Clémence Badie

Héloïse Tissot

Beniamino Sciacca

Maïssa K S Barr

Julien Bachmann

Christophe Vallee

Gaël Gautier

Thomas Defforge

Vincent Astié

Jean-Manuel Manuel Decams

Christophe Vallée

Vincent Astie

Mikhael Bechelany

Lionel Santinacci
email: lionel.santinacci@cnrs.fr

Conductive TiN thin films grown by plasma-enhanced atomic layer deposition: Effects of N-sources and thermal treatments

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

I. INTRODUCTION

Titanium nitride thin films are typically used in microelectronics for their metal-like electrical conductivity [START_REF] Knoops | Deposition of TiN and TaN by Remote Plasma ALD for Cu and Li Diffusion Barrier Applications[END_REF][START_REF] Cheng | Improvement of the properties and electrical performance on TiCl4-based TiN film using sequential flow chem. vap. deposition process[END_REF][START_REF] Shearrow | Atomic layer deposition of titanium nitride for quantum circuits[END_REF] . They can also constitute an efficient corrosion protection because they exhibit high thermal and chemical stabilities as well as good mechanical wear resistance for a relatively low cost [START_REF] Kong | Improved corrosion protection of CrN hard coating on steel sealed with TiO x N y -TiN composite layers[END_REF] . Recently, TiN films have been used in other fields such as energy storage [START_REF] Banerjee | Nanotubular metal-insulator-metal capacitor arrays for energy storage[END_REF][START_REF] Gao | Synergic Titanium Nitride Coating and Titanium Doping by Atomic Layer Deposition for Stable-and High-Performance Li-Ion Battery[END_REF] or plasmonic [START_REF] Fomra | Plasmonic colors in titanium nitride for robust and covert security features[END_REF] . Most of these applications request a homogeneous and pinhole-free layer that can accommodate the substrate topography. Atomic layer deposition (ALD) fulfills these requirements compared to physical and chemical vapor deposition (PVD and CVD) techniques [START_REF] Leskelä | Atomic layer deposition (ALD): from precursors to thin film structures[END_REF][START_REF] Johnson | A brief review of atomic layer deposition: from fundamentals to applications[END_REF][START_REF] Asundi | Opportunities for Atomic Layer Deposition in Emerging Energy Technologies[END_REF] . Historically, halides and mostly TiCl 4 with NH 3 constitute the main precursor association to grow TiN exhibiting a satisfying film quality (i. e. low carbon contamination ≤ 5% and low resistivity ≤ 1000 µΩ•cm). These processes are performed at high temperatures (300 to 500 • C) [START_REF] Ritala | Atomic Layer Epitaxy Growth of TiN Thin Films from Til 4 and NH 3[END_REF][START_REF] Ahn | Characteristics of TiN thin films grown by ALD using TiCl 4 and NH 3[END_REF] , that are not compatible with thermally sensitive substrates. In addition, the halide-based chemicals can be trapped within the deposit or back-react with the film and damage the pumping system. Organometallic compounds can be used at lower process temperatures (T ALD ) and they are less corrosive. They have thus been largely explored and have progressively become the most investigated Ti precursors. Few studies are dedicated to alkoxides (e. g. titanium tetraisopropoxide, TTIP [START_REF] Sowińska | Analysis of nitrogen species in titanium oxynitride ALD films[END_REF] ), because such oxygen containing ligands should be avoided to prevent oxygen contamination. Most of the literature focus on alkylamides such as tetrakis(dimethylamino)titanium (TDMAT), tetrakis(diethylamino)titanium (TDEAT), tetrakis(ethylmethylamino)titanium (TEMAT) or heteroleptic molecules such as biscyclopentadienyl titanium triisopropylguanidinate (Cp 2 Ti[(N i Pr) 2 CN(H) i Pr]) [START_REF] Min | Atomic Layer Deposition of TiN Films by Alternate Supply of Tetrakis(ethylmethylamino)-Titanium and Ammonia[END_REF][START_REF] Okada | Adsorption and desorption kinetics of tetrakis(dimethylamino)titanium and dimethylamine on TiN surfaces[END_REF][START_REF] Yun | Comparison of Tetrakis(dimethylamido)titanium and Tetrakis(diethylamido)titanium as Precursors for Metallorganic Chemical Vapor Deposition of Titanium Nitride[END_REF][START_REF] Caubet | Low-Temperature Low-Resistivity PEALD TiN Using TD-MAT under Hydrogen Reducing Ambient[END_REF][START_REF] Wasslen | A Family of Heteroleptic Titanium Guanidinates: Synthesis, Thermolysis, and Surface Reactivity[END_REF][START_REF] Chen | Plasma-Enhanced Atomic Layer Deposition (PEALD) of TiN using the Organic Precursor Tetrakis(ethylmethylamido)Titanium (TEMAT)[END_REF] . Among those organometallic precursors, TDMAT is currently the most used and investigated in research laboratories. The co-reactant should be sufficiently reactive to complete the ALD reaction by removing the ligands of chemisorbed metallic precursor to limit the contamination (i. e. ligands trapped into the film). Usually, NH 3 is used as N-source for thermal ALD process [START_REF] Elam | Surface chemistry and film growth during TiN atomic layer deposition using TDMAT and NH 3[END_REF][START_REF] Fillot | Investigations of titanium nitride as metal gate material, elaborated by metal organic atomic layer deposition using TDMAT and NH 3[END_REF][START_REF] Dube | Effects of Interfacial Organic Layers on Nucleation, Growth, and Morphological Evolution in Atomic Layer Thin Film Deposition[END_REF][START_REF] Longrie | Thermal and Plasma-Enhanced Atomic Layer Deposition of TiN Using TDMAT and NH 3 on Particles Agitated in a Rotary Reactor[END_REF] . Hydrazine is presented as highly reactive alternative but it is also highly hazardous [START_REF] Alvarez | Ultra-High Purity Hydrazine Delivery for Low Temperature Metal Nitride ALD[END_REF] .

Plasma-enhanced ALD (PE-ALD) enables the use of supplementary co-reactants. N 2 and H 2 can indeed lead to nitrides when they are injected separately or as a mixture in a plasma-assisted process. This deposition mode has resulted in various nitride thin films such as InN [START_REF] Haider | Low-temperature self-limiting atomic layer deposition of wurtzite InN on Si(100)[END_REF] , AlN [START_REF] Kim | Fabrication and properties of aln film on gan substrate by using remote plasma atomic layer deposition method[END_REF] , TaN x [START_REF] Chung | Plasma-Enhanced Atomic Layer Deposition of TaN Thin Films Using Tantalum-Pentafluoride and N 2 /H 2 /Ar Plasma[END_REF][START_REF] Langereis | Synthesis and in situ characterization of low-resistivity TaN x films by remote plasma atomic layer deposition[END_REF] , SiN x [START_REF] Leick | Surface reactions of aminosilane precursors during N 2 plasma-assisted atomic layer deposition of SiN x[END_REF] , HfN x [START_REF] Krylov | Electrical and structural properties of conductive nitride films grown by plasma enhanced atomic layer deposition with significant ion bombardment effect[END_REF] , ZrN x [START_REF] Krylov | Electrical and structural properties of conductive nitride films grown by plasma enhanced atomic layer deposition with significant ion bombardment effect[END_REF] and TiN [START_REF] Kim | Characteristics of TiN films deposited by remote plasma-enhanced atomic layer deposition method[END_REF][START_REF] Langereis | In situ spectroscopic ellipsometry study on the growth of ultrathin TiN films by plasma-assisted atomic layer deposition[END_REF][START_REF] Musschoot | Atomic layer deposition of titanium nitride from TDMAT precursor[END_REF][START_REF] Samal | Low-temperature (≤200•c) plasma enhanced atomic layer deposition of dense titanium nitride thin films[END_REF][START_REF] Burke | Low sheet resistance titanium nitride films by lowtemperature plasma-enhanced atomic layer deposition using design of experiments methodology[END_REF][START_REF] Sowińska | Plasma-enhanced atomic layer deposition of titanium oxynitrides films: A comparative spectroscopic and electrical study[END_REF][START_REF] Krylov | Role of reactive gas on the structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF][START_REF] Krylov | Obtaining low resistivity (≈100 µ ω•cm) TiN films by plasma enhanced atomic layer deposition using a metalorganic precursor[END_REF][START_REF] Belahcen | Control of ion energy during plasma enhanced atomic layer deposition: A new strategy for the modulation of TiN growth delay on SiO 2[END_REF] . Although NH 3 molecule exhibits high reactivity, several studies also report plasma-enhanced processes using this chemical because it may lead to reduced resistivity and contamination [START_REF] Kwon | Plasma-enhanced atomic layer deposition of ru-TiN thin films for copper diffusion barrier metals[END_REF][START_REF] Knoops | Status and prospects of plasma-assisted atomic layer deposition[END_REF][START_REF] Vallée | Plasma deposition-Impact of ions in plasma enhanced chem. vap. deposition, plasma enhanced atomic layer deposition, and applications to area selective deposition[END_REF] . In PE-ALD mode, additional parameters such as the generation type (inductively or capacitively coupled plasma), the geometry (remote or direct), the applied plasma power (P plasma ), dilution and ratio of the pulsed gases as well as sample bias can be adjusted. Comprehensive ex situ characterizations of the deposited films along with in situ analyses are therefore requested to get a better understanding of the parameter influence on the process and subsequently to improve the film quality. For instance, Krylov et al. demonstrated that minimizing the total plasma gases pressure (varying N 2 or NH 3 and Ar flows) is the key to obtain highly crystalline and low resistive TiN films (≤ 100 µΩ•cm) [START_REF] Krylov | Obtaining low resistivity (≈100 µ ω•cm) TiN films by plasma enhanced atomic layer deposition using a metalorganic precursor[END_REF][START_REF] Krylov | Effect of the substrate on structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF] .

The present work is dedicated to optimizing the growth rate of TiN PE-ALD and to study the co-reactant influence on the film properties such as resistivity. In a previous study [START_REF] Assaud | Highly-Conformal TiN Thin Films Grown by Thermal and Plasma-Enhanced Atomic Layer Deposition[END_REF] , we assessed conductivity and conformality of TiN films grown by thermal-and PE-ALD using TDMAT with NH 3 according to the recipes developed by the reactor manufacturer. Conformal films were deposited onto nanoporous anodic alumina (1:75) and resistivity ≤ 1000 µΩ•cm was achieved on planar silicon. Here, a special attention is drawn on the understanding of the plasma-assisted step using two different N-sources, NH 3 and N 2 , while the thermal-ALD process acts as reference. Using NH 3 plasma, N * and H * excited species will be generated, while using N 2 , no reducing species (H * ) will be formed, only N * . As mentioned above, NH 3 is intrinsically a highly reactive molecule, without plasma activation. Conversely, N 2 is an inert gas that does not lead to TiN growth by thermal-ALD (it is often used as vector gas in this mode). However, Krylov et al. [START_REF] Krylov | Role of reactive gas on the structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF][START_REF] Krylov | Obtaining low resistivity (≈100 µ ω•cm) TiN films by plasma enhanced atomic layer deposition using a metalorganic precursor[END_REF][START_REF] Krylov | Effect of the substrate on structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF] have reported that N 2 plasma activation can lead to TiN formation. They have also shown that using NH 3 instead of N 2 as well as by minimizing the gas flow (i. e. lower pressure) resulted to a higher crystallinity of the layers and, consequently, to a better conductivity. In addition, a slight improvement of the conductivity was also found when increasing P plasma during the deposition process [START_REF] Krylov | Obtaining low resistivity (≈100 µ ω•cm) TiN films by plasma enhanced atomic layer deposition using a metalorganic precursor[END_REF] . The present work attempts to discuss those results by in-depth studies of the influence of the gas flows and plasma power during the plasma-assisted step. Those investigations are coupled to comprehensive in and ex situ physico-chemical analyses of the TiN deposition on plane and structured surfaces.

The initial recipes optimization are performed by assessing the ALD sequence in order to improve the growth per cycle (GPC), to inject the appropriate amount of precursors and to shorten the process duration. Since it is well-documented, the first half-cycle, dedicated to Ti-source, is shortly described while the second half-cycle, devoted to N-source reaction, is deeply investigated. In situ spectroscopic ellipsometry (SE) and optical emission spectroscopy (OES) are respectively used to monitor GPC and to identify the plasma composition depending on the precursor chemistry, their dilution in the vector gas as well as the plasma power. Morphology, chemical composition, crystalline structure and electrical properties were studied after depositions and conformality was assessed on well-defined porous Si. The resistivity is then related to the deposition conditions and compared to the reference thermal process.

