriers either have to go through the cell very quickly or cooling rates have to be slowed in some way. If, instead of giving up their excess energy as heat, the high energy electron-hole pairs are used to create additional pairs, higher efficiency would be possible (Fig. 4). This can be achieved by using QDs because of the longer lifetime of the discrete excited states. The creation of more than one pair by high energy photons has attributed to impact ionization by the photoexcited carriers. In such a case, many electron-hole pairs could be generated by each incident photon (Fig. 5). [START_REF] Ellingson | Highly efficient multiple exciton generation in colloidal PbSe and PbS quantum dots[END_REF] In principle, MEG can give up to 75% efficient conversion. To the best of our knowledge, only the Auger principle has been demonstrated, but there may be different competing processes.

Charge cooling because of phonon coupling is, of course, one of them but also, if the absorption occurs into the wrong state, there will be no impact ionization. Auger recombination after decay can also happen.

Optical Properties of Thin Films Including Quantum Dots

The value of the complex refractive index or permittivity is needed to calculate the optical properties of photonic structures. The bulk value of the refractive index is not fitted when the materials are structured in small dimensions and the classical model [START_REF] Lorentz | Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte[END_REF][START_REF] Lorenz | Über die Refractionsconstante[END_REF][START_REF] Born | Chapter 2 in Principles of Optics[END_REF] cannot be used. A model for quantum wells can be used as a first theoretical model. Many works have already been performed on single and periodic quantum wells of square and nonsquare profiles. [START_REF] Carpena | Number of bound states of a Krönig-Penney finite-periodic Superlattice[END_REF][START_REF] Cohen-Tannoudji | Mécanique quantique[END_REF] The model presented in Ref. 72 based on the approximation of the effective mass, can be applied to single wells, coupled periodic, and nonperiodic multiple wells and to stacks of wells of any depth and any width. The main advantages of this model are its easy implementation and its small computation time. 6 Splitting of the highest energy level of electrons in coupled quantum wells, when considering from left to right 1, 2, 3, 4, and 5 identical coupled wells. Each well has a depth of 2.9 eV, a width of 1 nm and the distance between the wells is also 1 nm.

In a multiple well, a recurrent relation can be found for the function Q = 1 m dψ dx /ψ, where ψ is the electron wave function and x is the direction perpendicular to the well boundaries

Q j = k j Q j -1 -k j tan(k j a j ) Q j -1 tan(k j a j ) + k j , (1)
with j referring to a well boundary in the structure. The width of the jth layer is a j and k j = χ j for E > V j or -iγ j for E < V j . The function Q is continuous, crossing a potential barrier. So, the value of Q on one limit of a structure can be obtained as a function of Q on its other limit.

If the electron is confined in the structure, ψ is evanescent in the surrounding media. The equation describing the possible discrete energy levels is then derived from these considerations.

The wavefunction and the square of its modulus giving the electron density probability distribution, can then be calculated in the whole structure

j (x) = C + j e ik j x + ik j -Q j -1 Q j -1 + ik j e ik j (2x j -1 -x) , (2)
where C + j is a constant that can be normalized.

k j = χ j for E > V j or -iγ j for E < V j with χ 2 j = 2m/h 2 (E -V j ) and ρ 2 j = 2m/h 2 (V j -E).
For each possible value of the energy

|ψ| 2 dx = 1.
In identical coupled wells, the energy levels of the single well are split (Figs. 6 and7) and when there are a large number of coupled wells, they become energy bands. In nonperiodic structures, the electrons can be localized or delocalized like electrons in molecules (Fig. 8). It is worth noticing that the properties of the charges can then be engineered, to some extent, by adapting the size of the QDs and their coupling distance.

A lot of new applications are expected to come with such artificial molecules.

As for electrons, the energy levels of the holes can be calculated in quantum wells. The optical properties of quantum wells can then be deduced from the electron and hole energy distribution. An example of a model is proposed in Ref. 83 for the complex refractive index dispersion with a wavelength of interdiffused quantum wells. In the case of a film composed of quantum dot semiconductor embedded in a homogeneous matrix, the refractive index can be calculated from the refractive index of the quantum dots and the refractive index of the matrix by using the homogenization theory.

Knowing the energy levels of electrons and holes in a quantum structure [START_REF] Gaponenko | Optical Properties of Semiconductor Nanocrystals[END_REF] and taking into account the band broadening phenomena induced by size dispersion and coupling effects, the absorption dispersion with wavelength can be approximately calculated. But the amplitude of the absorption in QDs cannot be directly calculated as it depends on their absorption cross section, which has not yet been modeled. The absorption amplitude can then be only obtained by measurement. By measuring R(λ) and T (λ) of a thin film including quantum dots, the extinction coefficient dispersion is deduced from the absorption A(λ) = 1 -R(λ) -T (λ)

k(λ) = - λ 4πd Log (A (λ)) , ( 3
)
with d the thickness of the film. By using Kramers-Krönig relations, the dispersion of the real Q4 part of the refractive index of the TFIQD can be obtained. As to have quantum confinement, Fig. 8 Electron density of probability in a nonperiodic structure composed of five coupled wells of depth 2.9 eV, of widths 1, 1, 2, 1, and 1 nm, respectively. The distance between each well is 1 nm.

the bandgap of the QDs must be smaller than that of the surrounding material, the absorption is mainly due to the QDs for the wavelengths just shorter than the QDs bandgap. In this spectral region, the measured absorption dispersion with wavelength should correspond to the calculation.

So, research on quantum structures for photovoltaic cells is very active. In the near future, many progresses are expected to develop high-efficient and low-cost solar cells. The control of the material structures at a nanometer level is mandatory to fully benefit from the attractive Q5 properties of QDs of different sizes and made of different materials. Coupled QDs could also lead to new useful structures for many different applications.

