Pierre Margerit
email: pierre.margerit@ensam.eu

Extraction of dispersion surfaces: a review of applications, methods and open questions

The use of dispersion surfaces to describe the wave propagation characteristics of a structure is motivated using experimental data. A review of the different approaches allowing for a high-resolution estimation of the wave parameters from a full-field measurement is given. The sensitivity of each method with regard to measurement noise and model bias is compared throughout numerical examples.

Introduction

The development of dynamical full-field measurement techniques [START_REF] Baqersad | Photogrammetry and optical methods in structural dynamics-a review[END_REF] opens the door for the formulation of original structural identification procedures using the rich data available [START_REF] Avril | Overview of identification methods of mechanical parameters based on full-field measurements[END_REF]. More precisely, these new wideband methods aim to bridge the gap between modal analysis [START_REF] Reynders | System identification methods for (operational) modal analysis: review and comparison[END_REF] and ultrasound techniques [START_REF] Chimenti | Review of air-coupled ultrasonic materials characterization[END_REF], respectively confined to the low and high ends of the frequency spectrum. In addition, a particular effort is dedicated to inverse methods with formulations that are insensitive to boundary conditions and applied loads in order to open the door to in-situ implementations. Indeed, such methods focus on the characterization of the transfer mechanisms in the structure of interest, represented by its governing equations of motion, or, equivalently, its wave propagation characteristics.

The linear equations of free harmonic motion of any macroscopically invariant structure can be expressed in the form of a differential system of homogeneous equations L u, {p i } = 0, where u(x) is a kinematic field (e.g generalized displacement or velocity), x the spatial coordinate and {p i } is a set of coefficients sampling the structural viscoelastic behavior. The robust estimation { p i } from a partial and noisy measurement u remains an active field of research in different communities ranging from (multidimensionnal) system identification to (architectured) material characterization.

Two main approaches may be identified in the litterature: (i) direct minimization of the residual L u, { p i } , using either finite difference schemes (e.g Force Analysis Technique [START_REF] Pezerat | Identification of vibration sources[END_REF][START_REF] Leclère | Practical implementation of the corrected force analysis technique to identify the structural parameter and load distributions[END_REF][START_REF] Marchetti | Development of the corrected force analysis technique for laminated composite panels[END_REF]) or a weak formulation (e.g Virtual Fields Method [START_REF] Pierron | The virtual fields method: extracting constitutive mechanical parameters from full-field deformation measurements[END_REF][START_REF] Berry | Identification of dynamic loading on a bending plate using the virtual fields method[END_REF]); (ii) decomposition of the measured field u on a set of elementary solutions u e (x) of the operator L , using for example Green functions [START_REF] Cuenca | Measurement of complex bending stiffness of a flat panel covered with a viscoelastic layer using the image source method[END_REF][START_REF] Marchetti | Experimental methodology to assess the dynamic equivalent stiffness properties of elliptical orthotropic plates[END_REF] or plane waves [START_REF] Ferguson | The estimation of wavenumbers in twodimensional structures[END_REF][START_REF] Berthaut | K-space identification of apparent structural behaviour[END_REF][START_REF] Cherif | Damping loss factor estimation of two-dimensional orthotropic structures from a displacement field measurement[END_REF][START_REF] Geslain | Spatial laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems[END_REF][START_REF] Margerit | High resolution wavenumber analysis (hrwa) for the mechanical characterisation of viscoelastic beams[END_REF][START_REF] Margerit | The high-resolution wavevector analysis for the characterization of the dynamic response of composite plates[END_REF][START_REF] Boukadia | An inverse convolution method for wavenumber extraction (income): Formulations and applications[END_REF].

The first approach provides a direct relation between the measured data and the parameters to be identified and can be generally applied to a wide range of operators. However, the full kinematics of the structure have to be measured, thus reducing the application to simple models (e.g thin plates or beams). In addition, high-order derivatives have to be estimated (e.g fourth-order for the bending motion), leading to a relative sensitivity of the identified parameters to measurement noise. By contrast, the second approach using elementary solutions of the operator overcome the observability limitation, as only the projection of these solutions on a partially observed kinematic field is needed. However, the dependence of these solutions as a function of the parameters p is not direct, especially in the case of Green functions that are known analytically in very few situations (e.g elliptical orthotropy for thin plates). However, the approximation of a measured field with plane waves does not require a priori knowledge of the underlying operator, thus providing a very general framework for the identification of the system dispersion characteristics. This is of particular interest when the behavior of the structure of interest is not known, and may be viewed as an intermediate step in the identification process.

x z 1 z 2 e -i k x Ω x Ω z Ω (a) Beam z x 1 x 2 e - i k 1 x 1 e -i k 2 x 2 Ω x Ω z Ω k (b) Plate v 1 v 2 Ω x Ω z (c) Periodic honeycomb

Dispersion surfaces

Plane wave propagation is associated to the formalism of wave guides, that are characterized by a characteristic microstructure Ω z that is tiled over a macroscopic domain Ω x such that the full structural domain can be expressed as Ω = Ω z ⊗ Ω x (see the figure 1):

• Homogeneous waveguides with a charateristic section Ω z

• Periodic waveguides with a unit cell Ω z tiled with periodicity vectors {v i } such that x and z denote the separation of scales (resp. macroscopic and microscopic)

From the theoretical point of view, plane waves are generally expressed as:

u e (x, z) = Φ(z) e -i k•x (1)
where k is the so-called wavevector and Φ(z) the corresponding wave mode. Injecting the equation (1) in the differential operator L leads to a general expression of the dispersion equations:

D(k, ω)Φ(z) = 0 (2)
where the dependence on the frequency ω have been explicitly written in order to show that the solutions of the equation (2) form a discrete set of hypersurfaces in the (k, ω) Fourier space: the so-called dispersion surfaces.