II. EXPERIMENTS

A. Sample preparation and TiN deposition

TiN deposition was performed on n-type Si (100) (phosphorus doped, 1-10 Ω•cm, from Sil'tronix, France). The wafers were cut in 1×1 cm 2 pieces and were degreased by sonicating in acetone, isopropanol, methanol, and ultra-pure water, followed by a dip in HF 5% for 30 s. All chemicals were analytical grade from Sigma Aldrich. The samples were then vigorously rinsed in ultra-pure water, dried under a nitrogen stream and stored in an Ar-filled glovebox.

The conformality of the TiN depositions was assessed on microstructured Si wafers. Fig. 1 presents ordered macropores grown on p-type Si (100) (boron doped, 30-50 Ω•cm) by an anodic etching performed in HF 5% and cetyltrimethylammonium chloride (120 µg•L -1 ) at a constant current density of 15 mA•cm 2 for 7 h. This process has been previously described in details [START_REF] Defforge | Role of Electrolyte Additives during Electrochemical Etching of Macropore Arrays in Low-Doped Silicon[END_REF] .

Both thermal-and PE-ALD were carried out in a showerhead geometry reactor (Fiji 200, Veeco/Cambridge Nanotech) equipped with a remote inductive-coupled plasma source (Seren IPS Inc.), depicted in Fig. 2. During the PE-ALD processes, P plasma was ranged from 50 to 300 W at 13.56 MHz. TDMAT (99.99%, Strem Chemicals) was used as Ti precursor with two different N-sources, NH 3 or N 2 , associated with Ar as vector gas. All gases were electronic grade (≥ 99.999%, from Linde Electronics). Ar was simultaneously flowed at solid/liquid and gaseous precursor inlets (see Fig. 2a). Ar flow (φ Ar ) at the gaseous precursor inlet (where plasma is generated) was varied (from 0 to 200 sccm) to adjust N-source dilution in the vector gas. The pumping system operated at its maximum during the depositions. The pressure (P) in the reactor was not regulated. It ranged from 0.1 to 0.3 Torr depending on the gas flows (φ Ar , φ NH 3 and φ N 2 for Ar, NH 3 and N 2 , respectively). TDMAT was maintained at 70 • C and chamber temperature was varied from T ALD = 100 to 300 • C to screen the ALD window. If not specified, T ALD = 200 • C. ALD sequences consist of pulsing and purging successively TDMAT and N-sources for defined durations (x Ti :y Ti :x N :y N ). They are indicated in the following since optimization of the different processes are reported in the results section. The thickness of the films (t ALD ) was adjusted by varying the number of ALD cycles (N ALD ). For most of the depositions, wafers were positioned on the sample holder at the regular generator-to-sample distance (d plasma ) of 38.2 cm (Fig. 2a). In some cases, samples were lifted up using metallic cylinders to shorten d plasma . As indicated on Fig. 2b, the shorter distances were 16.2 and 20.2 cm, respectively. Some TiN films were encapsulated to prevent oxygen incorporation after air exposure during sample transportations. Thin ALD Al 2 O 3 and TiO 2 layers exhibiting thicknesses of 10 and 5 nm, respectively, were grown directly after the TiN process at 150 • C. Depositions of both oxides have been described previously [START_REF] Dufond | Quantifying the Extent of Ligand Incorporation and the Effect on Properties of TiO 2 Thin Films Grown by Atomic Layer Deposition Using an Alkoxide or an Alkylamide[END_REF][START_REF] Roy | Preventing Corrosion of Aluminum Metal with Nanometer-Thick Films of Al 2 O 3 Capped with TiO 2 for Ultraviolet Plasmonics[END_REF] . In other cases, TiN layers were annealed under a reductive and nitriding atmosphere (N 2 :H 2 :NH 3 , 480:60:60 sccm, respectively) using an AS-One furnace (Annealsys SAS) for 5 min at an annealing temperature (T an ) ranging from 600 to 800 • C under P = 50 Torr. All samples were sealed in an Ar-filled container directly after deposition and annealing for storage and transportation (they were exposed to air less than a minute).

B. Characterizations

TiN thickness was monitored during deposition by in situ SE using a M-2000V (J. A. Woollam Co.). Curve fitting was performed using the CompleteEASE 5.10 software (model was assessed by comparing simulations to thickness measurements performed on TEM cross section observations). GPC was deduced from thickness measurements achieved by SE. The root mean square roughness (R RMS ) of the layers was measured by atomic force microscopy (AFM) using a XE-100 PSIA system by scanning a 5×5 µm 2 area. Scanning and transmission electron microscopies (SEM and TEM) using a JSM 7900F (JEOL Ltd) and a JEM 2010 (JEOL Ltd), respectively, were used to investigate the morphology, confirm the thickness and assess the films conformality. Energy dispersive spectroscopy (EDS) was performed using a Quantax FlatQuad (Bruker). The TEM cross sections were carried out on samples prepared by ion milling with a precision ion polishing system (PIPS, Gatan M691).

Crystalline structure was analyzed by X-ray diffraction (XRD) using a high brilliancy rotating anode X-ray source (Rigaku RU 200BH) and a Xenocs Fox3D Cu 12_INF mirror (Osmic) coupled with an image plate detector (Mar345, Rayonix). The incident radiation was Cu K α (1.5418 Å) exhibiting a 0.5 × 0.5 mm 2 beam size.

X-ray photoelectron spectroscopy (XPS) was used to determine the film composition depending on their preparation procedure. Samples were studied after TiN deposition (asgrown, AG), after thermal treatment (annealed, An) and after protective encapsulation. The analyses were carried out on an ULVAC-PHI Quantera II system working at a base pres-sure of 7.5×10 -7 Torr and equipped with a monochromated Al K α source and a hemispherical energy analyzer. To prevent charging a combination of electron and ion neutralization was employed. Data processing was performed with CasaXPS analysis software (version 2.3.22PRO1.0) using a Shirley type background subtraction. The component peaks were defined by binding energy (BE), full width at half maximum (FWHM) and Gaussian-Lorentzian envelopes.

During TiN PE-ALD, plasma composition was monitored by in situ OES depending on injected N-source (NH 3 or N 2 ). Emitted light was collected at the plasma source (Fig. 2a) and analyzed between λ = 360 -900 nm by an UV-vis spectrometer (Avaspec ULS 3648 USB2 from Avantes) driven by the Avasoft Full software. During OES measurements, the optical fiber was located at the plasma generator window because no sufficient light flux was collected at the lower window (closer to the sample).

To assess film quality, electrical measurements were performed on the TiN layers by evaporating successively 10/150 nm thick Cr/Au pads (100 × 80 µm 2 , 100 µm-spaced). The resistivity was deduced from current-voltage curves obtained by 4-point probe (FPP) technique using a HP4140 Pi-coAmmeter/DC voltage source.

III. RESULTS AND DISCUSSION

A. Process optimization

As mentioned in the introduction, the three TiN ALD processes are investigated: thermal using NH 3 (th-NH 3 ) and plasma-assisted using NH 3 or N 2 (pl-NH 3 and pl-N 2 , respectively) both mixed with Ar. Since thermal process optimization led to slight modifications of the initial ALD sequence, it is reported in Fig. S1 in supplementary materials (see supplementary material at URL will be inserted by AIP Publishing for additional process optimization data and characterizations). Conversely, investigations performed on PE-ALD processes result in strong improvements of the initial recipes that are deeply described in Table S1. Studies are therefore focused on the second half-cycle (plasma-assisted N-source reaction) while the first half-cycle of both plasma-enhanced processes (TDMAT reaction) is also depicted in Fig. S2 in supplementary materials. For all processes, saturation curves for TDMAT reveal an optimum pulse time of 0.2 s coupled with an 6 s-long purge that have been applied in the following. ALD windows of all processes before and after optimization is presented and discussed in Fig. S3 in supplementary materials. The ALD window for the three processes seems to be around 150 ≤ T ALD ≤ 200 • C.

Effect of N-source dilution

The optimization of the two plasma-based recipes starts by studying the influence of N-source dilution within Ar. In initial recipe, Ar and NH 3 flows at plasma generator inlet are set to φ Ar = 200 and φ NH 3 or N 2 = 70 sccm, respectively. This situation is named diluted conditions in the following. It corresponds to a dilution ratio ρ φ < 1, with ρ φ defined by Eq.1:

ρ φ = φ NH 3 φ Ar or φ N 2 φ Ar (1)
Fig. 3a presents the influence of NH 3 dilution in Ar on GPC during the corresponding plasma processes. In diluted conditions, increasing φ NH 3 does not modify the GPC that remains almost constant around 0.6 Å/cycle (dark blue squares in Fig. 3a). The situation named equal flow conditions consists of setting co-reactant and vector gas flows to equal values: φ NH 3 = φ Ar then ρ φ = 1. In such situation, GPC significantly increases with φ NH 3 (dark blue circles on Fig. 3a). This behavior can be ascribed to the loss of pure ALD behavior due to a CVD contribution. When the pl-NH 3 purge is too short, TDMAT injected during the next ALD cycle can, indeed, react with the supplemental physisorbed species such as NH 3 or NH * 3 . To confirm this observation, the purge duration is extended to 20 s (light blue circles on Fig. 3a). This elongated evacuation step is then sufficient to remove all physisorbed species and GPC saturates around 0.85 Å/cycle when φ NH 3 ≥ 20 sccm. This indicates that, in equal flow conditions, to avoid CVD contribution, a pulse of 20 s must be associated to a 20 s purge instead of 5 s proposed in the initial recipe (Table S1). For lower flows (φ NH 3 < 20 sccm), GPC is lower because not sufficient molecules are introduced to react with all available surface active sites.

Fig. 3c shows GPC evolution with φ NH 3 in diluted conditions for th-NH 3 . It initially increases and reaches a plateau at 1 Å/cycle for φ NH 3 = 50 sccm corresponding to surface active sites saturation. This evolution is different than equivalent pl-NH 3 exhibiting a constant GPC (dark blue squares on Fig. 3a). Although a similar NH 3 and Ar mixture is injected in the re-actor at same temperature, ALD and PE-ALD do not exhibit similar GPC trends. To understand this difference, one has to consider the plasma properties. Species formed in this phase (i. e. ions, radicals, electrons) must travel from the generator to the surface. The mean free path ( ) is the distance covered by those species until they remain active. According to Eq. 2, decreases with P and d (kinetic diameter of the species).

= kT √ 2πd 2 P (2)
When P increases, more collisions occur leading to recombinations into inert species, such as N 2 generated by NH * 3 collisions, is then reduced. This leads to a decreased number of excited species. High P can therefore be detrimental to the deposition. This is the case in diluted conditions in which P is almost twice as high as compared to equal flow conditions: P = 0.31 and 0.18 Torr, respectively, for φ NH 3 = 20 sccm (see pressure profiles, Fig. S4, in supplementary materials). The mean free path should therefore decrease significantly. In addition, Ar is the main gas in the reactor (φ NH 3 < φ Ar ). This leads to a slower GPC and indicates that large NH 3 dilution within Ar is detrimental to deposition. Conversely, in equal flow conditions, for a similar φ NH 3 , P is lower, is extended and more excited NH * 3 species reach the surface and react. GPC is thus higher.

An analogous approach is applied to pl-N 2 . Fig. 3b shows GPC evolution with φ N 2 in diluted and equal flow conditions. Similarly to pl-NH 3 , diluted conditions lead to constant GPC around 0.6 Å/cycles, despite the φ N 2 increase (dark green squares on Fig. 3b). In contrast to pl-NH 3 , under equal flow conditions (φ N 2 = φ Ar ), GPC is significantly higher (up to 1.2 Å/cycle) at low flows (φ N 2 ≤ 20 sccm) and it stabilizes around 0.6 Å/cycle for φ N 2 ≥ 50 sccm (dark green cir- cles Fig. 3b). Such results indicate that large φ N 2 seems also detrimental to generate excited N-species while at lower φ N 2 , reactive species undergo fewer collisions leading to a longer . High GPC values at low φ N 2 suggest a CVD contribution. When a longer purge (20 s) is applied, for φ N 2 = φ Ar = 20 sccm, GPC indeed decreases to 0.7 Å/cycle because physisorbed reactants excess and/or byproducts are more efficiently pumped away (light green circles on Fig. 3b). These results demonstrate that gas mixture composition is crucial to control both PE-ALD processes. In equal flow conditions, higher GPCs are observed because more excited Nspecies exhibiting longer are generated and travel toward the surface to react. Note that residence times of the species in the gas phase or at the surface have also been considered but it appears that ρ φ and P are more likely to be the prevailing parameters (a detailed explanation is given in page 4 of supplemental materials).