Nanoplasmonics

The free electrons at a planar surface between a metal and a dielectric material can exhibit a collective oscillation of free electrons at a resonant frequency. The corresponding surfaceplasmon-polariton (SPP) electromagnetic wave of TM polarization propagates along the surface. This surface wave can only be excited with an evanescent wave. A totally reflecting prism can be used for that purpose. [START_REF] Kretschmann | Radiative decay of non-radiative surface plasmons excited by light[END_REF] When considering metallic nanoparticles of dimension of a few tens of nanometer, the SSP can be excited on the nanoparticle surfaces. [START_REF] Mie | Beiträge zur optik trüber medien, Speziell Kolloidaler Metallösungen[END_REF] The two key interest areas are 1. the ability to channel light into sub-wavelength zones and 2. the local optical electric field is strongly enhanced on the metal surface thanks to plasmon resonance. This can be used to increase the absorption in the surrounding material or to enhance fluorescence and Raman scattering.

As the propagation length of a SPP is generally longer than the circumference of the metallic nanoparticle, the electric field amplitude is modulated on the surface. If several nanoparticles are close enough to be coupled, the optical electric field is also increased in the medium between the nanoparticles. The resonance and its width depend, of course, on the metal used but also on the size, the shape of the nanoparticle, and the dielectric properties of the surrounding medium. It can take place in the visible or in the infrared. The SPPs has a TM polarization when the dielectric Q6 media is isotropic but multiple modes of SPP-wave propagation with different polarization states can be simultaneously supported by a metal/structured thin film interface. [START_REF] Polo | Surface electromagnetic waves: A review[END_REF] The plasmonics features have been extensively studied by the international scientific community and are considered as extremely promising. [START_REF] Kreibig | Optical Properties of Metal Clusters[END_REF] Optical nanoantennas based on metallic nanostructures will boost the efficiency of light-matter interactions. [START_REF] Novotny | Optical antennas tuned to pitch[END_REF] This electromagnetic enhancement plays a key role in surface enhanced techniques: surface enhanced Raman spectroscopy , surface enhanced fluorescence, surface enhanced infrared absorption, etc. Using Q7 these techniques, single molecule detection can now be achieved. [START_REF] Kneipp | Single Molecule Detection Using Surface-Enhanced Raman Scattering (SERS)[END_REF] The correlation between the observed electromagnetic enhancement and the plasmonic properties of these nanoantennas is heavily studied.

Highly efficient plasmonic nanostructures are also required for further development of new optical near-field techniques such as tip enhanced Raman spectroscopy . [START_REF] Yeo | Tip-enhanced Raman Spectroscopy-Its status, challenges and future directions[END_REF] Q8

Nanoplasmonic is also under study to make plasmonic optical circuits, [START_REF] Barnes | Surface plasmon subwavelength optics[END_REF] highly efficient light emitting diodes, [START_REF] Liedberg | Surface plasmon resonance for gas detection and biosensing[END_REF] and extremely sensitive chemical and biochemical sensors. [START_REF] Kwon | Surface-plasmon-enhanced light-emitting diodes[END_REF] Plasmonics has a great interest in the photovoltaic domain in order to improve the photonic absorption, thus, the solar cell efficiency. [START_REF] Stenzel | Enhancement of the photovoltaic conversion efficiency of copper phthalocyanine thin film devices by incorporation of metal clusters[END_REF] For this application, the aim of plasmonics is to excite localized plasmons by using metallic nanoparticles to trap or confine light inside the photoactive material, or to obtain beneficial resonant internal light scattering on these metallic particles. [START_REF] Derkacs | Improved performance of amorphous silicon solar cells via scattering from surface plasmon polaritons in nearby metallic nanoparticles[END_REF][START_REF] Pillai | Surface plasmon enhanced silicon solar cells[END_REF][START_REF] Konda | Surface plasmon excitation via Au nanoparticles in n-CdSe/p-Si heterojunction diodes[END_REF] This excitation depends on several parameters such as the metal nature, the surface density of the particles, their diameter, the incident angle of light, the wavelength, the polarization, etc. Light absorption in the active material can then be increased, in particular in the spectral ranges where the photovoltaic material only weakly absorbs the light. As an example, one of the best power efficiencies for an organic solar cell [6.5%, Ref. 99] is based on an active donor:acceptor bulk heterojunction, the P3HT:PCBM. Its low cutting wavelength (close to 650 nm, i.e., a bandgap around 2.1 eV) greatly limits the efficiency. Enhancing light absorption around this spectral range where solar intensity is maximum is the current challenge. As a consequence, a precise and highly efficient characterization of optical nanoantennas used in such devices is essential and can be seen as a real challenge. Unfortunately, it remains a "largely unexplored terrain." [START_REF] Kreibig | Optical Properties of Metal Clusters[END_REF] Today, many theoretical calculations [finite-difference time-domain (FDTD), for instance] have clearly demonstrated the nature and the spatial localization of the electromagnetic field in the vicinity of metallic nanoparticles. As an example, the electric field distribution calculated with the FDTD method in a poly [2-méthoxy-5-(2-éthyl-hexyloxy)-1, 4-phénylène-vinylène]: 6, 6-phenyl C61-butyric acid methyl ester (MEH-PPV:PCBM) bulk heterojunction including silver nanospheres is shown in Fig. 9. A comparison between the absorption with and without nanoparticles of such a heterojunction is given in Fig. 10. 100,101 Nevertheless, it remains extremely difficult to have an experimental nanometer localization of the enhanced electromagnetic field. In classical surface enhanced spectroscopy, the final optical signal is measured in far field and the precise localization of hot spots remains impossible due to the poor resolution. As a consequence, it appears extremely difficult to validate the nature of the enhanced electromagnetic field obtained with theoretical calculations by experimental measurements. Some experimental works have already been performed, but this remains an important topic of study.