Experimentally, the dispersion surfaces can be obtained by computing the frequency-wavevector spectrum as the multidimensional Discrete Fourier Transform (DFT) of the structure response measured over spatiotemporal grid of (N 1 , . . . , N d , N t ) points with uniform spacings (∆x 1 , . . . , ∆x d , ∆t), so that L i = N i ∆x i and T = N t ∆t.

In the figure 2, the experimental dispersion surfaces of a Sandwich plate made of an isotropic foam core between two Carbon Fiber Reinforced Polymer (CFRP) skins are displayed. The theoretical dispersion surfaces, computed with the Spectral Finite Element Method [START_REF] Bartoli | Modeling wave propagation in damped waveguides of arbitrary cross-section[END_REF], are superimposed. The different propagating wave modes are well separated: out-of-plane bending, in-plane shear and traction. Surprisingly, the signature of the fluid-structure interaction with the surrounding air also appears as an additional dispersion surface (see [START_REF] Margerit | The high-resolution wavevector analysis for the characterization of the dynamic response of composite plates[END_REF] for more details).

In the figure 3, the experimental spatio-temporal response of a PVC beam (L × S = 1m × 15 2 mm 2 , E = 5GPa, G = 2GPa, ρ = 1380kg/m 3 ) is displayed with its corresponding frequency-wavenumber spectrum. For vizualization purposes, the bending and torsion have been separated by measuring the structure response on several points over the beams width. The dispersion curves corresponding to the two motion mechanisms are well captured on the frequency-wavenumber spectrum (as hypersurfaces in a 2D domain) and show an excellent agreement with theoretical predictions (see [START_REF] Margerit | High resolution wavenumber analysis (hrwa) for the mechanical characterisation of viscoelastic beams[END_REF] for more details).

Hence, dispersion surfaces appear as an excellent candidate for the characterization of the dynamical behavior of structures: (i) they can easily be obtained as the Fourier transform of a full-field measurement of the structure kinematics; (ii) few assumptions are involved (only the system linearity), making the approach relatively general; (iii) they can be compared to theoretical predictions of the wave dispersion (Eq. ( 2)) without taking into account boundary conditions or applied loads, thus providing a framework for in-situ identification procedures. However, the use of the Fourier transform to obtain the frequency-wavenumber spectrum is submitted to the uncertainty principle:

∆x i ∆k i = 2π/N i (3)
which applies also to the couple (ω, t) and gives the maximum wavenumer k max i = 2π/∆x i and the wavenumber resolution ∆k i = 2π/L i . While k max i can be increased by a refinement of the measurement mesh, the wavenumber resolution ∆k i is bounded by the size of the structure of interest. This strong constraint thus motivates the development of dispersion surface extraction methods that overcome the resolution limitation.

Extraction methods

The development of alternatives to the Fourier transform for the extraction of dispersion surfaces is motivated by two needs: (i) overcoming the wavenumber resolution limit; (ii) allowing the extraction of the imaginary For the sake of readability, the following developments are given for the elementary case of a scalar field u(x) ∈ C measured along a unidimensional grid {x n = x 0 + n∆x, n = 0 . . . N -1} and corresponding to the harmonic response of a structure, such that the frequency ω is fixed and the waves are parameterized by a scalar wavenumber k ∈ C. The generalization of the reported methods to multidimensional waves and grids is, in most of the case, relatively straightforward and is not developed here (the reader is thus referred to the given references for implementation details).

Most method formulations start with a combination of plane waves s as signal model:

s({a r } , {k r } , x) = R r a r exp(i k r x) (4)
where the signal order R and the sets of wave amplitudes {a r } ∈ C R and wavenumbers {k r } ∈ C R have either to be estimated or assumed.

In the following, the review of available methods is separated into three approaches: (1) non-linear minimization, (2) low-rank approximation and (3) kernel identification procedures.

Non-linear regression

Given a signal order R, the set of wavenumbers {k r } can be estimated from the measured field using the following double minimization problem:

k r = arg max {kr}∈C R min {ar}∈C R n s({a r } , {k r } , x n ) -u(x n ) 2 (5)
which is solved as a two-step procedure: (i) for a given set of wavenumbers, determine {a r } using linear least-squares; (ii) find the set {k r } that minimizes the error residual.

In order to reduce the complexity of this non-linear search, one can assume: (i) a unitary signal order (R = 1), as in the IWC [START_REF] Berthaut | K-space identification of apparent structural behaviour[END_REF][START_REF] Cherif | Damping loss factor estimation of two-dimensional orthotropic structures from a displacement field measurement[END_REF] (Inhomogeneous Wave Correlation) and the SLaTCoW [START_REF] Geslain | Spatial laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems[END_REF] (Spatial LAplace Transform for COmplex Wavenumber) methods; (ii) a relation between the wavenumbers, as in the McDaniel method [START_REF] Mcdaniel | Estimation of structural wave numbers from spatially sparse response measurements[END_REF] where the four bending wavenumbers are searched simultaneously as {k, i k, -k, -i k}. While the latter strategy is limited to simple structures where the wavewnumber relation is known a priori, it is associated to an exact signal model thus is very robust to noise and overcome the wavenumber resolution (see [START_REF] Margerit | High resolution wavenumber analysis (hrwa) for the mechanical characterisation of viscoelastic beams[END_REF]). This is not the case of both IWC and SLaTCoW methods, that can be regarded as the maximization of a normalized Laplace Transform and inherit its resolution limitations. Despite their respective limitations, these approaches are relatively simple to implement and thus are a good candidate for a first extraction of wave characteristics from a measured structure response before the application of more sophisticated methods.