Depending on N-source and vector gas flows, color changes are observed at the plasma generator window. A magenta light is emitted under diluted conditions that turns to blue-purple and light-pink when equal flows are injected for pl-NH 3 /Ar and pl-N 2 /Ar, respectively. Emissions at various wavelengths correspond to specific transitions. Magenta is attributed to Ar while blue and pink are ascribed to pl-NH 3 and pl-N 2 , respectively [START_REF] Pearse | The identification of molecular spectra[END_REF] . In situ OES has thus been carried out during the plasma ignition to identify the created species.

Fig. 4 compares emission spectra measured in diluted and equal flow conditions for pl-NH 3 and pl-N 2 . In the case of pl-NH 3 (Fig. 4a), significant variations are observed depending on ρ φ . As expected, signals emitted by excited NH * 3 species are observed in any cases. Intense atomic hydrogen lines H α and H β are measured at λ = 656 and 486 nm, respectively, while peaks corresponding to N* species are almost merged in the background. As soon as Ar is injected in the reactor, emission peaks are measured at high wavelengths (λ = 680 to 880 nm). To quantify the emission of each gas, assuming that excitation coefficients (k NH 3 and k Ar ) do not depend on the discharge parameters (such as power, pressure, gas composition), we calculate, R H/Ar , that corresponds to the ratio between the peak intensities of H α and Ar(2p 1 ) at λ = 750 nm (assuming k NH 3 /k Ar constant):

R H/Ar = I H I Ar (3)
In equal flow conditions, R H/Ar = 5.8 while, in diluted conditions, it is significantly reduced to 2.2. Excited NH * 3 species signals seem to be enhanced in equal flow conditions with respect to pure pl-NH 3 . The ratio between H α and a residual peak at λ = 757.6 nm, taken as reference, is indeed 10% higher in equal flow compared to diluted conditions (see Table S2 in supplementary materials). Ar flow, i. e. dilution, is therefore a major parameter since it expands or dampens excited NH * 3 species emissions. At low φ Ar , excited Ar * as well as the numerous plasma generated electrons may indeed collide and transfer their energy to NH * 3 leading to enhanced emission. In diluted conditions, emissions arising from Ar excitations are conversely dominating. Since total pressure is high, many collisions and recombinations occur, then H α and H β peaks are quenched. A better reactivity and, consequently an optimized GPC are therefore expected in equal flow conditions.

Emission spectra variations in the case of pl-N 2 dilution is presented in Fig. 4b. pl-N 2 signals (molecular N 2 transitions, first and second positive system at λ = 350 to 450 nm and λ = 530 to 780 nm, respectively) are easily identified in equal flow conditions, φ N 2 = φ Ar = 20 sccm (i. e. ρ φ = 1), whereas they almost disappear in diluted conditions. This evolution is very similar to pl-NH 3 . Again, φ Ar is beneficial at low values but becomes detrimental at high flows.

As mentioned above, when using remote plasma generator 400 500 600 700 800 λ (nm) Measurement are taken at the plasma generator window with P plasma = 300 W using an acquisition time of 44 ms. The peaks have been identified according to literature [START_REF] Pearse | The identification of molecular spectra[END_REF][START_REF] Mackus | Optical emission spectroscopy as a tool for studying, optimizing, and monitoring plasma-assisted atomic layer deposition processes[END_REF][START_REF] Lock | Experimental and Theoretical Estimation of Excited Species Generation in Pulsed Electron Beam-Generated Plasmas Produced in Pure Argon, Nitrogen, Oxygen, and Their Mixtures[END_REF] .

Intensity (a. u.) NH 3 ( 20
geometry, reactions occur while excited species travel to the substrate. Since the OES measurements have been performed in the upper part of the chamber, they reveals reactive species that are produced at the generator and not necessarily those reaching the surface. This is especially suitable to optimize pl-N 2 process as it is an inert precursor without plasma activation. Assessing N-source flows and dilution give, however, valuable information about optimal conditions to promote reactive species transport from the generator to the surface. Ar dilution effect on GPC was also investigated for pl-NH 3 and pl-N 2 . Fig. 5 shows the GPC evolution against φ Ar while φ NH 3 and φ N 2 remain constant at 20 sccm. This corresponds to an increasing dilution in Ar. As expected from OES characterizations, GPC decreases with φ Ar in both cases. It confirms that large dilution is detrimental to the film growth for both N-sources. However, at low dilution (i. e. φ Ar ≤ 20 sccm), the presence of Ar does not influence significantly the GPC. A flow mixture of 20:20 sccm of both Ar and N-sources is then set as optimum in the following work.

Effect of N-sources pulse duration

The pulse duration determines, as well as dilution, the amount of injected precursors into the reactor and indicates when saturation occurs. Its influence on GPC is studied for both pl-NH 3 and pl-N 2 . Since maximal GPCs are measured in equal flow conditions at low flows, Fig. 6 presents GPC evolution against pulse duration with φ NH 3 or N 2 = φ Ar = 20 sccm. Fig. 6a shows GPC against pl-NH 3 pulse time. Corresponding GPC under diluted conditions is the horizontal dashed line at 0.6 Å/cycle plotted for comparison (dark blue dashed line). Measurements have been performed using 5 and 20 s-long purges (dark and light blue circles on Fig. 6a, respectively). In both cases, GPCs exhibit similar evolutions. They, first, increase steeply and, then, reach a plateau for a pl-NH 3 pulse of 5 s denoting that saturation is achieved. The highest GPC is around 1 Å/cycle with a 5 s purge (dark blue circles). However, CVD contribution should be considered for this short purge, since GPC drops significantly near 0.8 Å/cycle when a 20 s long purge (light blue circles) is applied. ALD conditions are, therefore, fulfilled when a 5 s pulse is associated to a 20 s purge but the overall cycle is long (31.2 s). 5 s purge is therefore selected to shorten the process (16.2 s). Although a pl-NH 3 pulse of 2 s does not lead to the highest GPC (saturation is not completely achieved), it is chosen to further shrink the total cycle duration (13.2 s). It also slightly compensates the possible CVD contribution while relatively high GPC of 0.86 Å/cycle is maintained. This balance is then chosen for the following work. The ALD sequence for pl-NH 3 is 0.2 : 6 : 2 : 5 s. It will be assessed by testing conformality on high aspect ratio substrates thereinafter (Fig. 9).

Similarly, Fig. 6b presents the influence of pl-N 2 pulse duration on GPC. It is investigated under equal flow (light green circles) and diluted conditions are plotted as reference (dark green squares). As expected, GPC increases and stabilizes after pulsing 10 s in the two conditions. As already reported, dilution in Ar hinders film growth (dark green). The plateau is, indeed, around 0.5 Å/cycle when pl-N 2 is diluted while it is beyond 0.7 Å/cycle under equal flow conditions (light green). These results indicate that TiN grows in ALD regime for the 0.2 : 6 : 10 : 20 s sequence. As already proposed for pl-NH 3 , pulse and purge are reduced to 5 s (dark green circle) to shorten the overall duration (from 36.2 down to 16.2 s) while keeping, as close as possible, deposition within ALD regime. In this case, saturation is almost reached, and CVD contribution is minor while GPC is maintained at a relatively high value 0.72 Å/cycle. The conformality of the films grown using the 0.2 : 6 : 5 : 5 s recipe is also evaluated thereinafter (Fig. 9).

Effect of plasma power

The plasma power is also a key parameter for PE-ALD. Its influence on deposition has thus been assessed on the whole range of the generator. Spectra shown in Fig. 7 indicate that, for both pl-NH 3 and pl-N 2 , optical emission increases significantly with P plasma but no strong spectra modifications are observed if power is increased from 100 to 300 W. This is further confirmed by comparing normalized spectra presented in Fig. S5 in supplementary materials. OES indicates that more excited species emitting light are generated, near the generator, when P plasma is higher. However a direct correlation between emission intensity and number of species reacting at the surface cannot be established. As mentioned above, one has to consider other parameters such as P or . This is sug-gested when analyzing GPC evolutions with P plasma for the two N-sources (Fig. 7c). In the case of pl-NH 3 , a constant GPC near 0.8 Å/cycle is measured from 50 to 300 W while a progressive increase could be expected. It is explained by the intrinsic reactivity of NH 3 molecule that allows for ALD of TiN without plasma activation. A GPC of 0.44 Å/cycle is, indeed, observed under same conditions without plasma ignition (P plasma = 0 W on Fig. 7c). It demonstrates that a thermal ALD component contributes to pl-NH 3 process. As soon as plasma is lightened (P plasma = 50 W), GPC reaches a plateau because a small quantity of additional energy is sufficient to generate a large amount of excited species. It should be noted that without plasma-activation GPC is quite low as compared to optimized th-NH 3 shown in Fig. 3c (i. e. GPC = 0.76 Å/cycle at φ NH 3 = 20 sccm). This discrepancy comes from the different ALD time sequences of the two processes that are 2 and 6 s pulse for pl-NH 3 and th-NH 3 respectively. According to literature, PE-ALD can lead to higher GPC than thermal ALD [START_REF] Profijt | Plasma-Assisted Atomic Layer Deposition: Basics, Opportunities, and Challenges[END_REF][START_REF] Detavernier | Thermal Versus Plasma-Enhanced ALD: Growth Kinetics and Conformality[END_REF] . Conversely, here, thermal mode exhibits a GPC near 1 Å/cycle whereas it is around 0.8 Å/cycle for plasmaenhanced process. Again, it could be ascribed to excited species recombinations into inert or less reactive molecules (e. g. N 2 ) during their transport to the surface: is smaller than the travelling distance (d plasma ). NH 3 molecule high reactivity and short also induce the saturation plateau observed in PE-ALD when P plasma exceeds 50 W.

In the case of pl-N 2 , GPC progressively increases with P plasma corresponding to initial expectations (i. e. correlation between P plasma and the reactive species amount). Since N 2 molecule is almost inert, no TiN growth is observed at P plasma = 0 W (GPC = 0.07 Å/cyles). It demonstrates that deposition requires an assistance to proceed. Plasma ignition provides, indeed, a substantial amount of energy to excite N 2 molecules into N * 2 excited species. When P plasma increases more excited species are generated (Fig. 7b) and, consequently, they can react at the surface. In opposition to pl- NH 3 , no thermal-activated processes participate to the deposition with pl-N 2 . A lower GPC is therefore observed and no saturation appears within the explored power range. It should occur at higher P plasma but the present generator is limited to 300 W. As described in experimental section (Fig. 2b), depositions at shorter distances (d plasma = 16.2 and 20.2 cm) were compared to regular spacing (38.2 cm). Inset in Fig. 7c shows the GPC evolution with d plasma at P plasma = 100 W. GPC is the highest (0.61 Å/cycle) for the shortest distance, d plasma = 16.2 cm and it decreases and reaches a plateau around 0.4 Å/cycle for longer d plasma . It demonstrates that excited species are not reaching the surface because is significantly shorter than the regular generator-to-sample distance. According to Eq. 2, decreases with d 2 . Since N 2 exhibits a larger kinetic diameter than NH 3 (d N 2 = 364 pm and d NH 3 = 260 pm), is expected to be twice shorter. This can partially explain the GPC gap between pl-NH 3 and pl-N 2 at regular sample position. Shortening d plasma , reduces detrimental collision probability and increasing plasma power, at constant P, generates more excited species with an unchanged recombination rate. More excited species are able to reach the surface, GPC is therefore enhanced.

Since a P plasma of 300 W was required to optimize pl-N 2 , similar power was applied in the following pl-NH 3 to keep comparable conditions. For similar reasons, regular d plasma was used in the following.

B. Characterizations of the films

In the previous section, plasma-assisted processes have been optimized depending on the N-source nature and then compared to the reference th-NH 3 deposition. In the following morphology, conformality, chemical composition, crystalline structure and electrical conductivity of films grown according to both PE-ALD are exhaustively investigated.