Conclusions

With an increasing capability of structuring materials at a smaller and smaller scale, one can access more and more precise control of light/material interaction. We can expect to organize the propagation of optical waves as well, close to the material in the near field as at any distance of it.

Random surface structures that control statistical properties are developed to shape, at will, the bidirectional reflectance distribution function (BRDF) of surface. Wide-band antireflection Q9 structures are rapidly progressing but also deeply etched surfaces like black silicon appear to exhibit remarkably strong absorption and antireflection properties even in the infrared. The inhomogeneities and the structure of thin films can now be partly controlled. This is used to make materials with defined refractive index with, in particular, birefringent properties of high interest for applications. Photonic crystals have also experienced a very strong development these last years in particular with the metamaterials. New developments are now taking place with nanomagnetic structures. But when the structure can be controlled at a smaller size of a few thousands atoms, Q10 a new field is opening with a control of intimate properties of matter.

Quantum confinement, which can be used to tune the properties of electrons, must lead to completely new applications where electrons and photons must be considered together. The concept of permittivity is then revisited. This concept is also different for metallic nanoparticles embedded in a dielectric material and the association of plasmon resonance and scattering again concerns both electron and photons.

Though not discussed in this paper, the dynamic properties of electrons and the nonlinear optical properties have yet to be jointly considered and a lot of research has yet to be performed in these fields. This should pave the way to more and more complex applications at low cost, well suited to nomadism, for every day life, energy, security, environment, and health.

Fig. 1

 1 Fig. 1 Low resolution SEM of the black silicon surface (Ref. 29).

Fig. 2

 2 Fig. 2 Model of anisotropy for columnar thin films.

Fig. 3

 3 Fig. 3 Absorption spectrum of ZnO nanocrystals in a solution of MeOH.

Fig. 4

 4 Fig. 4 Quantum yield versus the relative energy of photons in front of the material bandgap showing multiple exciton generation with PbSe nanoparticles.

Fig. 5

 5 Fig. 5 Sketch of a multiple exciton generation process.

Fig.

  Fig.6 Splitting of the highest energy level of electrons in coupled quantum wells, when considering from left to right 1, 2, 3, 4, and 5 identical coupled wells. Each well has a depth of 2.9 eV, a width of 1 nm and the distance between the wells is also 1 nm.

Fig. 7

 7 Fig. 7 Electron density of probability for the lowest energy level in a structure composed of nine identical coupled wells.

Fig. 9

 9 Fig. 9 Electromagnetic power density distribution in the plane {x, y} for an array of four silver nanoparticles (NPs) embedded in a dielectric material. Incident wavelength: (a) 450 and (b) 600 nm.

Fig. 10

 10 Fig. 10 Intrinsic absorption of MEH-PPV:PCBM (light gray), silver NPs (black), and absorption losses (reflection and transmission) (dark gray) for a structure with NP diameters of 100 nm and period of 200 nm.

Francois

  Flory graduated from Ecole Centrale Marseille in 1975, then received a PhD in optics in 1978 and a "Thése d'Etat" in 1987. He is currently a professor "classe exceptionnelle" at Ecole Centrale Marseille. He was deputy director of this institution for 6 years and co-founded the optical cluster POPsud. He performs his research at the Institute for Materials, Microelectronics and Nanosciences, Mareseille, France. His research interests are optical coatings and optical properties of nanostructured materials for different application fields such as solar cells, sensors, and light detectors. He is editor and co-author of the book Thin Films for Optical Systems (Marcel Dekker, 1995) and author of more than 200 publications, book chapters, patents, and conferences. He was the chairman of the conference Education and Training in Optics and Photonics (2005). He has been member of scientific committees of several international conferences and is a reviewer for several international scientific journals. Ludovic Escoubas graduated from Centrale Marseille and received a PhD in optics in 1997. He is now a professor at Paul Cezanne University, Marseille, France and leader of the Optoelectronics Components and Photovoltaics (OPTO-PV) Team of IM2NP (CNRS Laboratory). His current research interests are micro-and nano-optical components and solar cells. He has authored more than 150 papers and communications, and he holds 6 patents. Gérard Berginc received a Dipl. Engineer-Physicist degree from Ecole Centrale de Marseille and his doctorate in theoretical physics. He is currently a chief scientist and head of Advanced Research and Technology at Thales Optronique S.A. His research activities include high-frequency asymptotics, random volume and rough surface scattering, coherent effects in random media, and localization phenomena. He has authored or co-authored over 150 publications in books, journals and conferences, and is a Fellow Member of the Electromagnetics Academy. He holds more than 60 patents and was listed in Who's Who in Electromagnetism and Who's Who in the World.

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D Mangelinck
email: dominique.mangelinck@im2np.fr

T Luo

C Girardeaux

Reactive diffusion in the presence of a diffusion barrier: experiment and model

Keywords: Diffusion barrier, reactive diffusion, model, growth kinetics, Nickel silicide

Reactions in thin film and diffusion barriers are important for applications such as protective coatings, electrical contact and interconnections. In this work the effect of a barrier on the kinetics of the formation for a single phase by reactive diffusion is investigated from both experimental and modeling point of view. Two types of diffusion barriers are studied: (i) a thin layer of W deposited between a Ni film and the Si substrate and (ii) Ni alloy films, Ni(1%W) and Ni(5%Pt), that form a diffusion barrier during the reaction with the Si substrate. The effect of the barriers on the kinetics of -Ni2Si formation is determined by in situ X ray diffraction and compared to models that explain the kinetics slowdown induced by both types of barrier. A linear parabolic growth is found for the deposited barrier with an increasing linear contribution for increasing barrier thickness. On the contrary, the growth is mainly parabolic for the barrier formed by reaction between an alloy film and the substrate. The permeability of the two types of barrier are determined and discussed. The developed models fit well with the dedicated model experiments, leading to a better understanding of the barrier effect on the reactive diffusion and allowing to predict the barrier behaviour in various applications.