Low-rank approximation

The purpose of the following family of methods is to provide: (1) a criterion to estimate the signal order R;

(2) a direct (optimization-free) identification of the wavenumbers {k r }.

The vector s = ⊤ [s(x 0 ) . . . s(x N -1 )] ∈ C N representing the pure signal model ( 4) sampled along a uniform grid of points can be rewritten as follows:

s = V • a (6)
where V ∈ C N ×R is the so-called Vandermonde matrix with components V n,r = exp(i k r x n ) and a ∈ C R contains the wave amplitudes {a r }.

The Vandermonde matrix exhibits a so-called rotational or shift invariance between two lines -or sampling locations-such that V n+1,r = V n,r exp(i k r ∆x), or in matrix notation:

V ↑ = V ↓ • Z (7)
where V ↑ and V ↓ respectively contain the N -1 last and first rows of V and Z = diag ({exp(i k r ∆x)}) ∈ C R×R is the pole matrix. Hence the wavenumbers k r = -i log(Z r,r ) /∆x can be estimated once the Vandermonde matrix is known; which is in practice impossible.

Assuming a low signal order such that R < N , the signal s is of low rank; in particular, one can define a Hankel matrix H ∈ C M ×Q with components H m,q = s m+q that is rank-defficient so that the following subspace decomposition applies:

C ss = H H • H = H W • diag (λ) • W + H W ⊥ • diag (λ ⊥ ) • W ⊥ (8)
where W ∈ C Q×R is the signal subspace associated to non-zero eigenvalues λ ∈ C R and

W ⊥ ∈ C Q×(Q-R)
is an arbitrary orthogonal associated to vanishing eigenvalues λ ⊥ ∈ C (Q-R) . While the signal order R can be estimated by observing the number of non-zero eigenvalues, it can also be shown that W spans the same subspace as the Vandermonde matrix so that W = V • T with T ∈ C R×R a transfer matrix. Thus the pole matrix Z can be directly estimated from the signal subspace as the eigenvalues of the spectral matrix F:

F = W ↑ † • W ↓ = T -1 • Z • T (9)
where W ↑ and W ↓ also represent shifted versions of the signal subspace and • † is the pseudo-inverse. An important property of the signal subspace is that it is left unchanged by an additional white gaussian noise applied to the signal. This gives the preceding approach a relative robustness with regard to experimental noise and modeling error.

While the subspace decomposition was first used in the MUSIC algorithm (MUltiple SIgnal Classification) [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF] and the use of the shift invariance of the Vandermonde matrix proposed in the Matrix Pencil method [START_REF] Hua | Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise[END_REF], the ESPRIT algorithm (Estimation of Signal Parameters via Rotationnal Invariance Techniques) [START_REF] Roy | Esprit-estimation of signal parameters via rotational invariance techniques[END_REF] was the first to make use of both techniques via the equation [START_REF] Berry | Identification of dynamic loading on a bending plate using the virtual fields method[END_REF]. It thus represents an excellent candidate method for the direct estimation of a set of wavenumbers from a noisy measurement.

However, the ESPRIT algorithm, when applied to full-field measurement performed on a structure, suffers from the limitations associated to the signal model ( 4): (i) if the rank R of the signal is high; (ii) if the signalto-noise is low; (iii) if the wavefront is not plane, for example in the case of a thin isotropic plate excited by a point source. In these situations, the signal order R might be difficult to estimate even with the variety of signal order estimation criteria that have been proposed in the literature: MDL (Minimum Description Length) [START_REF] Grünwald | The minimum description length principle[END_REF], ESTER (ESTimation of ERror) [START_REF] Badeau | A new perturbation analysis for signal enumeration in rotational invariance techniques[END_REF], SAMOS (Subspace-based Automatic Model Order Selection) [START_REF] Papy | A shift invariance-based order-selection technique for exponential data modelling[END_REF], among others. In addition, the non-plane wavefront geometry is associated to systematic errors on the wavenumber estimation, that is mainly concentrated on the imaginary part of k r corresponding to the wave decay. As a consequence, in practice the precise identification of wave characteristics with ESPRIT is possible in situations where a signal model ( 4) with a low signal order R is a good approximation of the measured signal: (i) for structures with 1D propagation space -the macroscopic coordinate xthe signal model is exact; (ii) when their is a few number of components in a multidimensional signal (e.g if the reflections at the boundaries select particular wave propagation directions); (iii) in the far-field of a source, when the observed signal is close to a plane wave.

Spectrum relocalization

One strategy to overcome the high rank estimation problem is to artificially reduce the signal rank by filtering techniques. More precisely, a collection of M narrow-band-pass filters f (m) can be applied to the signal in order to obtain a collection of reduced-order signals ξ (m) that can be later processed by an ESPRIT algorithm with low order [START_REF] Ege | High-resolution modal analysis[END_REF].

The new problem that arises is then the choice of the filter parameters: central wavenumber k m , order Q. A systematic choice can be to choose the filter weights f (m) ∈ C Q as a windowed complex exponential:

f (m) q = w q exp(i k m q) (10)
with w ∈ R Q the window function and k m = 2π m/M . Then it is easy to see that the ξ (m) = f (m) * s are the rows of the weighted column-wise DFT of the Hankel matrix H: 1) . . .

χ = ⊤ [ξ (
ξ (M ) ] = F 1 H • diag (w) (11)
which drastically reduces the computational complexity of the filtering step. Finally, the ESPRIT method can be applied to the individual rows of χ.

