Adel M'foukh

Member, IEEE Marco G Pala

IEEE David Esseni Fellow

Full band quantum transport of heterojunction electron devices with empirical pseudopotentials

Keywords: Empirical pseudopotential, quantum transport, NEGF, heterojunctions, electronic devices

This paper presents the methodology, implementation and application of a full-band quantum transport model based on the non-equilibrium Green's function formalism and the empirical pseudopotentials. In particular the paper reports the treatment of heterojunctions between lattice matched semiconductors, comprising a gradual transition region described according to a virtual crystal approximation. Our approach entails several numerical techniques to make the full-band quantum transport method computationally affordable, and thus enable robust and efficient self-consistent device simulations.

Then we employ our simulation scheme for the analysis of some exemplary devices based on quantum tunnelling, such as an Esaki tunnelling diode, as well as n-and p-type heterojunction Tunnel FETs. In particular we investigate the influence on the current-voltage characteristics of the width of the heterojunction transition region. We observe that a gradual transition region mainly affects the device characteristics by lengthening the tunnelling path at the heterojunction, which has a different impact on device current depending on the external bias conditions.

I. INTRODUCTION

T HE aggressive shrinkage of FETs has effectively turned them into tiny quantum constrictions where quantum transport effects have become important, such as the sourcedrain tunnelling in MOSFETs [START_REF] Kim | Comprehensive Performance Benchmarking of III-V and Si nMOSFETs (Gate Length = 13 nm) Considering Supply Voltage and OFF-Current[END_REF], [START_REF] Esseni | Essential physics of the off-state current in nanoscale mosfets and tunnel fets[END_REF], or the band-to-bandtunnelling (BTBT) in Tunnel FETs [START_REF] Seabaugh | Low-Voltage Tunnel Transistors for Beyond CMOS Logic[END_REF], [START_REF] Esseni | A review of selected topics in physics based modeling for tunnel field-effect transistors[END_REF]. The interface to volume ratio has also steadily increased in 3D device architectures, culminating in CMOS stacked nanowire transistors [START_REF] Lizzit | Analysis of the Performance of n-Type FinFETs With Strained SiGe Channel[END_REF], [START_REF] Vasen | InAs nanowire GAA n-MOSFETs with 12-15 nm diameter[END_REF], [START_REF] Loubet | Stacked nanosheet gate-all-around transistor to enable scaling beyond finfet[END_REF], [START_REF] Badami | Benchmarking of 3-d mosfet architectures: Focus on the impact of surface roughness and self-heating[END_REF].

The use of heterojunctions is also an important technology option used, among other possible examples, in bipolar transistors [START_REF] Hamel | Technological requirements for a lateral sige hbt technology including theoretical performance predictions relative to vertical sige hbts[END_REF][10], as well as in Tunnel FETs [START_REF] Avci | Tunnel field-effect transistors: Prospects and challenges[END_REF], [START_REF] Brocard | Large On-Current Enhancement in Hetero-Junction Tunnel-FETs via Molar Fraction Grading[END_REF]. Moreover there is clear experimental evidence that fabricated heterojunctions are not abrupt, but instead feature a transition region extending for a few atomic layers [START_REF] Hudait | Strain relaxation properties of inas metamorphic materials grown on inp substrates[END_REF][14] [START_REF] Zhu | Structural, morphological, and defect properties of metamorphic in0.7ga0.3as/gaas0.35sb0.65 ptype tunnel field effect transistor structure grown by molecular beam epitaxy[END_REF], which solicits an atomistic description of the interfaces.

Actually an atomistic approach is nowadays perceived as the frontier of transport modelling, and different efforts have been reported for models based directly on first-principle calculations, which in turn have become available in several ab-initio packages [START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF], [START_REF] Giannozzi | Quantum espresso: a modular and open-source software project for quantum simulations of materials[END_REF].

An empirical pseudopotentials method (EPM) provides an interesting compromise between accuracy and computational burden, in that it can accurately and effectively describe both and nanostructured semiconductors by using only few fitting parameters [START_REF] Wang | Million-Atom Pseudopotential Calculation of Γ -X Mixing in GaAs/AlAs Superlattices and Quantum Dots[END_REF], [START_REF] Wang | Linear combination of bulk bands method for large-scale electronic structure calculations on strained nanostructures[END_REF], [START_REF] Chirico | Efficient Self-consistent pseudopotential calculation of nanostructured devices[END_REF], [START_REF] Esseni | Linear combination of bulk bands method for investigating the low-dimensional electron gas in nanostructured devices[END_REF], [START_REF] Van Der Steen | Validity of the parabolic Effective Mass Approximation in silicon and germanium n-MOSFETs with different crystal orientations[END_REF]. Some recent papers have explored the use of the EPM for transport modelling by using either a quantum transmitting boundary method [START_REF] Jiang | Quantum mechanical simulation of electronic transport in nanostructured devices by efficient self-consistent pseudopotential calculation[END_REF], [START_REF] Fang | Pseudopotential-based electron quantum transport: Theoretical formulation and application to nanometer-scale silicon nanowire transistors[END_REF], [START_REF] Fang | Theoretical Study of Ballistic Transport in Silicon Nanowire and Graphene Nanoribbon Field-Effect Transistors Using Empirical Pseudopotentials[END_REF], [START_REF] Van De Put | Scalable atomistic simulations of quantum electron transport using empirical pseudopotentials[END_REF], or a Non Equilibrium Green's Function (NEGF) method [START_REF] Pala | Full-band quantum simulation of electron devices with the pseudopotential method: Theory, implementation, and applications[END_REF], [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF], [START_REF] Pala | Unit cell restricted bloch functions basis for first-principle transport models: Theory and application[END_REF]. However, no attempts have been reported to our knowledge for the use of such transport methods for electron devices whose operation relies on the heterojunction of different semiconductors.

In this work we report a substantial extension of our recent work related to EPM based quantum transport, and we describe the treatment of lattice matched heterojunctions in nanoscale devices. We here propose a methodology for the description of either abrupt or gradual heterojunctions, where the EPM parameters are changed with the position so as to ensure a correct alignment of conduction and valence band edges along the heterojunction. The NEGF based formalism for quantum transport is also extended in order to duly account for a non homogeneus system. Two exemplary and yet technologically relevant case studies are then addressed, namely a III-V semiconductor based Esaki diode and both nand p-type heterojunction Tunnel FETs.

The paper is organized as follow. Section II illustrates the self-consistent transport model by focussing on the treatment of heterojunction within the EPM method. In Sec. III we report some simulation case studies for devices employing heterojunctions, and in Sec. IV we finally offer our concluding remarks.

II. MODEL DESCRIPTION

Here we briefly describe the full-band model employed in this work for band-structure and quantum transport calculations, which is based on the empirical pseudopotential method (EPM) [START_REF] Chelikowsky | Electronic structure of silicon[END_REF], [START_REF] Cohen | Electron Structure and Optical Properties of Semiconductors[END_REF]. The general methodology has been already described in previous publications [START_REF] Pala | Full-band quantum simulation of electron devices with the pseudopotential method: Theory, implementation, and applications[END_REF], [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF], so that we here focus on the aspects related to the treatment of heterojunctions that have been explicitly developed for this work.

We used a local formulation of the EPM where, for a diamond semiconductor (e.g. Si or Ge) or a zinc-blende compound (e.g. GaAs, InAs), the spectral components of the pseudopotential can be written as [START_REF] Chelikowsky | Electronic structure of silicon[END_REF], [START_REF] Cohen | Electron Structure and Optical Properties of Semiconductors[END_REF]

V L (G) = U S (|G|) cos[G • τ ] + iU A (|G|) sin[G • τ ] (1)
where G is a reciprocal lattice vector and τ =(1/8)a 0 (1, 1, 1) is the atomic basis vector. The material dependent U S (|G|), U A (|G|) denote respectively the symmetric and antisymmetric form factors, with U A (|G|) being null for diamond materials. It is well known that the full band relation of most technologically relevant semiconductors can be described well by using only three non null U S (|G|) and U A (|G|) form factors, corresponding to |G| 2 =3, 8, 11 [2π/a 0 ] 2 for U S (|G|) and to |G| 2 =3, 4, 11 [2π/a 0 ] 2 for U A (|G|) (with a 0 being the lattice constant) [START_REF] Chelikowsky | Electronic structure of silicon[END_REF].