Introduction

Thin film reactions and diffusion barriers are of interest in numerous applications of protective coatings in metallurgy, intermetallic control in aeronautics, interlayer in diffusion welding, and contact formation and interconnections in microelectronics.

In metallurgy, a diffusion barrier is a thin layer of metal usually placed between two other metals in order to protect either one of the metals from modifying the other [START_REF] Cahn | Physical Metallurgy[END_REF]. In aeronautics, the diffusion barrier may come from the segregation of a impurities contained in one of the materials and it could limit the growth of intermetallic [START_REF] Zhang | Effect of Ru on interdiffusion dynamics of β-NiAl/DD6 system: A combined experimental and first-principles studies[END_REF] [START_REF] Bai | Migration of sulphur in thermal barrier coatings during heat treatment[END_REF].

In microelectronics, barriers are usually introduced to chemically isolate semiconductors from metal interconnects, while maintaining an electrical connection between them. For instance, a layer of barrier metal must surround every copper interconnection in modern copper-based chips, to prevent diffusion of copper into surrounding materials [START_REF] Nicolet | Diffusion barriers in thin films[END_REF] [START_REF] Kaloyeros | Ultrathin Diffusion Barriers Liners for Gigascale Copper Metallization[END_REF].

In most of these cases, the growth of phases such as intermetallics occurs and reactive diffusion is involved. This is also the case in thin film reactions in which both diffusion and reaction are contributory.

Driven by the applications, a considerable amount of work has been done on reactive diffusion both from the experimental [START_REF] Heurle | Kinetics of formation of silicides: A review[END_REF][7] [START_REF] Colgan | A review of thin-film aluminide formation[END_REF] and theoretical aspect [START_REF] Kidson | Some aspects of the growth of diffusion layers in binary systems[END_REF] [10] [START_REF] Borgenstam | DICTRA, a tool for simulation of diffusional transformations in alloys[END_REF] [START_REF] Svoboda | Application of the thermodynamic extremal principle to the diffusional phase transformations[END_REF] [13] [START_REF] Erdelyi | Reactive diffusion and stresses in spherical geometry[END_REF]. In microelectronics, the formation of contact in devices is obtained by reaction between of a metal film and the Si substrate to form a silicide. The formation of these silicide films has been largely studied for thickness in the range of a few hundreds of nanometers [START_REF] Heurle | Kinetics of formation of silicides: A review[END_REF] [START_REF] Nicolet | Chapter 6 -Formation and Characterization of Transition-Metal Silicides[END_REF] and more recently for smaller thickness [START_REF] Mangelinck | Progress in the understanding of Ni silicide formation for advanced MOS structures[END_REF] [START_REF] Mangelinck | Chapter 9 -The Growth of Silicides and Germanides[END_REF]. These studies on silicide and other studies on different systems [START_REF] Barmak | Phase Formation Sequence for the Reaction of Multilayer Thin-Films of Nb/Al[END_REF] [START_REF] Emeric | Al-Co reactions in multilayer films: first stages and selective formation of new (metastable) phases[END_REF] have shown that the reactions in thin films (typically between 20 and 200 nm) [START_REF] Kaloyeros | Ultrathin Diffusion Barriers Liners for Gigascale Copper Metallization[END_REF] are usually characterized by a sequential growth, the absence of some equilibrium phases, and sometimes by the growth of metastable phases, while the simultaneous parabolic growth of all the equilibrium phases is usually observed in bulk interdiffusion couples. It was also shown that after the nucleation and the lateral growth that leads to a continuous layer, the thickening of the layer will occur by linear-parabolic growth [START_REF] Mangelinck | Chapter 9 -The Growth of Silicides and Germanides[END_REF]. This model of growth was first stated by Evans [START_REF] Evans | The corrosion caused by products of combustion of coal gas[END_REF] and then demonstrated by Deal and Grove [START_REF] Deal | General Relationship for the Thermal Oxidation of Silicon[END_REF] in the case of silicon oxidation. The growth of a single compound layer combine two types of process: (i) the rearrangement of the atoms at the interfaces required for the growth of the compound layer which may involve a reaction barrier and (ii) the diffusion of matter across the compound layer in which the diffusion flux slows down with increasing the layer thickness. If the interfacial reaction barriers control the kinetics, it is termed interface controlled (or reaction controlled) and the layer thickness increases linearly with time. If the diffusion process is rate limiting and controls the growth, the corresponding kinetics is termed diffusion controlled and the layer thickness increases proportional to the square root of time. Since the early works [19] [20], several studies deals with the linear parabolic growth as well as with its implication for the sequential growth both from experimental [START_REF] Nemouchi | Differential scanning calorimetry analysis of the linear parabolic growth of nanometric Ni silicide thin films on a Si substrate[END_REF] [22] [START_REF] Nemouchi | Simultaneous growth of Ni5Ge3 and NiGe by reaction of Ni film with Ge[END_REF] or theoretical [START_REF] Balogh | Interfacial Reaction and Phase Growth for Various Metal/Amorphous Silicon System[END_REF] [25] [START_REF] Gösele | Growth kinetics of planar binary diffusion couples: ''Thin-film case'' versus ''bulk cases[END_REF] point of view.