The equation ( 11) also shows that this particular choice of filters and the application of ESPRIT corresponds to the identification of the variation of the signal phase associated to each DFT bands as a function of the coordinate shift. This concept is at the basis of spectrogram reallocation methods [START_REF] Fulop | Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram, with applications[END_REF], where the phase derivative is associated to a local frequency. Equivalently, fixing R = 1 when applying ESPRIT corresponds to the estimation of the dominant wavenumber associated to each DFT band; a kind of spectrum relocalization technique.

While the spectrum relocalization technique provide a systematic method to get around the signal order estimation problem, the bias associated to the approximation of non-planar wavefront geometries with plane waves is still present, preventing a precise determination of the wave attenuation characteristics. In addition, the choice of the window, necessary to lower border effects, has an effect on the signal model which is no longer exact.

Kernel-based

Taking a step back to the Prony family of methods that resulted from its pioneering works [START_REF] Prony | Essai experimental et analytique[END_REF], it is well known that the 1D signal model in equation ( 4), when measured along a regular grid, satisfies a recurrence re-lation of the form s * g = 0, where g ∈ C R+1 is the signal kernel which coefficients define a characteristic polynomial P :

P (z) = R r=0 g r z r = g R R r=1 (z -z r ) (12)
where the {z r = exp(i k r ∆x)} are, as in equation ( 7), the signal poles. The two expressions below show that these signal poles are the roots of the z-transform of g, or equivalently that the wavenumbers {k r } are the roots of the Laplace transform of g. The vanishing convolution between s and g can be expressed as a system of homogeneous equations using, once again, an Hankel matrix H of size M × R:

H • g = 0 ( 13
)
and provides a way to estimate g from a measured signal either by (i) Ordinary Least-Squares (OLS) assuming that g 0 = 0 or (ii) Total Least-Squares (TLS) that identify g as the eigenvector w ⊥ associated to the smaller eigenvalue λ ⊥ of the covariance matrix C ss in equation ( 8).

This signal deconvolution technique has been widely studied in the system identification community [START_REF] Markovsky | Low rank approximation: algorithms, implementation, applications[END_REF], where it is applied to time data. In this 1D case, the equivalence between the plane wave decomposition (4) and zero-convolution kernel ( 13) is well established; moreover, the roots {z r } of the 1D kernel g can be estimated directly using the companion matrix of P (z) (12) [START_REF] Edelman | Polynomial roots from companion matrix eigenvalues[END_REF].

Recently, the INCOME algorithm (INverse COrrelation MEthod) [START_REF] Boukadia | An inverse convolution method for wavenumber extraction (income): Formulations and applications[END_REF] proposed to apply this method to multidimensional measurements using a TLS identification. In this case, the plane wave -zero kernel equivalence breaks: for example, a circular wave signal (e.g ponctual bending source in an isotropic thin plate) could fullfill the equation ( 13) with a small-order kernel g, while needing an infinite number of plane wave components to be approximated by ( 4). As a consequence, the kernel identification method is able to give an unbiased estimate of the wave attenuation part, even for multidimensional signals. Moreover, the corresponding multivariate polynomial P (z 1 , . . . , z D ) (with D the kernel dimension) is still an implicit definition of the structure's dispersion surfaces, that are the affine varieties of P (locations of its zeros) in C D . While the kernel estimation procedure is very simple to implement, determining the dispersion surfaces from g is in fact the hardest part, and involve some multidimensional complex root-finding schemes (or affine variety determination). In addition, the determination of g as the first eigenvector of the orthogonal subspace W ⊥ makes it highly sensitive to noise.

Numerical comparison 4.1 Noise Sensitivity

The noise sensitivity of each aforementioned wavenumber estimation methods is investigated here using an elementary uni-dimensional signal model ( 4) with a single component (R = 1) and a reduced number of samples N = 10. The variance of the wavenumber estimations is computed using 1000 Monte-Carlo random sampling of an additive noise corresponding to a signal to noise ratio (SNR) of 10 3 . In order to make the results more general, the obtained variances are compared to the Cramer-Rao Bound (CRB) of this particular signal model [START_REF] Kumaresan | Estimating the parameters of exponentially damped sinusoids and polezero modeling in noise[END_REF]. In particular, when the wavenumber is purely real (zero wave attenuation), the CRB is equal to:

CRB 0 = 12 SNR × N 3 (14)
Which shows that the wavenumber estimation should not be sensitive to its real part. This can be clearly observed in the figure 4a, where the normalized variance var(k) /CRB 0 is shown, for reach method, as a contour plot in the z = exp(i k) plane, and demonstrates that the results are invariant with Re{k} = arg(z).

The figure 4b thus focuses on the line Re{k} = 0 to show the effect of wave attenuation in the estimation variance. It can be observed that both Non-Linear Fit and ESPRIT methods are optimal in the sense that they provide estimates with variances on the lower Cramer-Rao Bound. By contrast, the Kernel-based estimation is characterized by a high variance (with a factor close to 2); this is due to the sensitivity of the leastsignificant eigenvectors of the signal covariance matrix with regard to noise. The least optimal method seems to be, in this particular situation, the spectrum relocalization method (RELOC); this can be related to the loss of signal model accuracy and data length associated to the pre-filtering technique.

Bias

The application of wavevector extraction methods on a multidimensional signal is submitted to systematic error, as the approximation of the measured kinematic response by a sum of waves (with a multidimensional version of the signal model ( 4)) is not exact, by contrast to the 1D case.