The quantum transport model used in this work relies on the Non Equilibrium Green's Function (NEGF) method formulated in a hybrid basis, xK yz , consisting of a discretized real space in the transport direction x and of plane waves in the (y,z) directions [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF]. In the hybrid basis the Hamiltonian matrix inside a given semiconductor takes the block tridiagonal form

       H(x 1 , x 1 ) H 0,1 0 0 • • • H † 0,1 H(x 2 , x 2 ) H 0,1 0 • • • • • • . . . 0 0 H † 0,1 H(x j , x j ) H 0,1 0 • • • • • • • • • • • •       
(2) where x j is a discretization point in the transport direction. For a 3D electron gas (i.e. no quantum confinement in directions y, z) the size of each blocks in Eq. ( 2) is

M 3D = 2N G /N d ,
where N G is the number of G vectors in EPM calculations (which is in turn set by the EPM cutoff energy), and N d is the number of discretization points inside a unit cell. For a 2D or a 1D electron gas the Hamiltonian matrix still has the tridiagonal form in Eq. ( 2), but the size of the blocks is increased [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF]. In all the calculations of this work we used N d =32, and an EPM cutoff energy E cut = 4.5 Ry corresponding to N G ≈800.

A. Modelling of heterojunctions with the EPM method

For a homogeneous semiconductor the diagonal blocks H(x j , x j ) in Eq. ( 2) depend on x j only through the local pseudopotential V L (x j , G yz -G yz ), which is obtained via a discrete Fourier transform from the G x to the x j and it is periodic of a 0 [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF]. In this work we studied lattice matched heterojunctions by using a virtual crystal approximation [START_REF] Bellaiche | Virtual crystal approximation revisited: Application to dielectric and piezoelectric properties of perovskites[END_REF], whose soundness has been proved also for quaternary alloys [START_REF] Vurgaftman | Band parameters for III-V compound semiconductors and their alloys[END_REF]. In this approach the EPM parameters U S (|G|), U A (|G|) change along the transport direction so as to describe either an abrupt heterojunction, or a gradual heterojunction having a transition width, W HJ , long up to a few lattice constants [START_REF] Borg | Inas/gasb heterostructure nanowires for tunnel fieldeffect transistors[END_REF]. More precisely, the Hamiltonian blocks of the two materials are first computed in the hybrid space xK yz , then they are patched at adjacent positions to form the heterojunction. This approach implies that at the heterojunction interface a discontinuity exists in the empirical pseudopotential. For the case at study in this paper, however, such a discontinuity is small because GaSb and InAs have similar EPM spectral components (see Tab. I) The coupling Hamiltonian matrix between adjacent blocks is simply given by the spatial discretization of the kinetic energy operator [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF], hence it is material independent. In the description of heterojunctions via the EPM method the correct alignment of conduction and valence band edges between the different materials must be ensured. At this regard, Eq. ( 1) shows that a non zero U S (0)=U S (|G|=0) simply results in a rigid shift of V L and thus of the bandstructure of a given material. Consequently, in the analysis of a homogeneous system one typically sets U S (0)=0, because U S (0) can be absorbed in the choice of the energy reference.

In our modelling of heterojunctions, however, U S (0) has been changed in order to ensure the correct alignment of the conduction and valence band edges of the constituent materials, which is an approach already discussed and validated in [START_REF] Friedel | Local empirical pseudopotential approach to the optical properties of si/ge superlattices[END_REF]. For devices including heterojunctions the position dependence of the EPM parameters contributes to the x j dependence of the diagonal Hamiltonian blocks H(x j , x j ) in Eq. ( 2), as it is illustrated in the sketch of Fig. 1.

Tab. I reports the EPM parameters for GaSb and InAs used in all the calculations of this work, and Fig. 2 illustrates the bandstructure of the two materials, as well as the corresponding density of states. As it can be seen, the GaSb-InAs heterojunction provides a broken bandgap system, in the sense that the bottom of the InAs conduction band is slightly below the top of the GaSb valence band. By varying the molar fraction x in GaSb 1-x InAs x the corresponding effective mass at the Γ minimum of the conduction band change as reported in Fig. 3 B. Quantum transport Our transport model is based on the NEGF method, where the transport equations are self-consistently solved with the Poisson equation for the overall device electrostatics. For ballistic transport the retarded Green's function can be formally defined as

G r (E)=[EI -H -Σ L -Σ R ] -1
, where H is the Hamiltonian matrix in Eq. ( 2), and Σ R/L are the left and right contact self-energies. Because only a small subset of the blocks of G r (E) are needed to calculate the carrier density and the current, we here employed the recursive Green's function algorithms based on Dysons's equations [START_REF] Anantram | Modeling of Nanoscale Devices[END_REF], which make the size of the Hamiltonian blocks the crucial parameter for the computational burden.

In this latter respect, we could obtain a substantial reduction of the size of such blocks by employing a mode-space transformation [START_REF] Venugopal | Simulating quantum transport in nanoscale transistors: Real versus mode-space approaches[END_REF]. Differently from Ref. [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF], the modes along the heterojunction direction change with the position x j , which we duly accounted for in the transformation of the Hamiltonian to the mode space.

In the case studies of this work we found that, by selecting only the lowest energy transverse modes in each device section, the size of the Hamitonian blocks could be vastly reduced compared to the blocks in Eq. ( 2). Even in heterojunction devices, in fact, a few tens of modes were sufficient to achieve results that agree within a few percent with those obtained without introducing the mode space reduction.

III. SELF-CONSISTENT DEVICE SIMULATIONS

In this section we report exemplary simulation results obtained with the EPM based simulation methodology developed in this work for two band-to-band-tunnelling (BTBT) based devices employing a GaSb-InAs heterojunction. In all simulations the NEGF transport equations were solved selfconsistently with the Poisson equation describing the device electrostatics. All simulations were carried at room temperature and for the transport direction [100].

A. Esaki tunnelling diode

The Esaki diode is sketched in Fig. 4 and consists of a ptype GaSb region and an n-type InAs region. The doping of the p and n-type regions was set to respectively N A =10 20 cm -3 and N D =3 × 10 19 cm -3 , which result in a degenerate carrier gas in both regions. We assumed periodic boundary conditions along the transverse directions and sampled the wave vectors k y and k z with a constant step of ∆k y = ∆k z = 0.05 2π/a 0 .

The current per unit area, I D , versus the applied bias V D of the Esaki diode is shown Fig. 4 for different widths, W HJ , of the heterojunction transition region. In the forward bias a negative differential resistance (NDR) is observed, which is more clearly illustrated in the inset of the figure. As it can be seen the peak current in the NDR region decreases for more gradual heterojunctions (i.e. for larger W HJ ), and it is maximum for the abrupt case. A similar dependence of I D on W HJ is observed also in the reverse bias region, where the current is still completely dominated by BTBT in the bias range explored in Fig. 4.

Fig. 5 illustrates the conduction and valence band profiles along the device for V D =0.1 V (i.e. approximately at the peak current in the NDR region), as well as the energy resolved current density J D . As expected the J D is practically relevant only in the narrow Fermi energy window identified by the Fermi levels E f,n and E f,p of respectively the nand pregion. Moreover, the narrower the transition region W HJ is the shorter is the BTBT path, which in turn leads to the corresponding I D increase in Fig. 4.

TABLE II DEVICE PARAMETERS FOR THE

T ch L G L S/D N A N D [nm] [nm] [nm] [cm -3 ] [cm -3 ] GaSb/

B. Ultra-thin body, heterojunction Tunnel FETs

Nanoscale FETs are often fabricated by using ultra-thin semiconductor films or nanowires, which can induce significant quantum confinement effects in the resulting low dimensional electron gas, as it is the case for the Tunnel FETs (TFETs) simulated in this section. The modelling of quantum confinement in our EPM based formalism has been described in Sec.II.A of reference [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF], and it is based on a pseudo-oxide material. The EPM parameters of the pseudooxide are tailored so as to obtain the appropriate discontinuity at the semiconductor-oxide interface of the conduction band minimum and valence band maximum, which in turn confine the carriers in the semiconductor region. For both the nand the p-type TFETs we used a physical oxide thickness of 1.5 nm and a relative dielectric constant κ=10, corresponding to an equivalent oxide thickness of 0.6 nm. The transverse wave vector k y was sampled with a constant step of ∆k y = 0.05 2π/a 0 . In this work we used the pseudo-oxide EPM parameters proposed in [START_REF] Pala | Quantum transport models based on NEGF and empirical pseudopotentials for accurate modeling of nanoscale electron devices[END_REF], which result in a conduction and valence band discontinuity between InAs and the pseudo-oxide of respectively 3.8 eV and 4.5 eV.

A sketch of the n-type heterojunction Tunnel FET is shown in Fig. 6 and the main device parameters are reported in Tab. II. nand p-type TFET the metal gate workfunction was adjusted so that the off current (i.e. the I D at V GS = 0 V and V DS = 0.3 V) is approximately 1 nA/µm. The transistor exhibits a very steep sub-threshold region with a sub-threshold swing below 60 mV/dec, which is an expected result for a device with no defects and with an semiconductor-oxide interface free of traps and microscopic roughness [START_REF] Conzatti | Strain-Induced Performance Improvements in InAs Nanowire Tunnel FETs[END_REF], [START_REF] Conzatti | Surface-roughness-induced variability in nanowire InAs tunnel FETs[END_REF].