The Ni thin film reaction with silicon can be considered as a characteristic system for reactive diffusion and presents a large interest for application. Most of the studies on the reaction between a pure Ni thin film and the Si substrate have shown the following sequence of phases during the reaction [START_REF] Nicolet | Formation and Characterization of Transition-Metal Silicides[END_REF]: δ-Ni2Si appears as the first phase and grows at low temperature (from 250°C) until full consumption of the Ni metallic film. The NiSi phase then grows at the expense of the δ-Ni2Si phase and is stable in a temperature range up to 700-800°C. The third phase NiSi2 grows after NiSi at a temperature normally higher than 700°C since it has difficulty in nucleation [START_REF] Heurle | Nucleation of a new phase from the interaction of two adjacent phases: Some silicides[END_REF]. With the downscaling of microelectronic devices, NiSi is widely used as contacts in complementary-metal-oxide-semiconductor (CMOS) transistors because of its low resistivity, low Si consumption and Ni diffusion controlled reaction [START_REF] Lavoie | Towards implementation of a nickel silicide process for CMOS technologies[END_REF].

However, its application is limited by the agglomeration of NiSi and the formation of the high resistivity NiSi2 at intermediate temperature (500°C~800°C) that leads to a degradation of the contact resistance.

In order to solve these issues, Ni has been alloyed with different elements to increase the stability of NiSi. [START_REF] Mangelinck | Enhancement of thermal stability of NiSi films on (100)Si and (111)Si by Pt addition[END_REF] [31] [START_REF] Lavoie | Effects of additive elements on the phase formation and morphological stability of nickel monosilicide films[END_REF]. Mangelinck et al. [START_REF] Mangelinck | Enhancement of thermal stability of NiSi films on (100)Si and (111)Si by Pt addition[END_REF] [31] [START_REF] Deduytsche | Formation and morphological stability of NiSi in the presence of W, Ti, and Ta alloying elements[END_REF] have shown that the additive 5 at. % Pt in the Ni film can increase the temperature of NiSi2 nucleation and delay the agglomeration of NiSi by more than 100°C. Lavoie et al. [START_REF] Lavoie | Effects of additive elements on the phase formation and morphological stability of nickel monosilicide films[END_REF] have systematically investigated the role of alloy on the stability of NiSi: the additions of Pt, Pd and Rh are most efficient to delay the formation of NiSi2 while elements such as W, Mo, Re, and Ta are amongst the most efficient elements to delay the NiSi agglomeration. Adding alloying element of Pt or W to Ni film could also significantly modify the Ni silicides formation at low temperature [34] [35] [36] [START_REF] Deduytsche | Formation and morphological stability of NiSi in the presence of W, Ti, and Ta alloying elements[END_REF] . However a clear understanding of this effect is still not fully achieved [START_REF] Mangelinck | Progress in the understanding of Ni silicide formation for advanced MOS structures[END_REF]. A common way to introduce an alloy element is to place a thin metal layer (interlayer) between the Ni film and the Si substrate instead to deposit an alloy [START_REF] Wang | Comparison of the thermal stabilities of NiSi films in Ni/Si, Ni/Pd/Si and Ni/Pt/Si systems[END_REF] These models should allow to design a barrier or an interlayer in a variety of applications.

Experimental procedure

Three types of samples were deposited on Si(100) at room temperature by magnetron sputtering system using Ni, W, and Ni(5at.%Pt) targets. Note that, in the following, the atomic fraction are given in at.% but will be expressed as % to simplify the notation. The Si substrate was immersed into a 5% dilute HF solution for 1 min to remove the native oxide prior loading into the sputtering chamber. The deposition was performed with a base pressure of ~10 -8 Torr using 99.9999% pure Ar gas flow. The two first types of samples are 50nm thick pure Ni films deposited either directly on the Si substrate or on a W barrier with two thicknesses (0.5 or 1 nm). The third type of samples consists of 50nm thick Ni alloys, Ni(1%W) and Ni(5%Pt), on Si(100). The Ni(1%W) film was obtained by co sputtering of the Ni and W targets while the Ni(5%Pt) was deposited from the Ni(5%Pt) target. In order to avoid oxidation during the heat treatment, a 20nm thick SiO2 layer was deposited by low pressure chemical vapor deposition on top of all the samples. In-situ XRD measurements were performed using a Cu Kα source, a rapid detector (PANalytical PIXcel) and a chamber equipped with a heating stage under a vacuum of 10 -5 mbar. During the in situ XRD measurement, the temperature was increased from room temperature to 270 °C at a rate of 35 °C/min and 4.6 min long XRD scans were then recorded continuously during various time at this temperature.

The XRD peak were then fitted using a pseudo-Voigt function in order to extract the variation of the XRD peak intensity as a function of the time. In order to compare the kinetics of -Ni2Si, the intensity of the XRD peaks obtained by fitting with a pseudo Voight function are reported as a function of time in Fig. 2. As the XRD intensity is proportional to the volume of the phase and as the texture is not changing noticeably during the isothermal heat treatment (Fig. 1), the intensity can be used as an estimate of the thickness of -Ni2Si. The XRD peak intensity was thus converted into thickness by normalisation and multiplication by the thickness of -Ni2Si expected from the atomic volume change for 50nm of Ni (i.e. 75nm) [START_REF] Nicolet | Chapter 6 -Formation and Characterization of Transition-Metal Silicides[END_REF]. show a parabolic growth as indicated by the t 1/2 slope while the barrier samples have a more complex behaviour close to a linear variation.