Morphology and Conformality

Film morphology has been observed before and after annealing under nitriding atmosphere. TEM cross sections of asgrown films are shown in Figs. 8a-c Films roughness is measured by AFM. Evolutions of R RMS against T ALD are plotted on Figs. 8d-f. Measured values are relatively low (slightly higher than initial Si wafers roughness R RMS = 0.3 nm) for all three routes (0.3 < R RMS < 0.8 nm). It attests the film quality and the limited impact of T ALD on the morphology. Thermal deposition shows a low linear increase with T ALD (Fig. 8d) while both plasma processes exhibit almost no variation (Fig. 8e and Fig. 8f). In a previous study [START_REF] Assaud | Highly-Conformal TiN Thin Films Grown by Thermal and Plasma-Enhanced Atomic Layer Deposition[END_REF] , we reported similar results. R RMS increased with N ALD for th-NH 3 while it was lower and constant for pl-NH 3 . Plasmaactivation could encourage the generation of smaller domains in greater numbers leading to smoother films as compared to the thermal activation.

As mentioned in experimental section, conformality tests have been carried out onto porous silicon exhibiting high aspect ratio of 1:30 (Fig. 1). Figs. 9a-c present pl-NH 3 deposit at different locations of the substrates. Assaud et al. already reported conformal coating on anodic alumina membranes using initial pl-NH 3 recipe [START_REF] Assaud | Highly-Conformal TiN Thin Films Grown by Thermal and Plasma-Enhanced Atomic Layer Deposition[END_REF] . The optimized process should lead to compact and uniform coating onto such structured surfaces. Large-scale observation confirms the presence of TiN coating in the whole porous layer, verified by EDS analysis presented in Fig. S7 in supplementary materials. This is further evidenced by observing pore tips (Fig. 9a) and high magnification imaging performed at the pore wall and apex (Figs. 9b,c). Measured t ALD is around 40 nm for a targeted value of 48 nm on planar Si wafer. A 2-s long pl-NH 3 pulse is therefore sufficient to coat such deep features and the short purge (5 s) does not degrade the conformality.

Figs. 9d-f present the corresponding layer grown using pl-N 2 for a similar targeted TiN thickness (t ALD = 50 nm). Whereas whole pores are uniformly coated by pl-NH 3 , Figs. 9e,f reveal a partial covering of the pore tips. There is clearly deposition at the apex but the film is not as homogeneous as using pl-NH 3 . Nevertheless, on the walls (Fig. 9d) and the upper part of the pores, the deposit is compact. It exhibits a thickness near the expectation on planar Si wafer (around 42 nm). The film integrity is lost only in the tip vicinity. Since channels are very long (> 200 µm), this effect can arise from the short pulse duration or it can also be ascribed to the short when pl-N 2 is used as predicted by Eq.2 (i. e. recombinations do not allow these species to reach the deepest part of the pores). Note also that potential CVD contribution induced by the short purge step (5 s) does not lead to localized TiN accumulation.

Chemical composition

The composition of the thin films is assessed by XPS. Figs. 10a-c compare Ti 2p spectra of as-grown films deposited by thermal ALD (AG-th-NH 3 ) and as-grown PE-ALD layers using pl-NH 3 and pl-N 2 (AG-pl-NH 3 and AG-pl-N 2 , respectively) after a 5 min-long Ar-sputtering. In all cases, wide peak indicates several contributions. Ti 2p doublet peak decomposition reveals three contributions: (I) Ti-N from TiN around 455 eV, (II) Ti-O-N, ascribed to an oxygen placed in the TiN lattice (i. e. an oxynitride) around 456 eV and (III) Ti-O attributed to TiO 2 around 458 eV [START_REF] Gall | In situ X-ray Photoelectron, Ultraviolet Photoelectron, and Auger Electron Spectroscopy Spectra from First-Row Transition-Metal Nitrides: ScN, TiN, VN, and CrN[END_REF] .

General Ti 2p peak shape measured for pl-N 2 indicates a larger TiN contribution compared to both NH 3 -based processes (Fig. 10c). This is confirmed by semi-quantitative analysis presented in Table S3 in supplementary materials. No accurate stoichiometry has been calculated due to background uncertainty. The background shape hinders, indeed the modeling of the region between 2p 3/2 and 2p 1/2 components which leads to possible confusion when integrating the area of every contributions. Beside this restriction, all peaks were analyzed with the same conditions.

Since deposited films are oxynitrides rather to pure nitrides an annealing under nitriding and reductive atmosphere (NH 3 and H 2 ) is performed to convert the remaining oxide and oxinitride into nitride. Conditions are inspired from nitride synthesis method that consists in nitriding the correspond- reactor to the furnace and to the XPS facility [START_REF] Park | Stability of Plasma Posttreated TiN Films Prepared by Alternating Cyclic Pulses of Tetrakis-Dimethylamido-Titanium and Ammonia[END_REF] . This is confirmed by XPS measurements performed on protected layers. A set of TiN films were encapsulated just after deposition by growing an Al 2 O 3 /TiO 2 bilayer at the top surface. XPS analysis has been carried out at different film depths using Ar sputtering. Fig. 11 presents pl-NH 3 case. As expected, the layer composition evolves from mostly an oxide, at the top surface, to an oxynitride in its bulk. Table S4 in supplementary materials summarize the %area of each element (Ti 2p, N 1s, O 1s and C 1s) for all three routes. Film encapsulation appeared to be an interesting technique to minimize the impact of sample transportation from ALD reactor to spectrometer. Nevertheless, to grow the protective Al 2 O 3 layer, water pulses are used and could also cause film oxidation before the alumina film covers fully the TiN deposit. It is therefore difficult to conclude but the oxygen content decreases with Ar-sputtering suggests that film oxidation originates from an external source (capping layer deposition or environment). Thus, oxygen is inserted after deposition when the sample is exposed to air. In situ measurements are the most reliable technique in such situation. These analyses carried out on layers grown by ALD or other techniques have also observed an oxynitride and support the present findings (see e. g. Refs. [START_REF] Ernsberger | Angular resolved x-ray photoelectron spectroscopy study of reactively sputtered titanium nitride[END_REF][START_REF] Saha | Titanium nitride oxidation chemistry: An x-ray photoelectron spectroscopy study[END_REF][START_REF] Colligon | Composition and chemical state of titanium nitride films obtained by different methods[END_REF][START_REF] Krawczyk | Studies of the hot-pressed TiN material by electron spectroscopies[END_REF][START_REF] Ponon | Effect of deposition conditions and post deposition anneal on reactively sputtered titanium nitride thin films[END_REF][START_REF] Hansen | Plasma Enhanced Atomic Layer Deposition of Plasmonic TiN Ultrathin Films Using TDMATi and NH 3[END_REF] ).

Crystalline structure

Fig. 12 presents diffractograms of the films before and after nitriding at T an = 600, 700 and 800 • C. Films grown by PE-ALD are mainly amorphous with a slight crystalline proportion and pl-NH 3 route seems to be slightly more crystalline than pl-N 2 . Thermal process leads to more polycrystalline layers when deposition is carried out at T ALD ≥ 200 • C. In all cases, after annealing, crystallinity increases, especially for T an ≥ 700 • C. Note that main diffraction peaks are not perfectly centered on the TiN reference (PDF 00-038-1420, black lines on Fig. 12) but mostly on the nearby TiO (PDF 00-008-0117, gray lines on Fig. 12). After annealing, peaks move closer to TiN references and are finally centered in between the two references. As discussed in section III.B.2, TiN surfaces are not stable in contact with air. They get therefore oxidized (Fig. 11) and diffractions peaks shift to a TiO compound [START_REF] Alhussain | Influence of ammonia gas exposure on microstructure of nanocrystalline titanium nitride powder synthesized from titanium dioxide[END_REF][START_REF] Liu | Mechanism on reduction and nitridation of micrometer-sized titania with ammonia gas[END_REF] . In opposition to tetragonal anatase (PDF 00-021-1272), this compound exhibits a cubic structure like TiN, as well as a close lattice parameter. This is in agreement with XPS data (Tables S3, S4 in supplementary materials) since films are mainly oxynitrides even after annealing.

Electrical properties

Film quality is assessed by determining their resistivity throught FPP measurements on evaporated gold contacts. In literature, bulk TiN resistivity is around 20 ± 10 µ Ω•cm while a resistivity ≤ 100 µ Ω•cm has been achieved using PE-ALD by Krylov et al. [START_REF] Krylov | Obtaining low resistivity (≈100 µ ω•cm) TiN films by plasma enhanced atomic layer deposition using a metalorganic precursor[END_REF][START_REF] Krylov | Effect of the substrate on structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF][START_REF] Pierson | Handbook of Refractory Carbides and Nitrides: Properties, Characteristics, Processing and Applications[END_REF] . Investigation on the plasma power impact on the film resistivity presents no significant evolution for both pl-NH 3 and pl-N 2 processes in Fig. S11. Resistivities of as-grown and annealed films are displayed in Fig. 13. AG-pl-N 2 film is clearly the most resistive (≈ 5000 µ Ω•cm). This can be due to its very low crystallinity. AG-th-NH 3 and AGpl-NH 3 films present similar resistivities (≈ 1300 µ Ω•cm) that are significantly lower. Compared to literature, these values are far from pure TiN. However, they remains relatively low for oxynitride films [START_REF] Ahn | Characteristics of TiN thin films grown by ALD using TiCl 4 and NH 3[END_REF][START_REF] Caubet | Low-Temperature Low-Resistivity PEALD TiN Using TD-MAT under Hydrogen Reducing Ambient[END_REF][START_REF] Elam | Surface chemistry and film growth during TiN atomic layer deposition using TDMAT and NH 3[END_REF] . Thermal treatments in nitriding atmospheres significantly modify the resistivity. A strong decrease is, indeed, observed when T an ≥ 700 • C. All layers exhibit then a resistivity in the range of 200 and 400 µ Ω•cm for pl-NH 3 and pl-N 2 processes, respectively. This can be related to the crystalline evolution at such temperature during annealing (crystallinity improvement reported in Fig. 12). Since TEM cross sections are not significantly different after annealing (Fig. S6), one can assume that crystalline domains remain in the nanometer scale embedded in amorphous TiN x O y matrix. Though conduction channels are created during this structural film modification. Those charge pathways are most likely to be grains boundaries as proposed by Lin et al. [START_REF] Lin | Titanium nitride thin film for temperature sensing and its conductive mechanism in the cryogenic region[END_REF] on polycrystalline TiN x films grown by magnetron sputtering. Since the composition does not evolve drastically after annealing, it seems it has no strong effect on the variation of the conductivity of the films.

IV. CONCLUSION

After optimizing the recipe (pulse/purge durations, gas flows, dilution and plasma power) and shortening the process duration, valuable information about deposition processes have been reported. First, as-grown films are not pure TiN but a mixture mainly composed of TiN and TiN x O y . XPS analysis coupled to reductive annealing or capping experiments tends to demonstrate that oxygen present in the layers is most probably incorporated after the deposition. The reductive annealing acts only as a thermal treatment that crystallizes the film.

The present study demonstrates also that gas mixture composition is crucial during both PE-ALD processes. The combined use of SE and OES is, indeed, highly efficient to study the effect of reactive gas dilution in the vector gas as well as the influence of P plasma . Dilution is seldom taken into account in recipes and the present methodology could, therefore, successfully be applied to many other PE-ALD processes. Here, it is demonstrated that dilution and chamber pressure are crucial parameters since they can be strongly detrimental to the deposition. Their effect on is critical since they can induce deleterious collisions of the reactive species causing a low GPC. Equal flow conditions and a relatively low P are found to be the most efficient conditions for both pl-N 2 and pl-NH 3 processes.

As reported earlier in literature, [START_REF] Krylov | Role of reactive gas on the structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF][START_REF] Krylov | Obtaining low resistivity (≈100 µ ω•cm) TiN films by plasma enhanced atomic layer deposition using a metalorganic precursor[END_REF][START_REF] Krylov | Effect of the substrate on structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF] NH 3 exhibits a higher reactivity than N 2 . For NH 3 , we show that plasmaactivation is effective at low applied power and deposition saturates quickly for higher values. A small plasma-activation is sufficient because a non-negligible thermal activation contributes to pl-NH 3 . Conversely, a large P plasma is required in pl-N 2 since this N-source is inert without plasma. A shorter is also evidenced for N 2 . The difference in reactivity between NH 3 and N 2 must therefore be put into perspective. Indeed, when using a shorter generator-to-sample distance, pl-N 2 is very similar to pl-NH 3 . This demonstrates the significance of the reactor design.

Although PE-ALD is often not considered to be suitable for step edge decoration, conformal depositions using pl-NH 3 and pl-N 2 are achieved on porous Si exhibiting a 1:30 aspect ratio. A better uniformity is, nevertheless, reached with pl-NH 3 . It could be explained by the lower reactivity of pl-N 2 and/or its shorter . However, this last consideration about has been obtained on planar surfaces. It should therefore be confirmed in specific pore geometry as it is difficult to establish a direct extrapolation of recombination phenomena occurring within channels since, there, gas flows and collisions can be strongly affected by the molecule confinement or depletion.