Quite interestingly the width W HJ of the GaSb-InAs heterojunction has no sizeable influence on the subthreshold current and swing but it does affect the on current, which is in fact degraded when the transition region is wider. Fig. 7 shows that the markedly different influence of W HJ on the off and on current is again related to its modulation of the band-to-bandtunnelling path. In the off-state, in fact, the impact of W HJ on the relatively long BTBT path is only modest, whereas in the on state the relative importance of W HJ is larger because the tunnelling path is much shorter. By leveraging the inherently full-band nature of our EPM based modelling approach, we also investigate a p-type heterojunction Tunnel FET, whose sketch is shown in Fig. 8 and whose parameters are reported in Tab. III. Due to the fairly low conduction band density of states of InAs, the source doping N D =2 × 10 19 cm -3 is large enough to induce a strongly degenerate electron gas in the source. Such a doping concentration is fairly consistent with the value recommended in Ref. [START_REF] Vandenberghe | A model determining optimal doping concentration and material's band gap of tunnel field-effect transistors[END_REF], which was the outcome of an optimization of the tunnelling junction operation. The current-voltage characteristics in Fig. 8 show that the p-type Tunnel FET can hardly achieve a sub-threshold swing below 60 mV/dec, with a degraded minimum swing compared to the n-type counterpart in Fig. 6. This result is not unexpected because the low density of states in the source region leads to a relatively large screening length of the electric field produced by the gate bias (compared to the p-type source of the n-type Tunnel FET) [START_REF] Lizzit | Performance Benchmarking and Effective Channel Length for Nanoscale InAs, In0.53Ga0.47As, and sSi n-MOSFETs[END_REF], which in turn degrades the sub-threshold swing [START_REF] Esseni | A review of selected topics in physics based modeling for tunnel field-effect transistors[END_REF].

T ch L G L S/D N A N D [nm] [nm] [nm] [cm -3 ] [cm -3 ] InAs/GaSb 3.7 16
As for the influence of the width W HJ of the heterojunction transition region on the current-voltage characteristic of the device, the main features observed for the n-type Tunnel FET are confirmed. Namely, W HJ hardly affects the subthreshold region, where the tunnelling distance is substantially larger than W HJ , whereas the on current of the device is progressively reduced for increasing W HJ . In the on state, in fact, a gradual transition region for the heterojunction results in a sizeable lengthening of the tunnelling path.

IV. CONCLUSION

We have presented the implementation and application of a full band quantum transport methodology for nanoscale electron devices with heterojunctions. The model was developed in the framework of the empirical pseudopotentials method, and the heterojunctions were described according to a virtual crystal approximation, which allowed us to investigate the influence on the device characteristics of a gradual interface between the constituent materials.

We then leveraged the inherently full-band nature of the model and reported results for an nand a p-type, ultra-thin body heterojunction Tunnel FET, as well as for an Esaki diode.

The interface at the heterojunction was varied from the abrupt case to a more realistic transition width extending up to a few unit cells. The tunnelling current in the Esaki diode was found to be sensitive to the width of the heterojunction both in reverse bias and in the negative resistance region. In the GaSb-InAs Tunnel FETs the transition width affects the current only in the on-state. In the sub-threshold region, in fact, the effective tunnelling path is much larger than W HJ and thus it is not appreciably modulated by heterojunction width.

The results of this work qualify our methodology based on the empirical pseudopotentials and the NEGF formalism as a viable approach for the modelling and design of homojunction and heterojunction nanoscale electron devices.

  A. M'foukh and M. G. Pala are with Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 10 Bd Thomas Gobert, 91120 Palaiseau, France D. Esseni is with DPIA, University of Udine, Via delle Scienze 206, 33100 Udine, Italy

  bulk

Fig. 1 .

 1 Fig. 1. Pictorial illustration of the blocks of the Hamiltonian matrix across a gradual heterojunction where, in order to simplify the notation, one end of the heterojunction has been placed at x=0. W HJ is the width of the heterojunction transition region and d=a 0 /N d is the discretization step along x. The notation is consistent with Eq.2. In the heterojunction region the EPM parameters change with the position and thus contribute to the position dependence of the diagonal Hamiltonian blocks, H HJ (x i , x i ).

Fig. 2 .

 2 Fig. 2. (Left) Highest valance bands and lowest conduction bands for GaSb and InAs along the main symmetry points inside the first Brillouin zone and (right) the corresponding density of states.

Fig. 3 .

 3 Fig. 3. (a) Electron effective mass m * at the Γ minimum point of the conduction band versus the molar fraction x of the alloy GaSb 1-x InAsx. (b) Top of valence band, bottom of conduction band and band-gap versus the molar fraction. The energy reference is the top of the InAs valence band.

Fig. 4 .

 4 Fig. 4. (Top) Sketch of the Esaki diode where red is used for InAs and blue is used for GaSb, and (bottom) corresponding current density, J D , versus the applied bias V D , for different widths, W HJ , of the heterojunction transition region. The inset shows a focus on the negative differential resistance region.

Fig. 5 .

 5 Fig. 5. (Left) Conduction and valence band profile along the Esaki diode for different widths W HJ of heterojunction transition region. The source Fermi level, E f,n in the n region is taken as the energy reference. (Right) Corresponding current spectrum at ky = kz = 0, J D , versus the energy and for different W HJ . The bias is V D = 0.1V

  HETEROJUNCTION n-TYPE TUNNEL FET SIMULATED IN FIG. 6. T ch , L G , L S/D , N A , N D DENOTE THE CHANNEL THICKNESS, THE GATE LENGTH, THE SOURCE/DRAIN LENGTH AND THE DOPANT CONCENTRATIONS IN THE SOURCE AND DRAIN REGION, RESPECTIVELY.

Fig. 6 .

 6 Fig. 6. (Top) Sketch of the simulated n-type HJ TFET. (Middle) Conduction and valence band profile at different V GS and Fermi level in the source region E f,S and drain region E f,D for a abrupt heterojunction GaSb-InAs n-type tunnel FET. (Bottom) Drain-current versus gate-voltage characteristics at V DS =0.3 V for different widths, W HJ , of the heterojunction transition region. The metal gate workfunction was set to 4.625 eV.

  Fig. 6 also reports the current versus gate voltage characteristics of the TFET for different widths, W HJ , of the GaSb-InAs heterojunction and for V DS =0.3 V. For both the

Fig. 7 .

 7 Fig. 7. Local density of states, LDOS, for the T-FET simulated in Fig.6. (a),(c) report the LDOS in the off state for V GS =0.0V and V DS =0.3V ; (b),(d) report the LDOS in the on state for V GS =0.4V and V DS =0.3V . Plots in (a) and (b) correspond to an abrupt heterojunction (i.e. W HJ =0), while plots in (c) and (d) correspond to W HJ =3a 0 .

14 5 × 10 19 2 × 10 19

 19

Fig. 8 .

 8 Fig. 8. (Top) Sketch of the simulated p-type HJ TFET. (Middle) Conduction and valence band profile at different V GS for a abrupt HJ GaSb-InAs ptype tunnel FET. (Bottom) Drain-current versus gate-voltage characteristics for different widths, W HJ , of the heterojunction transition region. The metal gate workfunction was set to 4.665 eV.

TABLE I EPM

 I PARAMETERS IN RY FOR GASB AND INAS EMPLOYED IN THIS WORK. U S (0) IS SET TO 0 FOR INAS AND TO 0.84 EV FOR GASB.

	GaSb	-0.22	0	0.05	0.06	0.05	0.01
	InAs	-0.22	0	0.05	0.08	0.05	0.03

U S (3) U S (8) U S (11) U A (3) U A (4) U A

(

[START_REF] Avci | Tunnel field-effect transistors: Prospects and challenges[END_REF]

TABLE III DEVICE

 III PARAMETERS FOR THE HETEROJUNCTION p-TYPE TUNNEL FET SIMULATED IN FIG. 8.


Mamoun Fellah
email: mamoun.fellah@yahoo.fr

Linda Aissani

Mohammed Abdul Samad

Alain Iost

Mohamed Touhami

Alex Zine

Corinne Montagne

Nouveau

• Touhami

Mohamed Zine

Alex Montagne

Corinne Nouveau

Effect of Replacing Vanadium by Niobium and Iron on the Tribological Behavior of HIPed Titanium Alloys

Keywords: Tribological behavior, Titanium alloys, Wear testing, Nanotribology, Milling time, Hot isostatic pressing (HIPing)

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

Introduction

Usage of titanium alloys is continuously increasing in biomedical applications due to their high biocompatibility and good mechanical properties compared to conventional biomaterials [1][2][3][4][5][START_REF] Fellah | Comportement e ´lectrochimique de deux prothe `ses totales de hanche en acier AISI 316L et en alliage de Ti-6A-7Nb[END_REF][START_REF] Feng | [END_REF]. Due to their excellent properties, they have also been applied to aerospace industry, petroleum industry and also in medicine [8][9][10]. It is known that Ti can be alloyed with variety of elements to further improve its properties, such as high-temperature performance and formability [1,11,12]. The strength of the alloy is much increased over that of pure titanium, when aluminum (Al) and vanadium (V) are added to pure titanium in small quantities [13].