To demonstrate the consequence of such a biased approximation, this second numerical study considers the case of a free harmonic response corresponding to an isotropic Helholtz operator:

L (u) = ∂u(x) ∂x 1 + ∂u(x) ∂x 2 + k 2 u(x) = 0 (15)
which, associated to a point source at x 0 , generate a response u(x) ∝ J 0 (k ∥x -x 0 ∥), where J 0 is the Bessel function of the first kind.

The result of the analysis is shown on the figure 5, where the relative errors associated to the real part κ and the decay factor γ in the estimation of the wavenumber k = κ(1 -i γ) = 6π L (1 -0.05 i) are given as a function of the distance d to the source x 0 . It can be observed that all methods based on a plane wave approximation (4) (NLFIT, ESRRIT and RELOC) provide poor estimates of the spatial decay γ (dashed lines), that is highly influenced by the geometrical decay and the cylindrical wavefront geometry. As the distance to the source increase, the far-field wavefront straightens thus the estimation error decreases. This model bias has a smaller influence on the real part of the wavenumber κ, which corresponding estimation error quickly decreases with the distance to the source. By contrast to plane-wave expansion methods, the KERNEL-based identification provides extremely good estimations of both real and imaginary parts of the wavenumber (purple curves in the figure 5). Indeed, no plane wave assumption is involved in this method; instead, the equation ( 13) is used to estimate the

that corresponds to a discrete version of the operator L in [START_REF] Geslain | Spatial laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems[END_REF]. The wavenumber is then estimated as the root of the Fourier transform of G that is closest to the real plane. Even in the absence of noise, the wavenumber estimation given by the KERNEL-based method is not exact; this can be related to two aspects: (i) in the near-field regime (close distnace to the source), the intrinsic dispersion of the chosen kernel which associated affine variety (locii of the roots of the Fourier is not perfectly isotropic; (ii) in the farfield regime, the signal invariance along the wavefront plane making estimation (13) badly conditioned, producing the error oscillations that can be observed on the figure 5.

Conclusions

In this communication, the use of dispersion surfaces as an intermediate step to structure identification is motivated throughout experimental examples. Then, a number of available approaches are presented, allowing to increase the wavevector resolution when few wavelengths are contained in the signal and/or an estimation the spatial wave decay is needed. Finally, the performance of all methods is compared with two numerical examples, giving insights on the sensitivities of the estimates to noise and signal modeling errors.

Despite its rather simplicity of implementation, the non-linear fitting technique (NLFIT) is limited to elementary cases (e.g when only one wave is present in the signal or when the different wavenumbers are related with in a known sequence). Thus the use of the more general ESPRIT method, that provides a closed-form estimate of the wavevectors, is advised when the measured signal can be well approximated by a relatively reduced number of plane waves: in one-dimensional signals, or for multidimensional structures in the far-field regime. In order to increase the maximum signal order that can be estimated with ESPRIT-type approaches, a spectrum RELOCalization technique was presented. However, the influence of the chosen window function on the obtained estimates is still to be investigated. Finally the KERNEL-based approach seems to be a very good candidate for the identification of dispersion surfaces of multidimensional structures. However, its high sensitivity to noise limits for now its application to experimental signals.

Figure 1 :

 1 Figure 1: Typical waveguides

Figure 2 :

 2 Figure 2: Dispersion surfaces associated to a sandwich CFRP plate.

Figure 3 :

 3 Figure 3: Dispersion of torsion and bending waves in an homogeneous PVC beam

0 Figure 4 :

 04 Figure 4: Variance of the wavenumber estimation normalized by the zero-attenuation Cramer-Rao Bound CRB 0 (14), for a model (4) with R = 1 and N = 10.

Figure 5 :

 5 Figure 5: Wavevector estimation bias associated to cylindrical wave field. Effect of the distance d to the source x 0 in the estimation of the isotropic wavenumber k = κ(1 -i γ) = 6π L (1 -0.05 i).


Katharina Häckl

Andrea Mühlbauer

Jesus Ontiveros

Sinisa Marinkovic

Boris Estrine

Werner Kunz
email: werner.kunz@chemie.uni-regensburg.de

Véronique Rataj
email: veronique.rataj@univ-lille1.fr

Jesús F Ontiveros

Véronique Nardello-Rataj

Carnitine alkyl ester bromides as novel

Keywords: Carnitine alkyl esters, Cationic, Ionic liquids, Hydrotropes, Surfactants

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

Introduction

Carnitine, also called 3-hydroxy-4-N-trimethylaminobutyrate, is a naturally sourced quaternary ammonium compound [START_REF] Bremer | Carnitine-metabolism and functions[END_REF]. It occurs in different biological materials, like animal tissue, plants, fruits and microorganisms, whereas mammal muscles are the richest source. In the organism, it can be partly bio-synthesized from the lysine and methionine amino acids. It is needed for fatty acid transportation through mitochondrial membranes [START_REF] Fritz | The effect of muscle extracts on the oxidation of palmitic acid by liver slices and homogenates[END_REF][START_REF] Friedman | Reversible enzymatic acetylation of carnitine[END_REF]. As a chiral compound, L-carnitine is the biologically active form. It is commercially available as a nutritional supplement and supposed to enhance fat burning. Carnitine is a zwitterion in its neutral state and combines the functional groups of choline (hydroxyl group) and betaine (carboxylic group). While choline and its derivatives are well known in the field of green alternative solvents, surfactants and catalysis [START_REF] Kunz | Low Toxic Ionic Liquids, Liquid Catanionics, and Ionic Liquid Microemulsions[END_REF][START_REF] Shahabi | One-pot synthesis of quinoline derivatives using choline chloride/tin (II) chloride deep eutectic solvent as a green catalyst[END_REF][START_REF] Zhao | Biocompatible Deep Eutectic Solvents Based on Choline Chloride: Characterization and Application to the Extraction of Rutin from Sophora japonica[END_REF][START_REF] Klein | Choline carboxylate surfactants: Biocompatible and highly soluble in water[END_REF][START_REF] Klein | Choline alkylsulfates-new promising green surfactants[END_REF], carnitine has only been scarcely studied in this context up to now. This quaternary ammonium, which is authorized in food, pharmaceutics and cosmetics, is therefore a relevant candidate for the design of novel biobased, biocompatible and non-toxic amphiphiles [START_REF] Kamm | Polycarnitine--a new biomaterial[END_REF].