Since this work deals with TiN x O y instead of pure TiN films, the measured conductivities are, obviously, not as high as pure nitrides (≤ 500 µΩ•cm). Annealing treatments demonstrate clearly the critical effect of the crystallinity on the conductivity. In previous reports, [START_REF] Krylov | Role of reactive gas on the structure and properties of titanium nitride films grown by plasma enhanced atomic layer deposition[END_REF][START_REF] Krylov | Obtaining low resistivity (≈100 µ ω•cm) TiN films by plasma enhanced atomic layer deposition using a metalorganic precursor[END_REF] the best conductivities were ascribed to the improvement of the crystalline structure of columnar layers. Here, the relation between crystallinity and conductivity is further obvious because as-grown samples are amorphous and they show a very high resistivity while, after annealing, they become crystalline and they turn to be significantly more conductive (≤ 400 µΩ•cm).

In the end, conformal conductive thin films are efficiently grown in a shortened experimental time. Further improvement could be achieved by using a reactor design exhibiting a smaller plasma generator-to-sample distance (especially for pl-N 2 ) and a better analysis of the plasma composition near the surface should be completed if OES measurements are performed closer to the sample.

FIG. 1 .

 1 FIG. 1. SEM observations of bare ordered macropores: (a) cross section and (b) top view. Pores opening and length are respectively around 7 and 215 µm, that corresponds to an aspect ratio of 1:30.

LiftFIG. 2 .

 2 FIG. 2. Schematic drawings of the shower-head reactor (Fiji 200) and its main constituting components. (a) The sample is located at its regular position d plasma = 38.2 cm and OES monitoring is shown near the plasma generator. (b) The sample position is lifted to shorten the plasma generator-to-sample distance: d plasma = 16.2 or 20.2 cm.

FIG. 3 .

 3 FIG.3. Evolution of GPC depending on gas flow conditions for (a) pl-NH 3 and (b) pl-N 2 . GPCs are plotted against (a) φ NH 3 and (b) φ N 2 under diluted (squares) and equal flow (circles) conditions. Purge time is varied from 5 to 20 s in equal flow conditions for pl-NH 3 and pl-N 2 . (c) Evolution of GPC against φ NH 3 under diluted conditions for the reference th-NH 3 process. In all cases, ALD sequence is 0.2:6:x N :y N (x N :y N are indicated on the plots). Under diluted conditions φ Ar = 200 sccm.

FIG. 4 .

 4 FIG.4. OES spectra of (a) pl-NH 3 and (b) pl-N 2 pure or mixed within different Ar flows. Measurement are taken at the plasma generator window with P plasma = 300 W using an acquisition time of 44 ms. The peaks have been identified according to literature[START_REF] Pearse | The identification of molecular spectra[END_REF][START_REF] Mackus | Optical emission spectroscopy as a tool for studying, optimizing, and monitoring plasma-assisted atomic layer deposition processes[END_REF][START_REF] Lock | Experimental and Theoretical Estimation of Excited Species Generation in Pulsed Electron Beam-Generated Plasmas Produced in Pure Argon, Nitrogen, Oxygen, and Their Mixtures[END_REF] .

FIG. 5 .

 5 FIG. 5. Influence of dilution in Ar on both pl-NH 3 and pl-N 2 . φ NH 3 and φ N 2 are constant at 20 sccm.

3 FIG. 6 .

 36 FIG.6. (a) pl-NH 3 and (b) pl-N 2 pulses optimizations associate to purge duration. Circles and continuous lines correspond to equal flow conditions while squares and dashed lines stand for diluted conditions. The recipe used is 0.2:6:x N :y N s. N-source pulse times, x N , correspond to abscissa while their purge times, y N , are indicated on the plot.

FIG. 7 .

 7 FIG. 7. OES spectra at different P plasma for both (a) pl-NH 3 and (b) pl-N 2 mixed within Ar in equal flow conditions (φ NH 3 or φ N 2 = φ Ar = 20 sccm and an acquisition time of 5 ms). Note that intensity of the pl-NH 3 spectrum at P plasma = 100 W has been magnify (×3) for better reading. (c) Influence of P plasma on GPC for pl-NH 3 and pl-N 2 processes under equal flow conditions (φ NH 3 or φ N 2 = φ Ar = 20 sccm). Inset shows the dependence of GPC against d plasma for pl-N 2 at P plasma = 100 W (corresponding to the vertical dashed line on the main plot).

FIG. 8 .

 8 FIG. 8. TEM cross sections of TiN layers grown at T ALD = 200 • C for (a) th-NH 3 , (b) pl-NH 3 and (c) pl-N 2 . R RMS of TiN layers measured by AFM after N ALD = 100 cycles depending on T ALD for (d) th-NH 3 , (e) pl-NH 3 and (f) pl-N 2 .

FIG. 9 .

 9 FIG. 9. SEM cross sections of porous Si coated by TiN films: (a,b,c) pl-NH 3 and (d,e,f) pl-N 2 . (a,f) show general views of pore tips while (b,e) and (c,d) show high magnification images of a single wall at half depth of the pore and at the pore tips, respectively. Scale bars in (b,c,d,e) correspond to 100 nm. Expected 50 nm-thick TiN layers grown at T ALD = 200 • C.

FIG. 10 .FIG. 11 .

 1011 FIG. 10. Ti 2p XPS peaks from (a) th-NH 3 , (b) pl-NH 3 and (c) pl-N 2 processes before and after annealing at T an = 700 • C (d,e,f), respectively.

FIG. 12 .

 12 FIG.12. XRD patterns of as-grown layers and their evolution with annealing temperature for the three deposition routes. Annealing in N 2 , NH 3 and H 2 mixture for 5 min at T a = 600, 700 and 800 • C. Black and gray vertical lines correspond to TiN and TiO references, respectively.

3 th-NH 3 pl-N 2 FIG. 13 .

 33213 FIG.13. Resistivity of TiN films grown by thermal-and PE-ALD (300 W, φ N-surce = φ Ar =20 sccm, P = 0.16 Torr) and effect of different annealings. TiN films were ≈ 50 nm thick.

ACKNOWLEDGMENTS

D. Chaudanson, A. Altie, V. Heresanu, B. Demirdjian, S. Lavandier, F. Bedu (CNRS) are acknowledged for their precious help within electronic microscopy, x-ray diffraction, OES, contacts deposition and electric measurements. The authors acknowledge the financial support of the French Agence Nationale de la Recherche (ANR) under grant ANR-17-CE09-0049-03 (project MENINA). The French research network RAFALD (GDR CNRS) is also acknowledged for lending the first OES spectrometer.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.


Davide Mattera
email: mattera@unina.it

Mario Tanda
email: tanda@unina.it

Maurice Bellanger
email: bellang@cnam.fr

On single-tap equalization for FBMC multicarrier systems in wireless channels

Keywords: FBMC, single-tap equalization, MMSE, PAM, dispersive channel

This paper deals with single-tap equalization for the recently introduced FBMC-PAM system, a multicarrier scheme able to meet the requirements of cognitive radio such as high level of adjacent channel leakage ratio and asynchronous communications. The optimum single-tap gain in the minimum mean square error (MMSE) sense is derived and its performance is compared with that of previously considered receivers. It is shown that when the optimum MMSE single-tap gain is multiplied (before the real part extraction) by a zero-mean complex noncircular random variable (that is, when the zero-mean output of the standard matched filter has a complementary variance different from zero) the achieved performance can be better than that of other single-tap receivers and similar to that of a more advanced previously considered equalization structure.

Introduction

Filter bank multicarrier (FBMC) techniques are potential alternatives to the current orthogonal frequency division multiplexing (OFDM) schemes for some emerging applications fields such as machine type communications or cognitive radio [START_REF] Medjahdi | On the road to 5G: comparative study of physical layer in MTC context[END_REF]. In these systems the prototype filter controls performance and a number of operational aspects. Long filters assure high of out-of-band attenuation and asynchronous user coexistence. However, for reduced system latency short prototype filters are required.

In [START_REF] Pinchon | Derivation of analytical expressions for flexible PR low complexity FBMC systems[END_REF][START_REF] Nadal | Design and evaluation of a novel short prototype filter for OFDM/OQAM modulation[END_REF] short prototype filters with overlapping factor K= 2 (i.e., only 2 adjacent multicarrier symbols overlap in time), have been proposed. Since FBMC offset-quadrature amplitude modulation (OQAM) combined with short filters assures limited performance, the complex lapped transform has been used for the design of the FBMC-PAM transceiver [START_REF] Mattera | Filter bank multicarrier with PAM modulation for future wireless systems[END_REF][START_REF] Sabeti | Performance Analysis of FBMC-PAM in Massive MIMO[END_REF][START_REF] Sexton | Enabling asynchronous machine-type D2D communication using multiple waveforms in 5G[END_REF]. For a special choice of the prototype filter it is much similar to the first analog multicarrier system introduced by Chang [START_REF] Chang | Synthesis of band-limited orthogonal signals for multichannel data transmission[END_REF]. The scheme is based on pulse amplitude modulation (PAM) combined with a sine prototype filter with overlapping factor K=2 and it achieves perfect reconstruction in the real field. Moreover, the sine prototype filter has a main lobe in the frequency domain whose width is 3 times the sub-carrier spacing while it is only 2 times for OFDM scheme. For this reason and for its better spectral decay, FBMC-PAM outperforms OFDM systems as well as FBMC-OQAM systems in terms of CFO sensitivity [START_REF] Mattera | CFO sensitivity and efficient estimation for the short filter multicarrier system FBMC-PAM[END_REF].

Several equalization structures have been considered for FBMC-PAM systems, among them the well known single-tap zero-forcing equalizer and the more advanced structures termed 2M-4M and 2M-6M [START_REF] Mattera | Filter bank multicarrier with PAM modulation for future wireless systems[END_REF]. In particular, the 2M-4M receiver evaluates the FFT of the received signal in a window of length 4M whose central part of length 2M is used by the single-tap receivers, divide this FFT by that of the channel in a window of the same length, performs the inverse FFT of the resulting data and, then, uses the central part of length 2M as the input of the receiver for the AWGN channel. This paper deals with single-tap equalization of FBMC-PAM signals in wireless dispersive channels. The optimum single-tap gain in the minimum mean square error (MMSE) sense is derived and it is shown that it assumes a more involved expression when the output of the standard matched filter (that multiplied by the considered gain and after real part extraction provides the decision variable) is noncircular [START_REF] Neeser | Proper complex random processes with applications to information theory[END_REF][START_REF] Picinbono | On circularity[END_REF][START_REF] Schreier | Bounds on the degree of impropriety of complex random vectors[END_REF][START_REF] Delmas | On the degree of second-order non-circularity of complex random variables[END_REF], that is when its complementary variance is different from zero. Moreover, it is shown through computer simulations, that in this case the optimum MMSE single-tap equalizer can outperform other well-known singletap structures and can assure a performance similar to that of the more advanced 2M-4M receiver.

The organization of the paper is as follows. In Section 2, the standard FBMC-PAM system is recalled. In Section 3 the MMSE single-tap gain expression is derived while in Section 4 and in Section 5, the single-path case and the asymptotic case (large number of subcarriers), respectively, are analyzed. Simulation results are reported in Section 6 and conclusions are drawn in the final Section.

It is shown that when the optimum MMSE single-tap gain is multiplied (before the real part extraction) by a zero-mean complex noncircular random variable (that is, when the zero-mean output of the standard matched filter has a complementary variance different from zero) the achieved performance can be better than that of other single-tap receivers and similar to that of a more advanced previously considered equalization structure.