However, the release of Al and V ions from the dissolution of the passive film on the alloy surfaces produces a toxic effect, causing severe health problems [START_REF] Fellah | Comparative tribological study of biomaterials AISI 316L and Ti6Al7Nb, in 40th leeds[END_REF][START_REF] Lopez | [END_REF]. For this reason, the non-allergic and non-toxic beta-eutectic (b) elements such as niobium (Nb) and iron (Fe) have been used as safe alloying elements for developing low elastic modulus and high-strength titanium alloys [16,17]. The Ti alloy with aluminum and iron is found to be more suitable for implant applications [17].

Ti is currently used in dentistry, and the Ti-6Al-4V alloy is used in many orthopedic applications. However, it is a subject of many disadvantages and limitations [18,19]. The most important obstacles to hinder the wider use of titanium-based alloys and limit their applications under sliding conditions and contact loads [20][21][START_REF] Fellah | Effect to replacement of V by Nb and Fe on the tribological behavior of biomedical titanium alloys, in 2 e `mes journe ´es scientifiques franco-maghre ´bines[END_REF] are poor tribological properties such as high friction coefficient and low abrasive wear resistance coupled with relatively low hardness [START_REF] Ao | [END_REF]. Therefore, in order to improve the tribological properties of titanium alloys, various methods have been applied for improving resistance to abrasive wear [24] and plastic shearing [25]. Moreover, tribological properties may also be improved or modified by surface coatings, changing microstructure of surface layers or thermochemical treatments [25][26][27][28]. Johnson and Eberhardt [29], for example, improved the wear resistance of Ti-6Al-4V using thermal oxidation. Although this is a low-cost processing route for wear performance enhancement, it suffers limitations as other processes such as nitriding and boriding [24,30,31].

Recently, a new processing route for the commercial production of a lower grade of titanium alloys has been developed [21], which should facilitate the usage of Ti, particularly in medical industry. However, up to now, lot of efforts have only been dedicated to evaluate the mechanical and microstructural properties of b titanium alloys with Fe and Nb addition, but very few studies have been systematically conducted to evaluate the tribological behavior and mechanical properties of a-type Ti-6Al alloy with Fe and Nb addition [START_REF] Fellah | Experimental study of new titanium alloy Ti-6Al-4Fe for biomedical application[END_REF].

Nanostructured titanium alloys have also been developed to improve the mechanical properties [START_REF] Bolzoni | [END_REF][34][35][36][37][38]. However, correlations among conditions of milling, microstructure and tribological behavior have not been very well studied for these nanostructured alloys [39][40][START_REF] Fellah | Comparative friction and wear behavior between titanium alloys Ti-6Al-7Nb and Ti-6A1-4V[END_REF].

In mechanical milling, one of the most important parameters is the milling time, as it significantly affects the wear resistance and durability of total hip prosthesis [START_REF] Wang | [END_REF][43][44]. Usually, the time is chosen so as to achieve a steady state between the fracturing and cold welding of the particles. The required time varies depending on the milling conditions. Therefore, it is always good to mill powders just for the necessary time and no more [START_REF] Fellah | Comparative Study on Tribological Behavior of Ti-6Al-7Nb and SS AISI 316L alloys, for Total Hip Prosthesis[END_REF].

Therefore, the present study was undertaken with the main focus of examining the effects of replacing vanadium (V) by niobium (Nb) and iron (Fe) on the tribological behavior of nanostructured milled and hot-isostatic-pressed (HIPed) titanium alloys. In addition, the correlation between the milling time, hardness, microstructure, lattice parameters and wear resistance will also be evaluated, and the possibility of using nanostructured Ti-6Al-4Fe and Ti-6Al-7Nb alloys as biomaterial alloys will be discussed.

Materials and Methods

Synthesis Process

Ti, Al, Nb, Fe and V powders with 99.94% purity and 10-50 lm average particle sizes are used. The powders were milled at 2, 6, 12 and 18 h, respectively, using a highenergy ball mill, Fritsch P7, under an argon atmosphere. The milling media consisted of 18-mm-diameter alumina balls confined in a 350-ml volume vial. The weight ratio of ball to powder was about 20. The milling was carried out in cycles of 28 min with a pause of 9 min in between the cycles. The vials were opened after a 35-to 40-min cooling period, after each milling operation.

The milled powders of Ti-6Al-7Nb (6 wt% Al, 7 wt% Nb and the rest Ti), Ti-6Al-4Fe (6 wt% Al, 4 wt% Fe and the rest Ti) and Ti-6Al-4V (6 wt% Al, 4 wt% V and the rest Ti) were pressed uniaxially at 100 MPa into 15-mmdiameter circular disks with a thickness of 5 mm by a rigid steel die followed by sintering in a high vacuum furnace at 3 9 10 -5 mbar at temperatures of 1150 °C for 1 h at a heating rate of 20 K min -1 to obtain a closed porosity [START_REF] Attanasio | [END_REF]. In order to produce a high-density nanocrystalline material (density of 99% measured using the Archimedes principle), the sintered samples subsequently were hot isostatically pressed (HIPed) using ASEA-HIP at 1050 °C at a heating rate of 20 K min -1 and at an isostatic pressure of 300 MPa for 30 min [47][48][49].

Microstructure Characterization

For microstructure characterization, the samples were polished with 2-nm diamond paste. After preparing the surfaces, samples were etched with Kroll's reagent (10HNO 3 -2HF-88H 2 O) and characterized by scanning electron microscopy (SEM, JEOL JSM-3C) to examine morphology and particle sizes. Transmission electron microscopy was used to observe and measure the average grain size of the starting powder morphology. The phase identification and structural evolution were examined using X-ray diffractometry with CuKa radiation. Also, the grain size in the samples was evaluated using Williamson-Hall formula equation [START_REF] Fellah | Synthesis, microstructural and tribological characterization of calcined nano-bioceramic a-al 2 O 3 , sintered at different temperatures[END_REF].

The Vickers hardness of the nanostructured titanium alloys was evaluated using a universal hardness testing machine (Zwik ZHV 2.5) with a Vickers diamond indenter for which elastic constant E i = 1140 GPa and Poisson ratio v i = 0.07.

Tribological Characterization

A ball-on-disk-type oscillating tribometer (TRIBOtester) was used for tribological tests, under wet conditions using physiological solution in accordance with the ISO7148 standard. The tests were carried out under a normal load of 6 N, with an AISI 52100 grade steel ball (E = 210 GPa, HV 0.05 = 840 and density of 7.85 g/cm 3 ) as a counter face. The HIPed disk specimens of 15 mm in diameter and 5 mm in thickness polished to 0.50-0.70 nm surface roughness were used for the tests. Before each test, the balls and the samples were ultrasonically rinsed in acetone. After the wear tests, the worn surface morphology was observed by SEM. Tested samples and the counter face balls were weighed before and after the tests, but a modest weight difference was observed.

Results and Discussion

Microstructural Characterization

Grain Size and Distribution

Morphology and particle size evolution of milled titanium alloy powders as a function of milling time are shown in Fig. 1. It is known that the process of milling consists of three main stages: (I) cold welding, (II) fracturing and (III) steady-state condition [START_REF] Suryanarayana | [END_REF]52]. At the first milling stage (2 h), the powder particles are found to be angular and irregularly shaped with a 4-to 30-lm size distribution and were dispersed uniformly throughout the surface. It is observed that the grain size is relatively small as well. It is clear that at 2-h milling, the particles were fine and had formed some agglomerates with 80 vol% of particle size less than 30 lm and 20 vol% with size less than 10 lm. It is known that during the milling process, the milling energy deforms the powders which are compressed by the process of microforging [53]. As shown in Fig. 1, irregular and elongated grains with pointed ends were observed. At 2-, 6-and 12-h milling, an increase in the quantity of the fine grains was observed.

However, for 18-h milled samples, agglomerates/cluster of particles appeared. It is observed that with an increase in milling time, the particle size decreased due to the excessive fracturing of particles as compared to that of the coldwelding process. Moreover, with increasing milling time, the chip morphology changed to equiaxed morphology [START_REF] Suryanarayana | [END_REF]52]. Compared to early milling stages (2 h), the particles size was more uniform after 18-h milling.

With the increase in milling time, the particles are deformed by the higher milling energy and they get compressed shapes and the particles size reduces to 2-5 lm, due to the fracturing of the powder before the cold-welding stage. It is observed that at 18-h milling time, the milled particles consist of 60 vol% with size \ 60 nm and 40 vol% with size \ 8 nm. However, while the particles with sizes less than 60 nm could be clearly seen, the particles with the sizes less than 8 nm are not seen as they may have agglomerated.