Results

Modelling of the barrier effect on reactive diffusion for a single phase

In order to better understand the experimental results, two models for the kinetics of formation in the presence of a barrier are now developed.

We will consider the following situation illustrated in Fig. 3: an intermetallic compounds,

q p B A   , is
growing by diffusion between two ends phases,  and  , in the presence of a barrier ,  , located between  and  . The  phase is either a pure element or an alloy

x x B A  1
. Moreover we will consider that atom A is the main diffusing specie and that the diffusion of B is negligible in all the phases. We will further assume that the diffusion of A in the ends phases is negligible. These assumptions allow to Since the reaction at the   / interface is:

q p B A p B p q A 1 ) (  
 the volume of  formed by atom of the mobile species is:

          A A c x p q p p 1 1 /          ,   ,  A
x ,and  A c are, respectively, the volume occupied by a formula unit, the atomic volume, the atomic fraction and the concentration of the diffusing species.

       A A A J c J dt dL 1 /    Eq. 2
As many intermetallic compounds exist only within very narrow limits of composition, and, when this is not precisely true, within poorly known limits of composition, growth kinetics cannot be properly interpreted in terms of Fick's law. It is thus more appropriated to consider diffusion in terms of the modified Nernst-Einstein equation:

z RT D c J A A A A                  Eq. 3
where the flux of A atoms, JA, is expressed as the product of the concentration of A atoms (

 A c ), their mobility, RT D A 
, and the gradient of chemical potential (i.e. the force on these atoms),

z A     . Furthermore, z A     , the gradient of chemical potential is taken equal to    L A  with   A
 , the chemical potential change per moving A atom [START_REF] Barge | Analysis of the diffusion controlled growth of cobalt silicides in bulk and thin film couples[END_REF]:

      L RT D c J A A A A            Eq. 4
Moreover, the use of the gradient of chemical potential would require complete knowledge of the free energies versus composition for the whole A/B system, and would become meaningful only if it was possible to determine the varying value of DA as a function of the chemical potential of A within the compound.

Similarly the flux trough the barrier is expressed as :

      L RT D c J A A A A            Eq. 5 where 2 3        A and 1 2        A

Assuming a steady state of the diffusion flux (

  A A J J 
) and equilibrium at the interfaces (equality of the chemical potential), one obtains the following expression:

                  1 3 2 A A J J Eq. 6
where

     L D c A A  and      L D c A A        RT D c L c D L RT c dt dL A A A A A 3 1 1 3 1 / 1 / 1 1                                Eq. 7
If the ends phases are the pure elements, the variation of the chemical potential can be related to the Gibbs energy of formation of the phase per mole of atoms,

) ( q P B A G  =  G  by    G p q p     3 1 .

Fixed thickness and fixed concentration for the barrier

The barrier may have a fixed thickness and a fixed concentration in several cases: for examples, when a layer is deposited between A and B or when a thin oxide layer is present at the A/B interface. In this case,  L ,  A c , and  A D are independent of time and Eq. 7 can be simplified to

  RT D L dt dL A 3 1 1 1                   Eq. 8 where     L c D c A A A   .
Equation 8 is formally equivalent to the Deal and Groove law [START_REF] Deal | General Relationship for the Thermal Oxidation of Silicon[END_REF]. For isothermal heat treatment, Eq. 8 can be integrated to obtain a linear parabolic equation for the growth of the  phase.

The parameter K depends mainly on two factors: the thickness of the barrier and its permeability defined as the product of the concentration and diffusion coefficient of A in the barrier:

   A A A D x P  .
In order to reduce the number of parameters, the permeability will be used in the following instead of the concentration and diffusion coefficient of A in the barrier.

  RT   , . / 01 . 0 3 at nm       , 3 / 2   A c
, and

s nm D A 2 1   .
Figure 4.a shows that the more the barrier is efficient to delay the diffusion, the more the growth kinetics is slowed down and the more it becomes linear. This is further more illustrated in Fig. 4.b where the K factor (Eq. 8) is plotted as a function of the barrier thickness. Note that the permeability and barrier thickness have an inverse role on the kinetics (Eq. 8): decreasing permeability leads thus to a very similar behavior than increasing barrier thickness.

Formation of one phase for a barrier built by consumption of an alloy

When the phase is formed from an alloy,

0 0 1 x x C A   
, the C element can be not incorporated in the forming phase and accumulates at the   / interface. As a first approximation, this case can be modelled as a barrier with a fixed concentration,

 x , i.e.    x x C A   1
, and a thickness that increases when the thickness of the  phase increases. If the amount of C in the  phase is negligible, the thickness of the barrier is then given by the conservation of matter:

        L c c c c c c L C A C A C A

 

Eq. 9

Taking Eq. 7 and 9, the growth rate is given by

                     A A C A C A A A C A A RTc D c c c c c c c D L c dt dL 3 1 1 2 1 1                Eq. 10
Since all the parameter in the parenthesis of Eq. 10 do not depend on time, one obtains:

  RT L D dt dL eff 3 1       Eq. 11
Eq. 11 shows that in the case of barrier coming from an alloy, the growth inversely proportional to the thickness of the growing phase and is proportional to an effective coefficient, eff D :

       A A A A A eff D D D D D D     1 1 1 where               2 2 2 1 1 1                                   A A A A A A A C A C A A A C x x x x x x x c c c c c c c
For isothermal heat treatment, the kinetics of formation will thus be parabolic. Figure 5.a shows the kinetics of formation for the same parameter than the one used in Fig. 4. For an atomic fraction of 0.1 in the alloy and for different values of the barrier permeability: the kinetics is always parabolic but decreased with decreasing permeability. Figure 5.b shows also that the effective diffusion depends strongly on the atomic fraction of C in the alloy.