System Model

Let us consider an FBMC-PAM system [START_REF] Mattera | Filter bank multicarrier with PAM modulation for future wireless systems[END_REF] with 2M subcarriers. The received signal in time-dispersive channel can be written as

r(t) = L-1 p=0 g p s(t -τ p ) + n(t) (1)
where s(t) is the transmitted FBMC-PAM signal, n(t) denotes the zero-mean circular complex white Gaussian noise with independent real and imaginary parts, each with two sided power spectral density N o , and τ p is the delay of the pth path with complex gain g p . The FBMC-PAM signal s(t) is equal to

s(t) = N s -1 i=0 2M-1 k=0 d k [i] e j π T (k+ 1 2 )(t-iT+ T 2 ) h(t -iT ) ( 2
)
where 2T is the FBMC-PAM symbol duration, N s is the number of payload symbols, d k [i] is the real information symbol transmitted on the kth subcarrier in the ith symbol interval, and h(t) is the real pulse-shaping filter. The data symbols are zero-mean statistically independent real random variables with mean-square value

P d , thus, E {d k [i]d m [l]} = δ[k -m] δ[i -l] P d . (3)
In (1) the complex gains g p are modeled as a zero-mean complex circular Gaussian random variables with variance σ 2 p and statistically independent of one another. Moreover, it is assumed that the following condition holds

L-1 p=0 σ 2 p = 1. ( 4
)
The received signal r(t) is filtered with an ideal low-pass filter with a bandwidth of 1/T s , where T s denotes the sampling period (the FBMC-PAM symbol duration is equal to 2T = 2MT s ), thus the discrete-time low-pass version of the received signal can be written as

r[l] = N s -1 i=0 2M-1 k=0 L-1 p=0 g p d k [i] T c [k, l -θ p -iM] + v[l] (5)
where

T c [k, l] = h[l] e j π M (k+ 1 2 )(l+ 1 2 + M 2 ) , (6)
θ p = τ p /T s is the normalized delay of the pth path and v[l] is a discrete-time zero-mean AWGN process with autocorrelation function

R v [m] = E {v[l]v * [l -m]} = 2N o T s δ[m] = σ 2 v δ[m]. (7)
It is assumed that the normalized delays θ p = τ p /T s in ( 5) are integer values. In [START_REF] Sexton | Enabling asynchronous machine-type D2D communication using multiple waveforms in 5G[END_REF] the real prototype filter h[l], equal to zero for l K 2M = {0, 1, . . . , 2M -1} and with energy

E h = 2M-1 l=0 h 2 [l], (8)
satisfies the following conditions

h[M + l] = h[M -l -1] ∀ l, (9)
+∞ m=-∞ h[l + mM]h[l + mM + 2rM] = δ[r] ∀ l, r (10)
that imply the orthogonality condition in the real field

       1 M 2M-1 l=0 T c [k, l]T * c [m, l -pM]        = δ[m -k] δ[p]. (11)
In this case, it follows that [START_REF] Mattera | Filter bank multicarrier with PAM modulation for future wireless systems[END_REF] in AWGN channel the optimum (in the maximum likelihood sense) decision variable for estimating statistically independent information symbols can be written as Dm [i] where

Dm [i] = 1 M 2M-1 l=0 r[iM + l]T * c [m, l] (12)
since conditions ( 9) and ( 10) assure the absence of intersymbol interference and intercarrier interference.

In the following it is considered the prototype filter

h[l] = sin π 2M l + 1 2 l ∈ K 2M , h[l] = 0 l K 2M . (13)
This prototype filter satisfies conditions ( 9) and ( 10) and simplifies the receiver as

shown in [START_REF] Mattera | Filter bank multicarrier with PAM modulation for future wireless systems[END_REF].

Optimum single-tap receiver

In this section we derive the optimum (in the minimum mean-square error (MMSE) sense) single-tap receiver for the signal model in [START_REF] Sabeti | Performance Analysis of FBMC-PAM in Massive MIMO[END_REF]. Specifically, we derive the expression of the complex gain a m such that the estimate of the symbol transmitted on the m-th subcarrier in the i-th interval

dm [i] = a m Dm [i] (14)
minimizes the mean-square error

E 2 = E dm [i] -d m [i] 2 . (15)
In ( 14) Dm [i] is the decision variable for AWGN (see [START_REF] Delmas | On the degree of second-order non-circularity of complex random variables[END_REF]) evaluated by the receiver and a m is the complex gain used by the single-tap equalizer to achieve satisfactory performance in a multipath channel. By substituting [START_REF] Lélé | Channel estimation methods for preamble-based OFDM/OQAM modulations[END_REF] in [START_REF] Mattera | On the performance of FBMC-PAM systems in frequency-selective Rayleigh fading channels[END_REF] and taking into account that

2 {A} = 1 2 A 2 + 1 2 | A | 2 (16)
it follows that

E 2 = 1 2 a 2 m E D2 m [i] + 1 2 | a m | 2 E Dm [i] 2 + P d -2 a m E d m [i] Dm [i] . (17)
A necessary and sufficient condition for a point a o m to be a stationary point for the real-valued function [START_REF] Matthaiou | The road to 6G: ten physical layer challenges for communications engineers[END_REF] analytic in both a m and a * m is that the derivative is zero in a o m when a m and a * m are considered as independent variables [START_REF] Brandwood | A complex gradient operator and its application in adaptive array theory[END_REF]. In particular, the real-valued function [START_REF] Matthaiou | The road to 6G: ten physical layer challenges for communications engineers[END_REF] can be minimized by solving the equation

a m E D2 m [i] + a * m E Dm [i] 2 = 2E d m [i] Dm [i] . (18)
In the derivation of ( 18) the following relationships

∂ ∂ a m a 2 m γ = ∂ ∂ a m 1 2 a 2 m γ + a 2 m * γ * = a m γ (19) ∂ ∂ a m {a m γ} = ∂ ∂ a m 1 2 a m γ + a * m γ * = 1 2 γ (20) ∂ ∂ a m γ | a m | 2 = ∂ ∂ a m γ a m a * m = γ a * m ( 21
)
have been accounted for, where γ is any complex constant.

The solution of (18) (see Appendix A) leads to the following expression for the complex gain to be used in the optimum MMSE single-tap receiver

a o m = 2E d m [i] Dm [i] E * D2 m [i] -2E * d m [i] Dm [i] E | Dm [i] | 2 E D2 m [i] 2 -E 2 Dm [i] 2 . ( 22
)
It is worthwhile to emphasize that the optimum gain a o m depends not only on the terms

E | Dm [i] | 2 and E d m [i] Dm [i] but also on the complementary variance E D2 m [i]
which is different from zero when the zero-mean complex random variable Dm [i] is noncircular [START_REF] Neeser | Proper complex random processes with applications to information theory[END_REF][START_REF] Picinbono | On circularity[END_REF][START_REF] Schreier | Bounds on the degree of impropriety of complex random vectors[END_REF][START_REF] Delmas | On the degree of second-order non-circularity of complex random variables[END_REF]. Note that when E D2 m [i] = 0 the solution of (18) can be immediately obtained and is coincident with (22) in the particular case E D2 m [i] = 0. By substituting (22) in [START_REF] Matthaiou | The road to 6G: ten physical layer challenges for communications engineers[END_REF] we obtain the expression of the minimum MSE value (see Appendix B)

E 2 min = P d +2 E * D2 m [i] E 2 d m [i] Dm [i] -E Dm [i] 2 E d m [i] Dm [i] 2 E 2 Dm [i] 2 -E D2 m [i] 2 . (23)
In particular, if we define

ρ = E D2 m [i] E Dm [i] 2 (24)
and

u = E 2 d m [i] Dm [i] E Dm [i] 2 (25)
we can write

E 2 min = P d - 2 1-| ρ | 2 | u | -ρ * u . ( 26
)
Let us note that the second term in the right-hand side of (26) is positive since | ρ |≤ 1 and, then,

ρ * u ≤ | ρ | | u | ≤ | u | .
Taking into account ( 12) and ( 5) it follows that

Dm [i] = 2M-1 k=0 q d k (q) w m,k (i -q) + N m (i) (27)
where

w m,k (i -q) = 1 M L-1 p=0 g p 2M-1 l=0 T c [k, l -θ p + (i -q)M] T * c [m, l] (28)
and

N m (i) = 1 M 2M-1 l=0 ν[l + iM] T * c [m, l]. (29)
In particular, the gain of the useful term in ( 27) is given by

g u = w m,m (0) = 1 M L-1 p=0 g p 2M-1 l=0 T c [m, l -θ p ] T * c [m, l] (30)
Since the noise samples ν[l] are zero-mean complex circular statistically independent Gaussian random variables with variance σ 2 ν , it follows that

E D2 m [i] = P d 2M-1 k=0 q w 2 m,k (i -q) (31) E d m [i] Dm [i] = P d w m,m (0) (32)
and

E Dm [i] 2 = P d         2M-1 k=0 q w m,k (i -q) 2 + 1 S NR         (33)
where

S NR = P d E h σ 2 ν . (34)
Thus, by using (31), ( 32) and ( 33) the optimum single-tap ( 22) can be obtained.

Moreover, the corresponding minimum MSE value can be obtained by substituting in ( 26)

ρ = 2M-1 k=0 q w 2 m,k (i -q) 2M-1 k=0 q w m,k (i -q) 2 + 1 S NR (35)
and

u = P d w 2 m,m (0) 2M-1 k=0 q w m,k (i -q) 2 + 1 S NR . ( 36
)

The single-path case

Let us consider the case of a single-path channel, that is L = 1 in ( 5) and let us assume, without a loss of generality, that

θ o = 0. It is shown in Appendix C that in this case 2M-1 k=0 q        1 M 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l]        2 = 0 (37) while it is shown in Appendix D that 2M-1 k=0 q 1 M 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l] 2 = 2 , (38)
therefore, accounting for (31), ( 28) and (37) we can write

E D2 m [i] = P d g 2 0 2M-1 k=0 q        1 M 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l]        2 = 0. ( 39
)
Moreover, from (32), ( 28) and ( 11) it results that

E d m [i] Dm [i] = P d g 0 1 M 2M-1 l=0 | T c [m, l] | 2 = P d g 0 ( 40
)
and from ( 33), ( 28) and (38) it follows that

E Dm [i] 2 = P d          | g 0 | 2 2M-1 k=0 q 1 M 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l] 2 + 1 S NR          = P d 2 | g 0 | 2 + 1 S NR . (41)
Since in this case E D2 m [i] = 0, the optimum single-tap value ( 22) reduces to

a flat m = 2 E * d m [i] Dm [i] E | Dm [i] | 2 = g * 0 | g 0 | 2 + 1 2 S NR (42)
and, moreover, since

ρ = 2M-1 k=0 q w 2 m,k (i -q) 2M-1 k=0 q w m,k (i -q) 2 + 1 S NR = 0 (43)
and

u = P d w 2 m,m (0) 2M-1 k=0 q w m,k (i -q) 2 + 1 S NR = P d g 2 0 2 | g 0 | 2 + 1 S NR (44)
the minimum MSE value results to be

E 2 flat min = P d              1 - | g 0 | 2 | g 0 | 2 + 1 2 S NR              . ( 45
)

The asymptotic case (M 1)

Let us now suppose that M is sufficiently large with respect to the maximum excess delay of the channel [START_REF] Lélé | Channel estimation methods for preamble-based OFDM/OQAM modulations[END_REF]. In this case, the prototype filter has contained time-variations, so that, for each value of θ p and l ∈ {0, 1, . . . , 2M -1} it follows that

T c [k, l-θ p ] = h[l-θ p ] e j π M (k+ 1 2 )(l-θp+ 1 2 + M 2 ) h[l] e j π M (k+ 1 2 )(l-θp+ 1 2 + M 2 ) = T c [k, l]e -j π M (k+ 1 2 )θp . (46)
By substituting ( 46) in (28), we obtain

w m,k (i -q) = 1 M L-1 p=0 g p 2M-1 l=0 T c [k, l -θ p + (i -q)M] T * c [m, l] 1 M L-1 p=0 g p e -j π M (k+ 1 2 )θp 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l] = H c 2k + 1 4M 1 M 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l] (47)
where

H c 2k+1 4M
is the frequency response of the multipath channel at frequency F = 2k+1 4M . Therefore, we can write

E D2 m [i] = P d 2M-1 k=0 q w 2 m,k (i -q) P d 2M-1 k=0 H 2 c 2k + 1 4M q        1 M 2M-1 l=0 T c [k, l + (i -q)M]T * c [m, l]        2 = P d 2M-1 k=0 H 2 c 2k + 1 4M        1 M 2M-1 l=0 T c [k, l + M]T * c [m, l]        2 +P d 2M-1 k=0 H 2 c 2k + 1 4M        1 M 2M-1 l=0 T c [k, l]T * c [m, l]        2 +P d 2M-1 k=0 H 2 c 2k + 1 4M        1 M 2M-1 l=0 T c [k, l -M]T * c [m, l]        2 . ( 48
)
In particular, the last equality holds since for the considered prototype filter (see [START_REF] Brandwood | A complex gradient operator and its application in adaptive array theory[END_REF]) the only three values of q to be considered are q ∈ {i -1, i, i + 1}. Moreover, in Appendix E, taking into account [START_REF] Sexton | Enabling asynchronous machine-type D2D communication using multiple waveforms in 5G[END_REF] and the considered prototype filter [START_REF] Brandwood | A complex gradient operator and its application in adaptive array theory[END_REF], it is shown that