Particles Powder Morphology

At different milling times, the X-ray diffraction patterns (Fig. 2) of milled titanium alloy powders show the existence of (a ? b) phases only. The diffractograms corresponding to the lattice planes for a-hcp (100), (002), ( 101), ( 102), ( 110), ( 101), ( 112), (004), ( 104) and for b-bcc (100), ( 110), (004), (220) are indicated. As observed in the patterns, the maximum peak positions of all titanium alloy powders as a function of milling time are similar but their width progressively increases with increasing milling time and their intensity decreases in consequence of the crystallite size refinement [54].

After 18-h milling, the peaks corresponding to (100), (110), ( 004) and ( 220) disappeared. This can be due to the gradual diffusion of Al, V, Fe and Nb atoms into the Ti structure, since they have enough solid-state solubility in Ti in contrast to zero solubility of Ti in Al, Fe, Nb and V [55].

Previous investigations have also concluded the diffusion of Al atoms in Ti lattice structure as the cause for the disappearance of the Al diffraction peaks after 4-h milling [56]. Trends of Ti peak shift are specifically indicated for the Ti (101) peak at 2-h milling (Fig. 2). This shift may have been caused by one element solution in Ti. The shift in Ti peaks toward lower angles occurs due to the diffusion of Al (r Al = 1.82 A ˚), V (r V = 1.43 A ˚), Nb (r Nb = 1.43 A ˚) and Fe (r Fe = 1.26 A ˚) atoms in the Ti (r Ti = 2 A ˚) interstitial spaces. By the end of this shift, a-Ti (002) and a-Ti (103) diffraction peaks are replaced with b-Ti (110) and b-Ti (211) peaks; subsequently, the stabilization of b-Ti phase in the primary a-Ti matrix is consequently guaranteed. The new b-Ti phase lattice parameter is identical to the literature values and corresponds to the bcc structure with about a b = 3.3 A ˚. Peak expansions can be indication of fine crystallite size and increased lattice strains [57]. Nevertheless, chemical heterogeneities and/or stress gradients can also cause peak expansion [57].

To evaluate the effects of Al, Fe, Nb and V atom dissolution on the Ti peak shift, pure titanium's (101) peak position at different milling times was also investigated, and no shifts were observed. In the present research, solidstate solubilities of Al, Fe, Nb and V could therefore be the only cause for the shift of Ti peaks. Thus, after 18-h milling, the solid-state in situ synthesis of nanostructured Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V alloys with The evolution of crystallite size {D} of milled (a ? b) titanium alloy systems versus milling time is shown in Fig. 3. It is expected that with increasing milling time, the grain size was decreased reaching a lowest critical value, as the mechanical milling is the result of the competition between the breaking procedures and cold fusions of components that cause the activation and particle refine. At the steady point, the breaking and fusion speed will be balanced [58,59].

As shown in Fig. 3, the crystallite size decreased slowly with milling time. The same dependency on milling time of lattice parameters and crystallite size was decrypted by others [60]. Mixed powder crystallite sizes at 2 h are 50.02, 54.10 and 56.20 nm, which then decreases to 24.12, 31.19 and 33.03 nm after 12 h reaching the values of 21.21, 29.10 and 30.11 nm after 18 h for Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V, respectively.

These results are attributed to the grain size refinement, followed by an increase in the internal strains. The a and b phase lattice parameters were determined from the 2h positions of their particular diffraction peaks. The lattice parameter, a, for the b phase was determined from the bcc peaks (100), and the lattice parameter, a, for the a phase was determined from the hcp peaks (110), assuming that the c/a ratio is equal to 1.5963, which corresponds to the value in the base metal at room temperature. These two peaks represent the strongest diffraction peaks for each phase and provided good measurements throughout the entire milling time of the lattice parameter for each titanium powder.

A significant lattice parameter increase is clearly observed in Fig. 4 after 2-h milling. Nevertheless, at 18 h only the (a ? b) phases exist. As clearly observed in Fig. 4, with increasing milling time from 2 to 18 h milling, the lattice parameters (a b , a a & c a ) increase.

The, a b , lattice parameter increases with increasing milling time (3.253-3.312 A ˚), (3.250-3.307 A ˚) and (3.248-3.300 A ˚), which represents 1.18%, 1.75% and 1.16% of cell volume expansion for Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V, respectively. The lattice parameter, a a , also increases with increasing milling time (2.950-2.990 A ˚), (2.950-2.988 A ˚) and (2.946-2.984 A ˚), representing approximately a cell volume expansion of 1.35%, 1.28% and 1.27% for Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V, respectively.

Moreover, the lattice parameter, c a , increases with increase in the milling time, assuming that the c/a ratio is equal to 1.5963, which is interpreted by the peaks shift to smaller angles 2h [61]. Usually, the cell parameters increase with the increasing milling time for the most part due to: (I) pound formation by mechanical milling and (II) the introduction of first-order constraints [START_REF] Fellah | Sliding friction and wear performance of the nanobioceramic a-Al 2 O 3 prepared by high energy milling[END_REF].

Microhardness

The HV 0.05 Vickers microhardness tests of milled titanium alloys after 2-, 6-, 12-and 18-h milling are shown in Fig. 5. As presented, with increasing milling time the microhardness increased. The highest values of 660, 588 and 535 HV 0.05 were obtained after 18 h and the lowest values of 335, 320 and 304 HV 0.05 were obtained after 2-h milling time, for Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V, respectively. It is important to point out that the crystallite size reduction may have contributed to the increase in hardness of the titanium alloys, due to the strengthening mechanism by grain refinement.

Tribological Characterization

Friction Coefficient Evolution

The friction coefficients evolutions of nanostructured titanium alloys milled for different times (2, 6, 12 and 18 h), at applied load of 6 N, and a sliding velocity of 15 mm s -1 , are registered and displayed in Fig. 6. As observed in Fig. 6, the Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V alloys exhibit similar frictional behavior under the present experimental conditions. The coefficient of friction increases quickly through the first few meters and then decreases. After this stage, the variations become lesser and the coefficient of friction continuously increases slightly throughout the test end.

At an advanced stage, tracks are smoothened out and the coefficient of friction reaches a steady state. At 2-h milling, the specimens exhibit more fluctuations, which can be attributed to the higher roughness of specimens. In addition to this, with increasing surface hardness as a function of milling time, the real contact area decreases; therefore, the friction coefficient decreases. It is clear from Fig. 6 that the coefficient of friction showed a decreasing trend as a function of milling time, and it was lesser for Ti-6Al-4Fe samples for all the milling times. At 18-h milling, all samples showed the lowest and the highest coefficient of friction at 2 h.

As shown in Fig. 7, the average mean values of the friction coefficient are in the range of 0.402-0.226 for Ti-6Al-4Fe, while it varies between 0.542 and 0.297 for Ti-6Al-7Nb, and 0.586 and 0.423 for Ti-6Al-4V as a function of milling time from 2 to 18 h, respectively. It is clearly observed that the coefficient of friction decreases with increasing milling time, which can be attributed to the

Evolution of Volume and Wear Rate

As shown in Fig. 8, the wear rate of HIPed titanium alloys tends to decrease with increasing milling time from 2 to 18 h. It varied within the range of 1.34 9 10 -2 -0.66 9 10 -2 , 1.44 9 10 -2 -0.87 9 10 -2 and 1.92 9 10 -2 -1.51 9 10 -2 lm 3 N -1 lm -1 for Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V, respectively. It seems that the milling time has a significant influence on the wear volume and wears rate of HIPed titanium alloys. All the estimations made range from 10 -1 to 10 -2 (lm 3 N -1 lm -1 ).

The above results confirm that the tribological behavior of titanium alloys not only depends on the milling time, but also depends on their chemical composition. As depicted in Figs. 7 and8, the samples milled for 18 h display a good wear resistance and tribological behavior. This behavior of titanium alloys (especially those alloyed with Fe) milled at 18 h can be attributed to the grain size refinement and enhanced mechanical properties. After 3100 m of sliding, the surface morphology was examined by SEM to evaluate the wear mechanisms.

As can be seen, both the friction coefficient and the wear rates are observed to decrease with increasing milling time (Figs. 7 and8). This may be attributed to the better compaction of the transfer layer because of the increased frictional heat. At the same time, the extent of coverage of the transfer layer also increases, which results in lowering of the friction and wear rate. The results of wear rate could also be explained on the basis of the hardness of the milled samples and strengthening mechanisms, since the hardness of the samples milled for 2 and 6 h is less, and hence, they show a relatively higher rate of wear in comparison with other milling time of 12 and 18 h.