A lot of literature can be found on acylcarnitine compounds regarding their biological functions [START_REF] Reuter | Carnitine and Acylcarnitines[END_REF][START_REF] Inazu | Physiological functions of carnitine and carnitine transporters in the central nervous system[END_REF][START_REF] Fonteles | Support of kidney function by long-chain fatty acids derived from renal tissue[END_REF][START_REF] Borrebaek | The role of acyltransferases in fatty acid utilization[END_REF]. Most reports are on their formation in the organism during fatty acid transport through the inner mitochondrial membrane. Only a few studies dealing with the surfactant properties of long-chain acylcarnitine compounds have been published [START_REF] Pande | Uneven distribution of palmitoyl carnitine in solutions because of migration to air/water interphase[END_REF][START_REF] Yalkowsky | Some micellar properties of long-chain acylcarnitines[END_REF][START_REF] Yalkowsky | Potentiometric titration of monomeric and micellar acylcarnitines[END_REF].

Goñi et al. have investigated the surfactant properties of palmitoylcarnitine, a zwitterionic amphiphile used as a biological detergent [START_REF] Goñi | Palmitoylcarnitine, a surface-active metabolite[END_REF]. The authors report its low critical micellar concentration and its ability to solubilize biological membranes. In contrast, literature on linear carnitine amphiphiles formed by modification of the carboxyl group is rather rare. De Maria et al. and Cipollone et al. have shown the formation of cubic, hexagonal and lamellar phases as well as multilamellar vesicles by different diesters of carnitine depending on the alkyl substituents [START_REF] De Maria | Lyotropic mesomorphism of alkyl esters of acyl-L-carnitines[END_REF][START_REF] Cipollone | Formation of micelles and liposomes from carnitine amphiphiles[END_REF]. A detailed study of the self-assembly of carnitine dodecyl amide (i.e. 3-(dodecylcarbamoyl-2-hydroxypropyl)-trimethylammonium) has been published by Patra et al. [START_REF] Patra | Cationic vesicles of a carnitine-derived single-tailed surfactant: physicochemical characterization and evaluation of in vitro gene transfection efficiency[END_REF] who described the spontaneous formation of cationic vesicles in water and examined the applicability of these aggregates to form complexes with DNA for gene delivery. Another paper of this group deals with the interactions between the cationic carnitine hexadecyl amide surfactant with the protein pepsin [START_REF] Ghosh | Solution Behavior and Interaction of Pepsin with Carnitine Based Cationic Surfactant: Fluorescence, Circular Dichroism, and Calorimetric Studies[END_REF].

Apart from these studies, carnitine has been rarely used for the design of amphiphilic molecules. In the present work, we have thus been interested in bromide salts of carnitine alkyl esters (Figure 1) with alkyl chain lengths ranging from C2 to C14. Depending on the chain length, different properties and behaviours were observed, some of them typical of hydrotropes and others of surfactants. Analogous to the critical micellar concentration (CMC) that characterizes surfactants, there is a certain hydrotrope concentration beyond which the solubilization power of hydrophobic compounds significantly increases. This concentration is often called the minimum hydrotrope concentration (MHC) [START_REF] Balasubramanian | Aggregation behavior of hydrotropic compounds in aqueous solution[END_REF].

Beside this, some ionic hydrotropes can also exhibit ionic liquid properties as reported by several authors. For instance, Claudio et al. reported the hydrotropic activity of several ionic liquids based on imidazolium, pyridinium, piperidinium, quaternary ammonium and phosphonium compounds as well as their ability to solubilize some antioxidant biomolecules.

In this way, they established a connection between the concept of ionic liquids and the hydrotropic solubility phenomenon [START_REF] Claudio | The magic of aqueous solutions of ionic liquids: Ionic liquids as a powerful class of catanionic hydrotropes[END_REF]. However, it should be noted that none of these positively charged headgroups are biobased.

In the present study, a series of alkyl carnitine ester bromides from C2 to C14 has been prepared in a simple one-step synthesis. Since esters are relatively easy to cleave, the (bio-) degradation products of them will mainly consist of products found in nature and as thus should be more easily biodegradable.

The pure compounds were characterized for their thermal properties. Their aqueous phase behaviour has been investigated by surface tension measurements and dynamic light scattering. The ability of the short-chain derivatives to solubilize the Disperse Red 13 hydrophobic dye was studied in order to examine their hydrotropic action and the solubility of vanillin was determined to demonstrate their usefulness in application. For such a purpose, the cytotoxicity of the newly developed compounds has also been determined and compared to two conventionally used cationic surfactants (DTAB, CTAB), one typical hydrotrope (SXS) and one ionic liquid ([C1C4Im]Br). Finally, the carnitine ester compounds were classified according to their hydrophobic/lipophilic behaviour using the PIT-slope method [START_REF] Ontiveros | A simple method to assess the hydrophilic lipophilic balance of food and cosmetic surfactants using the phase inversion temperature of C10E4/n-octane/water emulsions[END_REF] which is more powerful and more robust than the well-known HLB of Griffin [START_REF] Griffin | Calculation of HLB Values of Non-Ionic Surfactants[END_REF]. The results are discussed in terms of properties related to chain length and potential applications of these novel carnitine-based compounds.