1 M 2M-1 l=0 T c [k, l + M] T * c [m, l] = (-1) m j ξ(k -m) , (49) 1
M 2M-1 l=0 T c [k, l] T * c [m, l] = δ[k -m] + j 2 δ[k -m + 1] - j 2 δ[k -m -1] , (50) and 1 M 2M-1 l=0 T c [k, l -M] T * c [m, l] = (-1) k+1 j ξ(k -m) (51)
where

ξ(l) = 1 4M          sin π 2 (l + 1) sin π 2M (l + 1) + sin π 2 (l -1) sin π 2M (l -1)          . ( 52
)
Let us observe that ξ 2 (l) is periodic with period 2M and, moreover, as shown in Appendix F, it results

2M-1 l=0 ξ 2 (l) = 1 4 . (53)
By substituting (49), ( 50) and ( 51) in (48) it follows that

E D2 m [i] P d H 2 c 2m + 1 4M - 1 4 H 2 c 2m -1 4M - 1 4 H 2 c 2m + 3 4M -2 P d 2M-1 k=0 H 2 c 2k + 1 4M ξ 2 (k -m) . ( 54
)
Let us observe that, from (53) and (54) it follows that if the value of M is sufficiently large that the channel frequency response H c (F) is flat in a range of subcarriers such that ξ 2 (±l) becomes negligible we obtain

E D2 m [i] 0 . (55)
Moreover, taking into account (46) we can write

E d m [i] Dm [i] = P d w m,m (0) = P d 1 M L-1 p=0 g p 2M-1 l=0 T c [m, l -θ p ] T * c [m, l] P d H c 2m + 1 4M 1 M 2M-1 l=0 |T c [m, l]| 2 = P d H c 2m + 1 4M (56)
and

E Dm [i] 2 = P d         2M-1 k=0 q w m,k (i -q) 2 + 1 S NR         P d          2M-1 k=0 H c 2k + 1 4M 2 q 1 M 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l] 2 + 1 S NR          . ( 57
)
By substituting (49), ( 50) and ( 51) in (57) it follows that

E Dm [i] 2 P d        H c 2m + 1 4M 2 + 1 4 H c 2m -1 4M 2 + 1 4 H c 2m + 3 4M 2        +2 P d 2M-1 k=0 H c 2k + 1 4M 2 ξ 2 (k -m) + P d 1 S NR . (58)
Let us observe that, from (53) and (58) it follows that if the value of M is sufficiently large that the channel frequency response H c (F) is flat in a range of subcarriers such that ξ 2 (±l) becomes negligible we obtain

E Dm [i] 2 P d        2 H c 2m + 1 4M 2 + 1 S NR        . (59)
Thus, if M is sufficiently large that (55) and (59) hold, taking into account (56) the optimum single-tap (22) results to be

a asy m = H * c 2m+1 4M H c 2m+1 4M 2 + 1 2 S NR (60)
and, the corresponding minimum MSE results to be

E 2 asy min = P d                    1 - H c 2m+1 4M 2 H c 2m + 1 4M 2 + 1 2 S NR                    . ( 61
)
Thus, for high S NR values the optimum MMSE single-tap (for large values of M) a asy m in (60) is nearly equal to the gain adopted in the well-known zero-forcing single-tap receiver

a zf m = 1 H c 2m+1 4M (62)
based on the approximation of the gain in (30) for large values of M.

Simulation results

In this section the BER of the optimum MMSE single-tap receiver (labeled as SC MMSE) based on (22) in frequency-selective Rayleigh fading channel is assessed via computer simulations and compared with that of the single-tap equalizer (labeled as SC) based on (30), the exact expression of the gain of the useful term in the decision variable [START_REF] Mattera | On the performance of FBMC-PAM systems in frequency-selective Rayleigh fading channels[END_REF], that of the single-tap equalizer (labeled as SCA)

based on (62) and that of the asymptotic (M 1) expression of the optimum MMSE single-tap receiver (labeled as SCA MMSE) based on (22). In the figures the performance of the 2M-4M receiver equalizer is also reported, and, moreover, the BER in the ideal interference-free condition [START_REF] Mattera | On the performance of FBMC-PAM systems in frequency-selective Rayleigh fading channels[END_REF] is presented. The simulation results are obtained under the following conditions:

1. the transmitted data belong to a 2-PAM and a 8-PAM constellation and in the active subcarriers profile 1 (ASP1) the percentage of active subcarriers is 89%, moreover, in the active subcarriers profile 2 (ASP2), profile 1 is modified by inserting one virtual subcarrier between active subcarriers to simulate a scenario where a fragmented spectrum is considered;

2. the considered Extended Vehicular A (EVA) [16] multipath channel model has the following power/delay profile: relative power (in dB) equal to 4. each channel realization is assumed to be perfectly known and remains constant in the whole burst.

Figures 1 and2 show the BER of the considered receivers versus E b /N o for 2-PAM constellation and M = 64 in EVA channel, and for the active subcarriers profiles 1 and 2, respectively. The results show that, when ASP1 is considered the asymptotic versions of the single-tap structures, particularly that of the MMSE structure, perform worse than the other single-tap structures, and, moreover, the 2M-4M receiver provides the best performance. Note that taking into account (31), (32) and (33) the asymptotic version of the solution of (22) leads to an approximation of a many terms, this can lead to a performance degradation with respect to the other considered less complex single-tap structures. On the other side, when ASP2 is considered, the MMSE SC receiver outperforms the remaining single-tap receivers and, moreover, assures a performance slightly worse than that of the 2M-4M receiver. Thus, when ASP2 is considered the solution of ( 22) leads to a gain that can allow to exploit the noncircularity of the zero-mean output of the standard receiver based on the matched filter. Figures 3 and4 show the BER of the considered receivers for M = 128 in EVA channel and for the active subcarriers profiles 1 and 2, respectively. When ASP1 is considered all the receivers present a performance improvement and, moreover, the performance gap between the 2M-4M receiver and the other single-tap receivers is reduced. In the ASP2 case all the single-tap structures assure similar performance except for the MMSE single-tap receiver that, for high values of E b /N o slightly outperforms the other considered structures.

Conclusions

The FBMC-PAM transceiver with its capabilities in terms of spectral efficiency, asynchronous access and protection of adjacent users, has the potential to meet many requirements imposed by the future wireless systems [START_REF] Matthaiou | The road to 6G: ten physical layer challenges for communications engineers[END_REF].

In this paper the problem of single-tap equalization of FBMC-PAM signals in wireless dispersive channels has been analyzed. The MMSE single-tap receiver has been derived and its performance has been compared with that of other previously considered receivers. It has been shown that the MMSE single-tap gain assumes a more involved expression when the output of the standard matched filter (that multiplied by the considered gain, provides, after real part extraction, the decision variable) is noncircular. In this case the MMSE single-tap receiver can outperform other single-tap structures and can assure a performance similar to that of the more advanced 2M-4M equalizer. However, the observed performance improvement of the MMSE SC receiver with respect to the other considered singletap structures is more contained as the constellation size increases.

Appendix A

In this appendix we provide some detail about the solution of ( 18) which is reported here for readability

a m E D2 m [i] + a * m E Dm [i] 2 = 2E d m [i] Dm [i] . (A.1)
By using the definitions

E D2 m [i] = α R + jα I E Dm [i] 2 = β E d m [i] Dm [i] = ζ R + jζ I (A.1
) leads to the following system of two real equations

                 a R m α R + β -a I m α I = 2ζ R a R m α I + a I m α R -β = 2ζ I . (A.2)
where a R m = {a m } and a I m = {a m }. The solution of this system is

a R m = 2 E * D2 m [i] E d m [i] Dm [i] -2E Dm [i] 2 E d m [i] Dm [i] E D2 m [i] 2 -E 2 Dm [i] 2 (A.3) a I m = 2 E * D2 m [i] E d m [i] Dm [i] + 2E Dm [i] 2 E d m [i] Dm [i] E D2 m [i] 2 -E 2 Dm [i] 2 (A.4)
Therefore, we can write

a m = a R m + ja I m = 2 E * D2 m [i] E d m [i] Dm [i] -2E Dm [i] 2 E * d m [i] Dm [i] E D2 m [i] 2 -E 2 Dm [i] 2 .
(A.5)

Appendix B

In this appendix we provide some detail about the steps that lead to the expression of the minimum MSE value in (23) obtained by substituting ( 22) in [START_REF] Matthaiou | The road to 6G: ten physical layer challenges for communications engineers[END_REF] reported here for readability

E 2 = 1 2 a 2 m E D2 m [i] + 1 2 | a m | 2 E Dm [i] 2 + P d -2 a m E d m [i] Dm [i] . (B.1)
By using the definitions

E D2 m [i] = α (B.2) E Dm [i] 2 = β (B.3) E d m [i] Dm [i] = ζ (B.4)
we can write

E 2 = 1 2 a 2 m α + 1 2 | a m | 2 β + P d -2 {a m ζ} . (B.5)
Since the optimum complex gain results to be (see (A.5))

a m = 2α * ζ -2βζ * |α| 2 -β 2 = 2α * ζ -2βζ * δ (B.6)
where

δ = |α| 2 -β 2 (B.7)
the minimum value of the MSE results to be

E 2 min = 2 δ 2 (α * ζ -βζ * ) 2 α + 2 δ 2 |α * ζ -βζ * | 2 β + P d - 4 δ {(α * ζ -βζ * ) ζ} = 2 δ 2 α * ζ 2 | α | 2 -β 2 - 2 δ 2 β | ζ | 2 | α | 2 -β 2 + P d - 4 δ α * ζ 2 + 4 δ β | ζ | 2 .
Taking into account the definition of δ in (B.7) we can write

E 2 min = 2 δ 2 α * ζ 2 δ - 2 δ 2 β | ζ | 2 δ + P d - 4 δ α * ζ 2 + 4 δ β | ζ | 2 = P d - 2 δ α * ζ 2 + 2 δ β | ζ | 2 .
Finally, by using the definitions (B.2), (B.3), (B.4), and (B.7), the expression of the minimum MSE value in (23) immediately follows.

Appendix C

In this appendix we derive the relationship (37). Let us observe that, since 6)), we can equivalently show that

T c [2M + k, l + qM] = -T c [k, l + qM] (see (
2M-1 k=0 q        1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l]        2 = 0 . (C.1)
Taking into account ( 6) we can write

2M-1 k=0 q        1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l]        2 = 1 M 2 2M-1 k=0 q e j π M (m+k+ 1 2 )qM 2        2M-1 l=0 h[l + qM] e j π M (m+k+ 1 2 )(l+ 1 2 + M 2 ) h[l] e -j π M (m+ 1 2 )(l+ 1 2 + M 2 )        2 = 1 M 2 q e jπq 2M-1 l 1 =0 2M-1 l 2 =0 h[l 1 + qM] h[l 2 + qM] h[l 1 ] h[l 2 ] 2M-1 k=0 e j π M k(l 1 +l 2 +1+M) . (C.2)
Let us observe that

2M-1 k=0 e j π M k(l 1 +l 2 +1+M) =                  2M if l 1 + l 2 + 1 + M = 2M r 0 otherwise . (C.3) Since l 1 , l 2 ∈ {0, 1, . . . , 2M -1} the sum in (C.3) is different from zero only for l 2 = M -1 -l 1 with l 1 ∈ {0, 1, . . . , M -1}, and for l 2 = 3M -1 -l 1 with l 1 ∈ {M, M + 1, . . . , 2M -1}, thus we can write 2M-1 k=0 q        1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l]        2 = 1 M 2 q e jπq        M-1 l 1 =0 h[l 1 + qM] h[l 1 ] h[M -1 -l 1 ] h[M -1 -l 1 + qM] + 2M-1 l 1 =M h[l 1 + qM] h[l 1 ] h[3M -l 1 -1] h[3M -l 1 -1 + qM]        . (C.4)
Taking into account the symmetry condition (9) it follows that

h[M -1 -l 1 ] = h[M + l 1 ] (C.5) h[M -1 -l 1 + qM] = h[M -1 -(l 1 -qM)] = h[M + l 1 -qM] (C.6) h[3M -1 -l 1 ] = h[M + 2M -1 -l 1 u ] = h[M -1 -u] = h[l 1 -M] (C.7) h[3M -1 -l 1 + qM] = h[M + 2M -1 -l 1 + qM u ] = h[M -1 -u] = h[l 1 -M -qM] (C.8) By substituting (C.5) -(C.8) in (C.4) we obtain 2M-1 k=0 q        1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l]        2 = 1 M 2 q e jπq        M-1 l 1 =0 h[l 1 + qM] h[l 1 ] h[l 1 + M] h[l 1 + M -qM] + 2M-1 l 1 =M h[l 1 + qM] h[l 1 ] h[l 1 -M] h[l 1 -M -qM]        (C.9)
By using the position l = l 1 -M in the last sum in (C.9) we obtain

2M-1 k=0 q        1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l]        2 = 1 M 2 q e jπq        M-1 l 1 =0 h[l 1 + qM] h[l 1 ] h[l 1 + M] h[l 1 + M -qM] + M-1 l=0 h[l + M + qM] h[l + M] h[l] h[l -qM]        (C.10)
and, then, we can write

2M-1 k=0 q        1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l]        2 = 1 M 2        M-1 l 1 =0 h[l 1 ] h[l 1 ] h[l 1 + M] h[l 1 + M] + M-1 l=0 h[l + M] h[l + M] h[l] h[l] - M-1 l 1 =0 h[l 1 -M] h[l 1 ] h[l 1 + M] h[l 1 + 2M] - M-1 l=0 h[l] h[l + M] h[l] h[l + M] - M-1 l 1 =0 h[l 1 + M] h[l 1 ] h[l 1 + M] h[l 1 ] - M-1 l=0 h[l + 2M] h[l + M] h[l] h[l -M]        = 1 M 2        2 M-1 l=0 h 2 [l] h 2 [l + M] - M-1 l=0 h 2 [l] h 2 [l + M] - M-1 l 1 =0 h 2 [l 1 ] h 2 [l 1 + M]        = 0 (C.11)

Appendix D

In this appendix we show the relationship (38). A similar result has been shown in the Appendix of [START_REF] Lélé | Channel estimation methods for preamble-based OFDM/OQAM modulations[END_REF] with reference to the FBMC-OQAM signal case.