Figure 9 shows the worn surface morphologies of typical as-received titanium alloys and titanium alloys prepared after milling for 18 h. A moderate wear track was observed from the SEM micrograph samples. Moreover, wear debris particles with small sizes are also observed, which almost certainly were detached by plastic deformation. The enhancement in wear resistance of the 18-h milled titanium alloys may be attributed to its high hardness values. A A significant wear debris which is build up on the inside and the external periphery (Fig. 9) of the wear track was observed. In most cases of titanium alloys, it was observed that the weight loss was caused by adhesive wear and lamination defect.

As shown in Fig. 9, the extensive grooves in the case of as-received and HIPed samples can be seen (Fig. 9a), suggesting an abrasive wear mechanism. In the case of 18-h milled titanium-based alloys, the adhesive wear mechanism and lamination defect were observed (Fig. 9b). The extensive formation of slip bands possibly leads to the observed cracking in all the investigated materials.

In case of Ti-6Al-4Fe, the abrasion severity is much less compared to the investigated Ti-6Al-4V and Ti-6Al-6Nb alloys. Moreover, the smooth appearance of the worn surface correlates well with the low friction coefficient obtained with this material. It can be observed that in all wear tracks (Fig. 9), both adhesive and abrasive wear mechanisms existed. At a lower milling time (Fig. 9a), the predominant wear mechanism was abrasion, while at 18 h, it was adhesion (Fig. 9b). As indicated, the enhanced wear resistance of titanium-based alloy (especially Ti-6Al-4Fe) milled at 18 h can be attributed to its good proprieties (Fig. 5).

The titanium alloy microstructure also has a great influence on the material's properties, leading to a significant effect on the tribological properties as well. It is quite possible that Ti-based alloys undergo oxidation to form TiO 2 -rich layer during the wear process. Also, steel wear debris is transferred from the counter body. However, the non-protective nature of the oxide tribo-film leads to severe wear of the investigated materials [START_REF] Fellah | [END_REF]64].

After the wear tests, the wear debris corresponding to the as-received powder samples of Ti-6Al-4V is a mixture of more larger chip-like metallic particles and some fine wear particles, probably composed of elements from sample with some mixed iron particles from the counter face. Similar observations are made with regard to the wear debris corresponding to the milled Ti-6Al-7Nb sample. However, in this case the particle size is a bit finer and the chips are also of relatively smaller size. Some larger agglomerates of the debris that are flaked off from transfer layer could also be observed in the case of Ti-6Al-TNb (Fig. 9a).

The wear debris of sample milled for 18 h (Fig. 9b) presents some larger agglomerates flaked off from transfer layer apart from fine debris particles. Some larger lumps composed of very fine particles of white color probably of Al could also be observed in the micrograph. Under the conditions used in wet sliding, the mechanism of wear appears to be a mixture of adhesion and oxidation, as is evident from the presence of some metallic chip-like particles and fine oxide particles as shown in Fig. 9b.

Wear debris particles are generated by metal transfer or delamination mechanism or by plowing by the asperities of the counter face. Some of them get ejected out of the interface, whereas others may get trapped in between the sliding surfaces. The entrapped particles undergo repeated deformation before being oxidized. As the wear debris particles are repeatedly fragmented and oxidized, the fine particles generated become adherent to the wear surfaces and get agglomerated in some regions, particularly in grooves. These agglomerated clusters of particles are subjected to compression and compaction during further sliding, resulting in a compacted layer of wear debris. The debris generated during oxidative wear may affect the wear process in two different ways: (1) the oxide particles may get locked between the sliding surfaces and promote three-body wear, which should enhance volume loss in wear, and (2) these particles get compacted on the sliding surface to form a protective transfer layer, so that there are no scratch marks on it. A relatively brighter compacted oxide transfer layer could be observed in the SEM micrograph of worn surfaces as shown in Fig. 9.

However, the transfer layer is deeply torn on the surface of the base alloy.

It may be inferred that the beneficial role of wear debris in promoting wear resistance through formation of transfer layer is more prominent than its contribution to increased volume loss through three-body wear as only a few scratch marks have been observed on the surface of specimens. The presence of transfer layer has been observed on the surface of the all milled samples, but it appears to be more

Conclusions

Nanostructured titanium powder alloys were successfully prepared with varying milling times. The effect of replacing (V) by (Fe) and (Nb) on the tribological properties of HIPed Ti-6Al-4V is evaluated. Nanoscale grain sizes were produced in order to enhance its structural and tribological proprieties. Particular attention was paid to evaluate the milling time effects on the frictional behavior of these alloys. The following conclusions can be drawn from the study:

1. An increase in lattice parameters as a function of milling was observed, which reached the values of about a b = 3.312 A ˚and a a = 2.99 A ˚, a b = 3.307 A ånd a a = 2.988 A ˚and a b = 3.30 A ˚and a a = 2.984 A for Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4V, respectively, whereas the crystallite sizes decreased up to 21.21, 29.10 and 30.11 nm for Ti-6Al-4Fe, Ti-6Al-7Nb and Ti-6Al-4Fe, respectively. 2. Milling of titanium powder for 18 h significantly enhanced the hardness and wear resistance due to the grain size refinement. 3. With increasing milling time, the wear rate decreased, which is attributed to the enhanced mechanical properties of the powders due to their particle size refinement. 4. At longer milling time, the wear rate and the friction coefficient decreased significantly. At 18 h, the Ti-6Al-4Fe showed the lowest wear rate and friction coefficient as compared to Ti-6Al-4V which showed the highest values. 5. The chemical composition, grain size and crystallite played an important role in controlling the wear rate. The higher resistance of samples milled at 18 h (especially Ti-6Al-4Fe) is attributed to its enhanced mechanical and structural properties. 6. The wear mechanisms of nanostructured titanium alloys were established to be adhesion, abrasion and delamination defect.

Fig. 1

 1 Fig. 1 SEM micrographs of a Ti-6Al-4V, b Ti-6Al-7Nb, c Ti-6Al-4Fe powders particles after different milling times

Fig. 2

 2 Fig. 2 XRD patterns of milled titanium powders alloys versus: a 2-h and b 18-h milling

Fig. 3

 3 Fig. 3 Average crystallite sizes hDi (nm) of titanium powder alloy mixtures versus milling time

Fig. 4

 4 Fig. 4 Lattice parameters (A ˚) a and c for hcp and a for bcc versus milling time for titanium powder alloys

Fig. 5

 5 Fig. 5 Vickers hardness values of titanium alloys versus milling time

Fig. 6 Fig. 7

 67 Fig. 6 Friction coefficients of milled titanium alloys versus sliding distance

Fig. 8

 8 Fig. 8 Evolution of a wear volume (910 6 lm 3 ), b wear rate (910 -2 lm 3 N -1 lm -1 ) of milled titanium alloys under different milling times

Fig. 9

 9 Fig. 9 SEM micrographs corresponding to the worn surfaces of a as-received, b 18-h milled titanium alloys


Y Oussaiti¹

²׳ D Rideau¹

J R Manouvrier¹

V Quenette¹

H Wehbe-Alause¹

B Mamdy¹

A Lopez¹

G Mugny¹

M Agnew¹

E Lacombe¹

J Grebot¹

P Dollfus²

M Pala²

Verilog-A modeling of avalanche dynamics and quenching in Single-Photon Avalanche Diodes

INTRODUCTION

ingle-Photon Avalanche Diodes (SPADs) are reversebiased p-n junctions particularly profitable in Geiger mode. At this regime and when a large voltage is applied, the electric field in the depletion layer becomes strong enough that the charge carriers may acquire sufficient energy to trigger an avalanche through impact ionization mechanisms. Hence, when an absorbed photon leads to a photo electronhole pair generation, an avalanche current rises swiftly and it has to be quenched using external readout electronics (Fig. 1). Literature enumerates different types of quenching circuits for which a specific application is best-suited [START_REF] Cova | Avalanche photodiodes and quenching circuits for single photon detection[END_REF], and the SPAD related physics is reviewed by Spinelli and Lacaita [START_REF] Spinelli | Physics and numerical simulation of Single Photon Avalanche Diodes[END_REF]. Nowadays, these devices are gaining popularity in automotive industry (Light Detection and Ranging systems [START_REF] Takai | Single-photon avalanche diode with enhanced NIR-sensitivity for automotive LIDAR systems[END_REF]) and time-of-flight imaging [START_REF] Della Rocca | A high dynamic range SPAD pixel for time of flight imaging[END_REF]. This paper introduces a new VerilogA model to assess the SPAD devices buildup and its statistics. For accurate behavior description, we developed a Verilog-A diode model including the underlying physics and calibrated on Technology Computer Aided-Design (TCAD) [5]. Simulations confronted with measurements show an impact on the pulse width distribution, which will be discussed. Additionally, we investigated the voltage swing on the SPAD sensing node. This phenomenon is due to carriers overflow and has a value larger than the excess bias as it has been also demonstrated in [START_REF] Inoue | Modeling and Analysis of Capacitive Relaxation Quenching in a Single Photon Avalanche Diode (SPAD) Applied to a CMOS Image Sensor[END_REF]. This voltage drop impacts the power consumption and experimental data attests its relevance. Consequently, we discuss the qualitative outputs which in turn will allow us to evaluate the validity of the approach.