Discussion

In order to compare these models to experiments, the XRD in situ measurements have been fitted by Eq. 8 for the Ni film with or without W barrier and by Eq. 11 for the Ni alloy film, Ni(1 at.% W) and Ni(5 at.% Pt). Figure 6 shows that the simulation reproduce well the experimental variation of -Ni2Si thickness for the different sample. In the fitting procedure, the kinetics for the Ni film without W barrier was first used to determine the diffusion coefficient of Ni in -Ni2Si (

s nm D A / 04 . 0 2  
). The permeability of the barrier was then adjusted to fit the variation of -Ni2Si thickness for the samples with W barrier (Eq. 8) and the NiPt sample (Eq. 11). The permeability for each sample are reported in table I and the following values were used:

C T   270 , . / 01 . 0 3 at nm       , gram at kJ G . / 45     .
Table 1 shows that the value of the permeability are relatively close to each other for the three samples.

The similar value for the two samples with W barrier validates that the linear term, K, in Eq. 8 is indeed inversely proportional to the barrier thickness. It appears that the permeability (i.e. the "weakness") of the barrier is similar for W and Pt which is coherent with former results concerning the formation of Ni silicide from Ni(Pt) and Ni(W) alloy where the growth behaviour of this phase were similar.

Samples

  s nm D x P A A A 2     50nm Ni / 0.5 nm W 1.3 10 -2
50nm Ni / 1 nm W 1.0 10 -2 50 nm Ni1%W 0.8 10 -2 50 nm Ni5%Pt 0.9 10 -2

Table 1: Permeability obtained from the fit of the experimental kinetics. The following parameters were taken for the fit:  

15 / 3 1   RT   , . / 01 . 0 3 at nm       , 3 / 2   A c
, and becomes large in the case of the Ni(Pt) film. This is certainly an indication of the limitation of the model for the alloy film. Indeed the assumption of a constant concentration and a constant diffusion coefficient is quite restrictive since the real situation may be more a gradient in the Pt concentration as indicated by APT measurement [START_REF] Cojocaru-Mirédin | Snowplow effect and reactive diffusion in the Pt doped Ni-Si system[END_REF].

s nm D A / 1 2   .
Note that Eq. 8 and 11 do not take into account the usual reaction barrier in the linear parabolic growth that account for the rearrangement of the atoms at the interfaces required for the growth of the compound layer [START_REF] Deal | General Relationship for the Thermal Oxidation of Silicon[END_REF]. This term can easily be included in these equations but should have a much lower impact on the kinetics that the barrier. The alloy element or the barrier element may also change the diffusion in the growing phase by segregating at the grain boundaries [START_REF] Mangelinck | Three-dimensional composition mapping of NiSi phase distribution and Pt diffusion via grain boundaries in Ni2Si[END_REF] and further work may be needed to separate the two effects. Grain growth in the barrier might also induce a change in the effective diffusion coefficient taking into account both inter and intra grain diffusion in the barrier.

The relatively small change in P for the two W barriers (30% difference) may be attributed to the intermixing that is usually observed when a metal film is deposited on Si. This intermixing has been observed to occur over on 1 or 2 nanometers for Ni. For a refractory metal such as W, the intermixing may be more limited but could explain the 30% difference. Moreover a common assumption is that the grain size scales with the thickness of the layer. One could thus expect a difference in the grain size of the barrier layer. As diffusion is expected to occur via grain boundaries in the barrier, this change in grain size would also contribute to the measured 30% change in P.

The models used in the simulation are based on the assumption of one-dimensional growth. However, nucleation at the interface and/or at the triple junction, as well as the lateral growth along interface play an important role in the formation of silicide. This is particularly true for very thin films [START_REF] Kousseifi | Direct observation of NiSi lateral growth at the epitaxial θ-Ni2Si/Si(1 0 0) interface[END_REF] and may lead to much more complex microstructures. Anyway, for thickness larger than about 10 nm, the nucleation and lateral growth end to a continuous layer which grows then essentially in a one-dimensional manner.

Nevertheless, even if the two models are simple, they allow to reproduce the experimental behaviour and in particular the difference in kinetics regime: linear-parabolic for a deposited barrier and parabolic for a barrier resulting from an alloy consumption. Furthermore the similarity between the permeability allows to validate the models. Further investigations are in progress to compare the permeability value for other alloy elements.

These models are the basis for a simulation method to interpret the barrier effect and they match well with our dedicated model experiments that can be much more difficult to analyze for more complex systems. They should enable to better understand and to predict the barrier behaviour both in fundamental studies and in some applications. They also allow to characterize the efficiency of a barrier by simple parameters such as the barrier permeability. Even if these parameters may depend on the barrier microstructure (for example grain size, grain structure, texture…), our first experiment shows that, for a given process (sputtering in our case), these parameters could be materials constants and our models may thus help to design and to process barriers or interlayers for applications in metallurgy, microelectronics, aeronautics…

Conclusion

The effect of a barrier on the kinetics of the formation for a single phase by reactive diffusion has been investigated from the experimental and modeling point of view. Two types of diffusion barrier have been studied: barrier deposited as a thin layer of W between a Ni film and the Si substrate or barrier built during the reaction between Ni alloy films, Ni(1%W) or Ni(5%Pt), and Si. The kinetics of -Ni2Si formation have been measured by in situ XRD measurement for the two type of barrier and compared to the one of pure Ni. Models have been developed for the two cases and fitted to experiments. The main conclusion are: (i) the two types of barrier decreases the rate of the -Ni2Si phase formation (ii) the linear contribution in the linear parabolic growth becomes more important for increasing deposited barrier thickness while the growth is mainly parabolic for a barrier coming from an alloy (iii) the two model fit well the experiments (iv) the measured permeability are similar for different thickness of the barrier and for the two types of barrier. The developed models make it possible to design barriers or interlayers in a variety of applications.