Experimental Section

Materials

L-carnitine (99%), bromoethane (98%), 1-bromooctane (98%), 1-bromodecane (98%) and 1bromododecane (98%) were purchased from Alfa Aesar. 1-Bromobutane (99%), 1bromohexane (98%), 1-bromotetradecane (97%), dodecyltrimethylammonium bromide (99%) and 1-butyl-3-methylimidazolium bromide (> 97%) were delivered by Sigma Aldrich.

Disperse Red 13 came from Acros and vanillin (> 99%) from Roth. Sodium xylene sulfonate (> 90%) and cetyltrimethylammonium bromide (> 99%) were purchased from Fluka. NMR solvents D2O, d-DMSO and CDCl3 had an isotopic purity of 99.95% and were from Eurisotop.

Synthesis of carnitine alkyl ester bromides [CnCar]Br (n = 2 to 14)

The carnitine ester bromides, abbreviated as [CnCar]Br, were synthesized via the reaction of L-carnitine with 1-bromoalkanes (R = CnH2n+1 with n = 2 to 14) in acetonitrile under reflux. L-Carnitine (1 equiv.) and 1-bromoalkane (1.5 equiv.) were mixed in acetonitrile under inert conditions (argon-gas) before heating under reflux for at least 12 h. The reaction was monitored by NMR analysis. After evaporation of the solvent, the crude product was washed several times with diethyl ether and dried under vacuum. The product yields were > 90%. All NMR data are given in the Supporting Information (S2).

Surface tension measurements

Surface tension measurements were performed with a KRÜSS K100CC tensiometer with

Wilhelmy plate geometry at room temperature. Before each measurement, the platinum plate was intensively cleaned and dried to avoid any contamination. The sample solutions were prepared in volumes between 10 and 50 mL in millipore water and surface tension was measured after equilibration. The resulting values were mean quantities of at least two measurements. The critical aggregation (micelle) concentrations (CAC and CMC) and the minimum surface tensions (min) were taken from the surface tension curves.

Dynamic light scattering (DLS)

The concentrations of the DLS samples were chosen to be approximately 0.1 M (see Density measurements were performed on a DMA 5000M densitometer from Anton Paar (Graz, Austria).

Solubilization of Disperse Red 13

The solubilization of the hydrophobic dye Disperse Red 13 was measured using an Agilent

Technologies Varian Cary 60 UV-visible spectrophotometer. To this purpose, aqueous stock solutions (ca 2 M) of [CnCar]Br were prepared with millipore water and Disperse Red 13 was added until saturation was reached and powder remained in suspension. The solutions were stirred for 24h at 25 °C. To eliminate excess hydrophobic dye, each solution was filtered and the amount of dissolved Disperse Red 13 was determined by UV-visible absorption at 503 nm, where the absorbance was in the range between 0.5 and 2.

Solubilization of vanillin

[CnCar]Br aqueous solutions were prepared at different concentrations taking into account their MHC values and the water-solubility of the compound. Vanillin was added in excess to each solution. The solutions were stirred at 700 rpm at 25 °C for at least 72 h until saturation.

The samples were centrifuged in an EBA III centrifuge of Hettich (Tuttlingen, Germany) for 25 min with maximum speed to separate the solid excess of vanillin. The liquid phase was collected and diluted in ultrapure water. The solubilized amount of vanillin was quantified by UV-visible spectroscopy using a Lambda 18 UV/Vis spectrometer from Perkin Elmer (Neuried, Germany) at 280 nm. A calibration curve was previously established with ethanol as solvent (Supporting Information, Fig. S3). Reference samples with L-carnitine, SXS and

[C4C1Im]Br were prepared and measured in the same way.

Cytotoxicity tests

Cytotoxicity tests were performed in vitro with human skin keratinocytes (HaCaT cells) in form of a PrestoBlue assay. Detailed information can be found in the Supporting Information (S1.6).

PIT-slope method

The reference system used for this method was the tetraethyleneglycol monododecyl ether C10E4/n-octane/water system. For all samples the water fraction fwater was constant at 0.5. The detailed procedure is given in the Supporting Information (S1.7).

Results and Discussion

Temperature-dependent phase behaviour of the pure [CnCar]Br compounds (n = 2 to 14)

Ionic liquids (ILs) are salts with low melting temperature. Of particular interest are room temperature ionic liquids (RTILs). They often show high solubility and/or miscibility, because the crystallization energy does not need to be payed when they are already in a liquid state.

. They are characterized by a wide liquid range and a negligible vapour pressure [START_REF] Meindersma | Ullmann's encyclopedia of industrial chemistry[END_REF][START_REF] Wasserscheid | Ionic Liquids in Synthesis[END_REF].

Thermal properties (i.e. thermal phase transitions and thermal stabilities) of the [CnCar]Br compounds were determined and evaluated regarding ionic liquid characteristics. Thus, melting (Tm) and glass transition (Tg) temperatures were studied by differential scanning calorimetry (DSC) and degradation (Tdeg) temperatures by thermogravimetric analysis (TGA) (Table 1). [START_REF]Encyclopedia of materials: Science and technology[END_REF]. In contrast, the longer chain carnitine bromides with n = 8 to 14 have melting points of 34, 59, 69 and 71 °C, respectively. Per definition, they are ILs, too. Their different temperature-dependent phase behaviours can be explained by an increase of van der Waals interactions with increasing alkyl chain length [START_REF] Xu | Effects of alkyl chain length on properties of 1alkyl-3-methylimidazolium fluorohydrogenate ionic liquid crystals[END_REF]. There is an interesting break in degradation temperatures, where [CnCar]Br compounds with n ≥ 10 seem to be significantly more stable towards temperature than those with n ≤ 8. This can be attributed to stronger van der Waals interactions present in the long chain derivatives.