II. MODEL DESCRIPTION

As a representative device, we consider a SPAD with a large collection volume [START_REF] Veerappan | A Low Dark Count p-i-n Diode Based SPAD in CMOS Technology[END_REF]. The currents follow: (𝑣 𝑠 is the carriers saturation velocity, q the elementary charge and 𝑤 the effective multiplication region width) can initiate the avalanche. Hence, the total current 𝐼 𝑇 = 𝐼 𝑒 +𝐼 ℎ that is a succession of ionizations 𝐼 𝑇 =𝐼 0 (1+M+M²+M³+…) can be written in a familiar way as:

𝐼 𝑇 = 𝐼 0 1-M
where 𝑀 = 𝑀 𝑒 +𝑀 ℎ .

The coefficient 𝑀 𝑖 =𝛼 𝑖 . 𝑤 𝑖 depends on 𝑤 𝑛 and 𝑤 𝑝 [START_REF] Rees | Nonlocal impact ionization and avalanche multiplication[END_REF], as well as on the electric field through: 𝛼 𝑖 =𝑎 𝑖 .exp (

-𝑏 𝑖 𝐹 ) (𝑎 𝑖 and
𝑏 𝑖 are constants determined Van Overstraeten-de Man model [START_REF] Van Overstraeten | Measurement of the Ionization Rates in Diffused Silicon p-n Junctions[END_REF]). Therefore, an accurate estimation of the maximum electric field variation with the diode biasing is compulsory.

Even though analytical models exist in certain simple cases (e.g. abrupt junction), we used an empirical method that reproduces well-calibrated TCAD predictions on more complex SPAD architectures. As can be seen in Fig. 2, a change in the slope of the field versus voltage is observed at ∼17V, which occurs when the device starts to deplete its collection volume. Fig. 3 shows 𝑀 as a function of voltage, with 𝑤 𝑛 = 0.0736 µ𝑚 and 𝑤 𝑝 = 0.092 µ𝑚, one obtains M=1 for a voltage 𝑉 𝐵 = 17.1 V (25°C) which is close to the experimental breakdown voltage value. Fig. 4 (top) shows the SPAD current and voltage for various applied biasing as a function of time: the buildup takes typically ∼ 150𝑝𝑠 at operational voltage (∼23V), and can be even larger for lower biasing which is in line with TCAD simulations (not shown).

After the avalanche triggering, the rising current is lowered below the breakdown voltage and the diode rearmed. Fig. 4 (bottom) shows the SPAD current-voltage characteristics as well as the output pulse width. As can be noticed, the current swing exceeds the excess bias by a factor ∼ 1.4, which has also been recently reproduced in Ref. [START_REF] Inoue | Modeling and Analysis of Capacitive Relaxation Quenching in a Single Photon Avalanche Diode (SPAD) Applied to a CMOS Image Sensor[END_REF].

S

Fig. 1. SPAD diode connected to a high quench resistance 𝑅 𝑄 ∼ 500 𝐾𝛺 from the cathode node K. After avalanche triggering, generated charges accumulate and give rise to a measurable current which will be lowered and the device brought to its quiescent state. The inverter component allowing a square output pulse, reflects the swing voltage and hence, the overall time constant of the circuit.

III.

RESULTS

III. RESULTS

This swing voltage denoted 𝑉 𝐾𝑀𝐴𝑋 , affects the charge per pulse 𝐶 𝑃𝑃 (reflecting the power consumption) since 𝐶 𝑃𝑃 = 𝐶 𝐾 . 𝑉 𝐾𝑀𝐴𝑋 where 𝐶 𝐾 𝑖𝑠 the total capacitive charge in the cathode node (including parasitic). Fig. 5. shows the evolution of 𝐶 𝑃𝑃 with voltage 𝑉𝐻𝑉: as 𝑉 𝐾𝑀𝐴𝑋 increases with applied excess bias, 𝐶 𝑃𝑃 increases as well. Similarly, the Pulse width distribution, which can be inferred from the statistics of the buildup dynamics itself, is accounted for using a statistical distribution of α as testified in Fig. 6.

IV. CONCLUSION

We have developed a physics-based Verilog-A model for SPADs accounting for the buildup of the avalanche and its statistics. The model compares favorably with more complex TCAD mixed-mode predictions, but also with experimental data. This SPAD model allows SPICE simulations of realistic circuits, including quenching and sensing transistors. It provides an accurate way to estimate the power consumption and the voltage swing on the sensing node. (d)

  𝑒 . 𝐼 𝑒 + 𝑀 ℎ . 𝐼 ℎ -𝐼 𝑒 ]/𝜏 𝑒 , for electrons 𝑑𝐼 ℎ 𝑑𝑡 = [𝑀 𝑒 . 𝐼 𝑒 + 𝑀 ℎ . 𝐼 ℎ -𝐼 ℎ ]/𝜏 ℎ , for holes 𝜏 𝑖 is the transit time and 𝑀 𝑖 the multiplication factor (𝑖 represents 𝑒 for electrons and ℎ for holes). When an electron-hole pair is created, a current 𝐼 0 = 𝑞.𝑣 𝑠 𝑤

Fig. 2 .

 2 Fig. 2. The empirical model of the electric field (red dashed line) implemented in a Verilog-A code and expressed through a polynomial exploiting the high and low values of the field range. This model is calibrated on TCAD (solid).

Fig. 3 .

 3 Fig. 3. The mean multiplication factor versus voltage, for two temperatures. M= 1 corresponds to 𝑉 𝐵 = 17.1 volts at 25°C,

Fig. 4 :Fig. 6 .

 46 Fig.4: (a) The current buildup evolution with applied voltage (b) The voltage drop seen from the SPAD cathode exhibiting a swing. (c) SPAD I-V curve. (d) The output pulse 𝑣 𝑜𝑢𝑡 .


Maxence Ferrari
email: maxence.ferrari@univ-tln.fr

Hervé Glotin
email: glotin@univ-tln.fr

Marina Oger

Ricard Marxer

Mark Asch

Valentin Gies

Franc ¸ois Sarano

3D DIARIZATION OF A SPERM WHALE CLICK COCKTAIL PARTY BY AN ULTRA HIGH SAMPLING RATE PORTABLE HYDROPHONE ARRAY FOR ASSESSING INDIVIDUAL CETACEAN GROWTH CURVES

Passive acoustics allow us to study groups of animals and obtain information that could not be gathered by other means. In this paper we demonstrate a method using an ultra high velocity portable recorder, and apply it to a set of audiovisual recordings of a Sperm whale pod. We thus localise and segregate nearly all the clicks emitted by the animals that are filmed. To our best knowledge, this is the first characterization of these vocalizations correlated to visually identified individuals. This approach allows the construction of growth curves of the biggest toothed cetacean and opens avenues to possible acoustic individual signatures.

INTRODUCTION

Passive acoustics allow us to study groups of animals and obtain information that could not be gathered through other methods. In this paper we study a set of audiovisual recordings of a Sperm whale pod, acquired with a specific ultra high-frequency and small aperture antenna, shown in Figure 4. This antenna is equipped with a Go-Pro video camera and the JASON DAC [START_REF] Barchasz | Qualilife highblue: a novel smart low-power high speed and precision recorder for long term biodiversity surveys[END_REF][START_REF] Fourniol | Low-power wake-up system based on frequency analysis for environmental internet of things[END_REF], capable of recording up to 5 channels at a 2 MHz sampling rate. In this experiment it was set to record at 300 kHz. In 2018, the antenna had 3 hydrophones, thus increasing the number of cases that can be distinguished. This high velocity recording allows to reduce the size of the array, while keeping a good angle of arrival resolution. We then demonstrate how Time Difference of Arrival (TDoA) and a nonlinear solver can produce the localisation of nearly all the clicks emitted by the animals that are filmed. To our best knowledge, this is the first characterization of these vocalizations correlated to individuals that have been identified visually. It allows the construction of growth curves for individual whales, and opens avenues to possible acoustic individual signatures.

SPERM WHALE CLICK EMISSION

When a Sperm whale emits a click, it propagates through the water and reaches each hydrophone at a different time. By inverting the propagation equation, it is thus possible to obtain the location of a Sperm whale relative to an acoustic antenna [START_REF] Glotin | Whale cocktail party: real-time multiple tracking and signal analyses[END_REF]. The difficulty of this task resides in the estimation of the various times of arrival, and having good measurements of the speed of sound and of the shape of the antenna. Once the positions are obtained, the clicks can be clustered by individuals when locations of multiple individuals are sufficiently spaced. Note that this is the case when Sperm whales are foraging. While passive acoustics alone can obtain some degree of individual labeling with this technique, the obtained labels cannot keep the association with one individual over multiple dives. The association between multiple encounters being out of the question. The use of a video channel solves this issue since an expert can identify the same Sperm whale over multiple years using visual cues. The use of the video thus allows us to study the variation of an acoustical individual feature, such as the inter pulse interval (IPI) over the years, or perform behavioural studies.