Figure captions

  [START_REF] Huang | Effect of a thin W, Pt, Mo, and Zr interlayer on the thermal stability and electrical characteristics of NiSi[END_REF] [40][START_REF] Zhang | Impact of an ultra-thin Ti interlayer on the formation of NiSiGe/SiGe[END_REF]: this was shown to change the stability of the Ni silicide as well as their formation. The presence of a thin SiO2 layer can also modify greatly the reactive diffusion and in particular the kinetics of formation or the phase sequence of the Ni silicide [42] [43]. The goal of this work is to investigate the effect of a diffusion barrier or an interlayer on the reactive diffusion and more precisely on the kinetics of formation of a single phase by reactive diffusion. The kinetics of the δ-Ni2Si phase formation has been measured by in situ X ray diffraction during isothermal annealing of pure Ni film, Ni film with a W barrier, and Ni(5%Pt) films. Two models are developed to explain the change in kinetics in the presence of a deposited barrier or of a diffusion barrier that is built during the consumption of an alloy. A comparison between the model and the experimental result allows to validate the two models and to obtain experiment values for the permeability of the barrier which is the product of the diffusion coefficient and the atomic fraction of the diffusing species in the barrier.

Figure 1

 1 Figure 1 shows the XRD intensity as a function of the 2 theta angle and the time for (a) 50 nm Ni, (b) 50nm Ni / 0.5 nm W, (c) 50nm Ni / 1 nm W, and (d) Ni(5%)Pt films on Si(100) during the in situ annealing at 270°C. The in situ XRD measurement for the Ni(1%W) samples is not presented here but they are intermediate between the one of pure Ni and the one of Ni(10%Pt). In all cases, the -Ni2Si phase is growing at the extent of the metal film (Ni, Ni(1%W), or Ni(5%Pt)). When the metal film has been consumed, NiSi grows (this can only be seen in Fig. 1.b but it has been also observed for the other cases in other in situ XRD experiments not shown here). Note that the time scale is not the same for all the measurements and that the time to complete the formation of -Ni2Si is largely increased in the case of the presence of a barrier or the alloy metal film. The difference in texture for the samples without deposited barrier (50 nm Ni, 50 nm Ni(1%W) and Ni(5%Pt) films) and with deposited barrier (50nm Ni / 0.5 nm W, 50nm Ni / 1 nm W) may be due to difference in alloying element concentration, intermixed layer…

  Figure 2.a shows the -Ni2Si thickness as a function of time and confirms that the W barrier or the presence of Pt in the Ni film delays to a large extent the formation of -Ni2Si. Moreover the log-log plot in Fig. 2.b indicates that the variation of the thickness with time show a different behaviour for the samples with W barrier than for the pure Ni, Ni(1%W), and Ni(5%Pt) samples. Indeed the pure Ni, Ni(1%W), and Ni(5%Pt) samples

J

  simplify greatly the equations. The first assumption is valid for -Ni2Si in which Ni diffuses much faster that Si. The diffusion of Ni in Si is fast but its solubility is very low in Si. The flux of Ni in Si can thus be consider as negligible compared to the ones in the barrier and in -Ni2Si validating the second assumption.In these conditions, the growth rate of the  phase is related to the flux of A atoms in the  phase, are the thickness of  and the flux of A through  .    /  is the volume of  formed by atom of the mobile species (A in this case) at the growing interface.

Figure 4 .

 4 Figure 4.a show the variation of the thickness as a function of time for different thickness. The following typical values were taken for this figure and will be used also in the next paragraph:   15 / 3 1

Figure 6 .

 6 Figure 6.d also shows that the simulation fits less well the experiment when the thickness of -Ni2Si

Figure 1 :

 1 Figure 1 : in situ XRD measurement at 270°C for (a) 50 nm Ni, (b) 50nm Ni / 0.5 nm W, (c) 50nm Ni / 1 nm W, and (d) Ni(5%)Pt films on Si(100).

Figure 2 :

 2 Figure 2 : Kinetics of -Ni 2 Si growth for the different samples (a) thickness as a function of time (b) log(thickness) as a function of log(time) : only the part corresponding to the growth of -Ni 2 Si is shown and the linear (green line) and parabolic (yellow line) behavior are indicated at the origin to interpret the kinetics law. Note that the decrease in the -Ni 2 Si thickness observed in (a) is due to the consumption of -Ni 2 Si by NiSi.

Figure 3 :

 3 Figure 3: Scheme of the growth of the  phase in the presence of a barrier layer,  ,

Figure 5 :

 5 Figure 5: Kinetics of formation with a barrier resulting from an alloy (a) Variation of the thickness with time for different barrier permeability (b) Effective coefficient of diffusion as a function of atomic fraction of C in the alloy for a barrier permeability equal to 10 -2 nm 2 /s). The following parameters were taken :   15 / 3 1   RT   ,

Figure 6 :.

 6 Figure 6: Comparison between experience and simulation for (a) 50 nm Ni, (b) 50nm Ni / 0.5 nm W, (c) 50nm Ni / 1 nm W, and (d) Ni(5%)Pt films on Si(100). The following parameters were taken :   15 / 3 1   RT  

Acknowledgements

The French ministry for education and research is acknowledged for providing the scholarship for the PhD of T. Luo. The authors would like to thank M. Descoins and M. Bertoglio for technical assistance.