Let us observe that, since

T c [2M + k, l + qM] = -T c [k, l + qM] (see (6)), we can equivalently show that 2M-1 k=0 q 1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l] 2 = 2 . (D.1)
Taking into account [START_REF] Sexton | Enabling asynchronous machine-type D2D communication using multiple waveforms in 5G[END_REF] we can write

2M-1 k=0 q 1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l] 2 = 1 M 2 q 2M-1 l 1 =0 2M-1 l 2 =0 h[l 1 + qM] h[l 2 + qM] h[l 1 ] h[l 2 ] 2M-1 k=0 e j π M k(l 1 -l 2 ) . (D.2) Let us observe that 2M-1 k=0 e j π M k(l 1 -l 2 ) =                  2M if l 1 -l 2 = 2M r 0 otherwise . (D.3)
Since l 1 , l 2 ∈ {0, 1, . . . , 2M -1} the sum in (D.3) is different from zero only for l 2 = l 1 with l 1 ∈ {0, 1, . . . , 2M -1}, thus we can write

2M-1 k=0 q 1 M 2M-1 l=0 T c [m + k, l + qM] T * c [m, l] 2 = 2 M q 2M-1 l 1 =0 h 2 [l 1 + qM] h 2 [l 1 ] . (D.4)
Taking into account condition [START_REF] Picinbono | On circularity[END_REF] for r = 0 it follows that 2 M

2M-1

l 1 =0 h 2 [l 1 ] q h 2 [l 1 + qM] =1 = 2 M 2M-1 l 1 =0 h 2 [l 1 ] = 2 . (D.5)
Finally, from (D.5) and (D.4) equation ( 38) follows.

Appendix E

In this appendix, with reference to the adopted prototype filter in (13), we evaluate the explicit expression of the terms

t(k, m, i -q) = 1 M 2M-1 l=0 T c [k, l + (i -q)M] T * c [m, l] (E.1)
in (48). Let us observe that since the length of the prototype filter in ( 13) is

L h = 2M
, it immediately follows that t(k, m, iq) 0 only for iq = -1, 0, 1.

Let us evaluate t(k, m, 0), taking into account (6) we obtain

t(k, m, i) = 1 M 2M-1 l=0 T c [k, l] T * c [m, l] = 1 M 2M-1 l=0 h 2 [l]e j π M (k-m)(l+ 1 2 + M 2 ) . (E.2)
Since, for l = 0, 1, . . . , 2M -1,

h 2 [l] = 1 2 - 1 2 cos π M l + 1 2 it follows that 1 M 2M-1 l=0 h 2 [l]e j π M (k-m)(l+ 1 2 + M 2 ) = 1 M 2M-1 l=0 1 2 - 1 2 cos π M l + 1 2 e j π M (k-m)(l+ 1 2 + M 2 ) = 1 2M e j π M (k-m)( 1 2 + M 2 )        2M-1 l=0 e j π M (k-m) l - 1 2 e j π 2M 2M-1 l=0 e j π M (k-m+1) l - 1 2 e -j π 2M 2M-1 l=0 e j π M (k-m-1) l        = e j π M (k-m)( 1 2 + M 2 ) δ[k -m] - 1 2 e j π 2M δ[k -m + 1] - 1 2 e -j π 2M δ[k -m -1] = δ[k -m] - 1 2 e -j π 2 δ[k -m + 1] - 1 2 e j π 2 δ[k -m -1] . + 2M-1 k=0 1 M 2M-1 l=0 T c [m + k, l + M] T * c [m, l] 2 A 3 = 2 . (F.2)
Moreover, taking into account (E.6) we can write

1 M 2M-1 l=0 T c [m + k, l -M] T * c [m, l] = (-1) k+m+1 j ξ(k) , (F.3) from (E.3) it follows that 1 M 2M-1 l=0 T c [m + k, l] T * c [m, l] = δ[k] + j 2 δ[k + 1] - j 2 δ[k -1] , (F.4)
and, finally, taking into account (E.5) it follows that

1 M 2M-1 l=0 T c [m + k, l + M] T * c [m, l] = (-1) m j ξ(k) (F.5)
where (the expression of ξ(•) is reported here for readability)

ξ(l) = 1 4M          sin π 2 (l + 1) sin π 2M (l + 1) + sin π 2 (l -1) sin π 2M (l -1)          . (F.6)
Thus, from (F.2), (F.3), (F.4) and (F.5) we obtain

A 1 = 2M-1 k=0 1 M 2M-1 l=0 T c [m + k, l -M] T * c [m, l] 2 = 2M-1 k=0 ξ 2 (k) (F.7) A 3 = 2M-1 k=0 1 M 2M-1 l=0 T c [m + k, l + M] T * c [m, l] 2 = 2M-1 k=0 ξ 2 (k) (F.8) A 2 = 2M-1 k=0 1 M 2M-1 l=0 T c [m + k, l] T * c [m, l] 2 = 3 2 (F.9)
and, then, from (F.2) it follows that

Notation: j = √ - 1 ,

 1 superscript (•) * denotes the complex conjugation, [•] the real part, [•] the imaginary part, δ[k] the Kronecker delta, | • | the absolute value, ∠[•] the argument of a complex number in [-π, π) and E[•] denotes statistical expectation.

[0 - 1 . 5 - 1 . 4 - 3 . 6 -

 151436 0.6 -9.1 -7 -12 -16.9] and excess tap delay (in discrete samples) [0 1 3 6 7 14 22 35 50]. Moreover, the considered multipath channel model Extended Typical Urban (ETU) [16] has the following power/delay profile: relative power (expressed in dB) equal to [-1 -1 -1 0 0 0 -3 -5 -7] and excess tap delay (expressed in discrete samples) equal to [0 1 2 4 5 10 32 46 100]; 3. the BER values are obtained by averaging over all the active subcarriers of a given multicarrier symbol in a burst and over 10 4 independent channel realizations in the 2-PAM case and over 10 3 independent channel realizations in the 8-PAM case;

Figures 5 and 6

 6 Figures 5 and 6 show the BER of the considered receivers for 2-PAM constellation and M = 128 in ETU channel, and for the active subcarriers profiles 1 and 2, respectively. Moreover, figures 7 (ASP1) and 8 (ASP1) show the BER for M = 256. The results show that, in the ASP1 case the 2M-4M receiver outperforms all the single-tap structures, moreover, for M = 128 a performance degradation of the structures designed for large values of M (SCA and SCA MMSE) is observed while for M = 256 only the degradation of the SCA MMSE structure is present. In the ASP2 case when M = 128 the MMSE single-tap structure clearly outperforms the other single-tap structures and presents a contained performance degradation with respect to the 2M-4M receiver. Furthermore, when M = 256 the MMSE single-tap receiver assures a performance practically coincident with that of the 2M-4M structure and both outperform for high values of E b /N o the other considered single-tap receivers. Figures 9 and 10 show the BER of the considered receivers versus E b /N o for 8-PAM constellation and M = 64 in EVA channel, and for the active subcarriers profiles 1 and 2, respectively. The results show that when ASP1 is considered the 2M-4M receiver clearly outperforms the other considered structures while when ASP2 is considered a performance improvement of the SC MMSE receiver with respect to the other considered single-tap structures can be observed. However, this improvement is not observed for larger values of M. Moreover, figures 11 (ASP1) and 12 (ASP2) show that for M = 256 only the 2M-4M receiver assures a contained performance degradation with respect to the interference-free curve in the whole range of values of E b /N o . Finally, figures 13 and 14 show the BER of the considered receivers for 8-PAM

Fig. 1

 1 Fig. 1 BER versus Eb/No over EVA channel for 2-PAM constellation, M = 64 and active subcarriers profile 1.

Fig. 2 Fig. 3

 23 Fig. 2 BER versus Eb/No over EVA channel for 2-PAM constellation, M = 64 and active subcarriers profile 2.Fig. 3 BER versus Eb/No over EVA channel for 2-PAM constellation, M = 128 and active subcarriers profile 1.

Fig. 4

 4 Fig. 4 BER versus Eb/No over EVA channel for 2-PAM constellation, M = 128 and active subcarriers profile 2.

Fig. 5

 5 Fig. 5 BER versus Eb/No over ETU channel for 2-PAM constellation, M = 128 and active subcarriers profile 1.

Fig. 6

 6 Fig. 6 BER versus Eb/No over ETU channel for 2-PAM constellation, M = 128 and active subcarriers profile 2.

Fig. 7

 7 Fig. 7 BER versus Eb/No over ETU channel for 2-PAM constellation, M = 256 and active subcarriers profile 1.

Fig. 8

 8 Fig. 8 BER versus Eb/No over ETU channel for 2-PAM constellation, M = 256 and active subcarriers profile 2.Fig. 9 BER versus Eb/No over EVA channel for 8-PAM constellation, M = 64 and active subcarriers profile 1.Fig. 10 BER versus Eb/No over EVA channel for 8-PAM constellation, M = 64 and active subcarriers profile 2.

Fig. 11 BER

 11 Fig. 11 BER versus Eb/No over EVA channel for 8-PAM constellation, M = 256 and active subcarriers profile 1.

Fig. 12 BER

 12 Fig. 12 BER versus Eb/No over EVA channel for 8-PAM constellation, M = 256 and active subcarriers profile 2.

Fig. 13 BER

 13 Fig. 13 BER versus Eb/No over ETU channel for 8-PAM constellation, M = 128 and active subcarriers profile 1.

Fig. 14 BER

 14 Fig. 14 BER versus Eb/No over ETU channel for 8-PAM constellation, M = 128 and active subcarriers profile 2.

Fig. 15 BERFigure 1 :Figure 2 :Figure 3 :Figure 4 :

 151234 Fig. 15 BER versus Eb/No over ETU channel for 8-PAM constellation, M = 512 and active subcarriers profile 1.Fig. 16 BER versus Eb/No over ETU channel for 8-PAM constellation, M = 512 and active subcarriers profile 2.

Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 5678 Figure 5: BER versus Eb/No over ETU channel for 2-PAM constellation, M = 128 and active subcarriers profile 1.

Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :Figure 15 :Figure 16 :

 910111213141516 Figure 9: BER versus Eb/No over EVA channel for 8-PAM constellation, M = 64 and active subcarriers profile 1.

Therefore, we can write

Let us now evaluate t(k, m, 1), taking into account [START_REF] Sexton | Enabling asynchronous machine-type D2D communication using multiple waveforms in 5G[END_REF] we obtain

Let us now evaluate t(k, m, -1), taking into account (6) we obtain

where

.

By using the position l 1 = l -M we obtain

Therefore, we can write

where

and, then, taking into account (E.4) we obtain

Since q ∈ {-1, 0, 1} (see ( 6)), it follows that

List of figures