Figure 1 shows the acoustic paths that a pulse follows. After being emitted at the base of the black arrow, the pulse will bounce back and forth in the spermaceti (light green in Fig. 1) according to the bent horn model [START_REF] Norris | A theory for the function of the spermaceti organ of the Sperm whale (Physeter catodon)[END_REF][START_REF] Møhl | The monopulsed nature of Sperm whale clicks[END_REF][START_REF] Zimmer | Off-axis effects on the multipulse structure of Sperm whale usual clicks with implications for sound production[END_REF]. This bounc- ing generates the multiple pulses that form a Sperm whale click as seen in Fig. 2. The nominal IPI is the time difference of arrival between P2 and P1. The difference between P1 and P0 depends on the angle θ (see [START_REF] Abeille | Estimation robuste par acoustique passive de l'intervalle-inter-pulse des clics de Physeter macrocephalus: méthode et application sur le parc national de Port-Cros[END_REF][START_REF] Ferrari | Study of a Biosonar Based on the Modeling of a Complete Chain of Emission-Propagation-Reception with Validation on Sperm Whales[END_REF] for details on the automatic estimation of the IPI and the effects of the angles). The nominal IPI is related to the length of the head of the individual, thus related to its body length. The length of the head function of IPI is linked to the sound celerity in the spermaceti. The extrapolation to the body size has been studied in [START_REF] Gordon | Evaluation of a method for determining the length of Sperm whales (Physeter c.) from their vocalizations[END_REF][START_REF] Rhinelander | Measuring Sperm whales from their clicks: Stability of interpulse intervals and validation that they indicate whale length[END_REF][START_REF] Growcott | Measuring body length of male Sperm whales from their clicks: the relationship between inter-pulse intervals and photogrammetrically measured lengths[END_REF].

CLICK DETECTION

The collected recordings include a cocktail party of various types of click trains from several animals as shown in Fig. 5.

For this study, the clicks were detected using the Teager-Kaiser energy operator [START_REF] Kandia | Detection of sperm whale clicks based on the Teager-Kaiser energy operator[END_REF] after a bandpass filter. Local maxima in sliding windows of 20 ms are then extracted. Only the maxima that are above a threshold, set by a rolling median, are kept. This rolling median is used to estimate the noise level.

TDOA ESTIMATION AND SOURCE LOCATION

In order to estimate the time difference of arrival (TDoA), we use a technique called geometric steered response power [START_REF] Ferrari | Efficient artifacts filter by density-based clustering in long term 3D whale passive acoustic monitoring with five hydrophones fixed under an autonomous surface vehicle[END_REF]. This technique improves the TDoA estimation by combining all the possible cross correlations. Since this technique does not use the shape of the antenna, some spurious clicks can be filtered out by using the constraints for the antenna shape [14]. Finally, using the shape of the antenna seen in Fig. 3 and Fig. 4, the TDoA and the position of the source are estimated.

DOA PLOTTING

Since we have the position of the source, we compute the Direction of Arrival (DoA) of the click at the position of the GoPro and compare it to its optical axis. Thus, we were able to plot the position of the source in the video [START_REF] Ferrari | Study of a Biosonar Based on the Modeling of a Complete Chain of Emission-Propagation-Reception with Validation on Sperm Whales[END_REF], allowing us to know if the Sperm whale that emitted the click is present in the video or not, and which one it is (Fig. 6). Some of these videos can be viewed at http:// sabiod.univ-tln.fr/pub/Sarano_2018/. The whales are then identified by Franc ¸ois Sarano's team, leading to a database of Sperm whale clicks with individual labels over multiple years. However, some parts of this data flow are still to be fully automated, and a large part of our database still lacks this labeling.

CONCLUSION

This database led to multiple studies. Here we present the measurements of the nominal IPI growth of young Sperm whales over four years. To our knowledge this is the first time that IPIs are followed over four years for the same individuals.

The IPIs in Fig. 7 are identified using the video recording where the youngs are present (Fig. 6). Then for each click linked to the right individual, the IPIs were manually measured [START_REF] Oger | Etude acoustique d'une population de cachalots, Physeter macrocephalus, de l'ile Maurice: croissance des jeunes individus par détermination de leur Intervalle Inter Pulse nominal[END_REF]. We then estimate the size of the heads from these estimated nominal IPIs, for a sound celerity of 1405m/s, by assuming spermaceti temperature of 30 • C [START_REF] Goold | Sound velocity measurements in spermaceti oil under the combined influences of temperature and pressure[END_REF], (see the right axes Fig. 7).

Because we know their birthdates (Tache Blanche: June 2011 ; Eliot: March 2011 ; Zoé: December 2013), we notice they have different sizes at 5 years old, and that Tache Blanche grows faster than Eliot whereas they have the same father, and their mothers are of the same size (respectively Delphine and Adélie [START_REF] Sarano | Underwater photo-identification of marine megafauna: an identity card catalogue of Sperm whales (Physeter macrocephalus) off Mauritius island[END_REF]). These differences might be due to individual and environmental conditions, such as the quality of the milk they still drink at 5 years old. Following this method, we are currently adding other individuals, juveniles and adults, that we recorded over the following years with our system. These growth curves will help to better understand the growth of Sperm whales, the biggest toothed predator at the top of the trophic chain. 1

ACKNOWLEDGEMENTS

We thank V. Sarano for her discussions, M. Mercier for discussions on ultra high sampled recordings, Jean-Luc [14] Y. Kuang, E. Ask, S. Burgess, and K. Åström, "Understanding ToA and TDoA network calibration using far field approximation as initial estimate.," in ICPRAM (2), pp. 590-596, 2012.

Figure 1 .

 1 Figure 1. Pulse path in the head of the Sperm whale in the leaky bent horn model. The spermaceti is in the green volume.

Figure 2 .

 2 Figure 2. Example of a Sperm whale click demonstrating its multipulsed structure P0, P1, P2. The nominal IPI is the time lapse between P1 to P2.

Figure 3 .

 3 Figure 3. Franc ¸ois Sarano holding the 2018 antenna near Physeters m.

Figure 4 .

 4 Figure 4. F. Sarano holding the antenna, version 2018, designed and built by H. Glotin (Image: F. Guerin). The hydrophones are three Cetacean Research, at the top two CR55 spaced by 60 cm, at the bottom a CR305.

Figure 5 .

 5 Figure 5. Spectrogram of Physeter macrocephalus, P.m. in near field measured by our system showing click trains up to 50 kHz. Stereo recording of the 14 th of April 2018, at 300 kHz Sampling Rate, 16 bits, hydrophones CR57 (Cetacean Research Tech., USA). Graphic with the help of M. Mercier (TripinLab).

Figure 6 .

 6 Figure 6. Group of four Sperm whales recorded in 2019, with two sources being pointed out (red dots): from Tache Blanche at the right, and from Alexander at the center (3 months old). Chesna is at the back-right (born 2018), Eliot is at the left (born 2011). Note that for visibility the dot stays on for 7 frames (at 25 frames per sec.). Hence two sequential click trains can be seen on the same frame, as in this example. Videos are available for viewing-see text.

  Jung's team (UBO) for the DNA analyses of mothers, fathers and juveniles. We thank for financial support AID French Defense Agency, Region Haut de France, Programma Interreg Italia-Francia Marittimo 2014-2020 Progetto GIAS, ANR-20-CHIA-0014-01 national Chair in Artificial Intelligence ADSIL (H. Glotin), and ANR-18-CE40-0014 SMILES. We thank SMIoT UTLN and INPS for soundcard co-development. This work is part of a longitudinal study directed by Longitude 181 Association under the Maubydick program initiated by Marine Megafauna Conservation Organisation. Special thanks to Label Bleu, who provided most of the video, to Navin Rishinand Boodhonee and the Blue water diving center, as well as all the ecovolunteers of MMCO. Mauritian public authorities greatly helped the Maubydick project, in particular the Mauritian Prime Minister Office, the Continental Shelf, Maritime Zones Administration & Exploration (CSMZAE, Dr Réza Badal and his team), the Albion Fisheries Research Center (AFRC, Chief Scientific officer Mr Satish Kadhun), the Mauritius Film Development Corp. (MFDC, Mr Sachin Jootun & Mrs Eliana Timol) and the Tourism Authority (TA, Mrs Khoudijah Boodoo, ex-Director). We thank Olivier Adam's team for joint studies.

Figure 7 .

 7 Figure 7. Measurements of the nominal IPI of three youngs, over 3 years for two of them, showing their growth. Respective birth dates are: Tache Blanche June 2011; Eliot March 2011; Zoé December 2013.