Self-aggregation of [CnCar]Br compounds (n = 2 to 14) in water

With an amphiphilic and ionic structure, a certain organization of the alkyl carnitine esters in water can be expected, similar to that of hydrotropes in the case of short alkyl chains and of surfactants in the case of longer alkyl chains. As a consequence, either pure or in aqueous solution, such molecules are indeed expected to exhibit some solubilizing properties of hydrophobic compounds: solvent-like if we consider the pure IL and hydrotropic or micellar in aqueous solution. Hence, the self-aggregation behaviour of the series of carnitine ester bromides [CnCar]Br was first studied by measuring the equilibrium surface tensions () in dilute aqueous solutions at different concentrations (Figure 3). M, respectively. Th+us, with increasing alkyl chain length and therefore increasing hydrophobicity, the CAC decreases and the surfactant efficiency, that goes along with the surfactant concentration necessary for reaching a minimum surface tension, increases [START_REF] Rosen | Surfactants and interfacial phenomena[END_REF].

The CACs of the long chain carnitine bromides are in the millimolar region that is typical of the CMC of a charged surfactant. This critical threshold indicates the start of the formation of micellar aggregates in the water bulk phase [START_REF]Surfactants and interfacial phenomena[END_REF]. With CMC values of 1.4 x10 -2 , 2.7x10 -3 and 7.1x10 -4 M, respectively, [C10Car]Br, [C12Car]Br and [C14Car]Br show surfactant efficiencies in the range of the common cationic surfactant cetyltrimethylammonium bromide (CTAB) with a CMC of  1x10 -3 M [START_REF] Neugebauer | Detergents: An overview[END_REF], thus presenting a relevant green alternative to such petro-based cationic surfactants still used as conditioning agents for example. In contrast, the short chain carnitine bromides have their CACs in the molar range, which is typical of the minimum hydrotropic concentration (MHC) of a hydrotrope. They are unlikely to form welldefined spherical micelles, as their hydrophobic tail is too short [START_REF] Lumb | Phase equilibria in mixtures of alcohols with aqueous hydrotropic salt solutions[END_REF].

As expected, there is a linear relationship between log (CAC) and the alkyl chain length n (see Supporting Information, Fig. S2). This is in accordance with the observations of Klevens who first reported this linear relationship for homologous series of surfactants [START_REF] Klevens | Critical Micelle Concentrations as Determined by Refraction[END_REF]. In contrast, Eastoe et al. found a distinct kink in an analogous plot for n-alkylbenzoates between n = 0 to 3 and n = 4 to 8. This phenomenon was related to a definite break between hydrotropic and surfactant behaviours of these species [START_REF] Hatzopoulos | Are hydrotropes distinct from surfactants?[END_REF]. However, according to the CAC values obtained for the [CnCar]Br compounds, a continuous transition from hydrotrope-to surfactant-like behaviour is observed with increasing chain length.

The effectiveness of a surfactant is independent of the surfactant concentration and only depends on the minimum surface tension that can be reached: the lower the surface tension, the higher the effectiveness [START_REF] Klevens | Critical Micelle Concentrations as Determined by Refraction[END_REF]. The minimum surface tension shows a chain length dependence according to alkyl chain densities: the longer the alkyl chain, the closer the surfactant can pack on the surface, the lower is the surface energy [START_REF] Klevens | Critical Micelle Concentrations as Determined by Refraction[END_REF][START_REF] Pitt | The relationship between surfactant structure and limiting values of surface tension, in aqueous gelatin solution, with particular regard to multilayer coating[END_REF]. The increase of the alkyl chain length values slightly increases the surface excess concentration (m), indicating a closing packing in the air/water interface. In this study, the surface tension values at the CAC were taken as minimum surface tension σCAC. While significantly higher σCAC values were determined for n = 2, 4, they were approximately at the same level for n = 6-14. Thus, although higher concentrations of middle-chain hydrotropes/surfactants (e.g. n = 6) are necessary for the reduction of the surface tension of water, the minimum surface tension can be as low as for long-chain compounds (e.g. n = 14), i.e. while the efficiency shows strong chain length dependence, the effectiveness can be similar [START_REF] Queste | Short chain glycerol 1monoethers -a new class of green solvo-surfactants[END_REF][START_REF] Bauduin | Temperature dependence of industrial propylene glycol alkyl ether/water mixtures[END_REF][40]. The maximum surface excess concentrations of carnitine alkyl ester bromides are higher than the reported data for 1- As can be seen in Figure 4, well-defined correlation functions are obtained for n = 6 to 14.

The correlation functions became less pronounced with decreasing alkyl chain length. This finding suggests that the long-chain compounds are more likely to form aggregates than the short-chain ones, which, nevertheless, show also a moderate tendency to form aggregates, as we can expect for hydrotropes [START_REF] Balasubramanian | Aggregation behavior of hydrotropic compounds in aqueous solution[END_REF]. As for the other measured properties of this series of surface-active molecules, there is no significant break in the evolution when the chain length is increased. SXS and [C4C1Im]Br aqueous systems were measured as reference systems.

According to the results obtained from DLS, they show a weak aggregation behaviour similar to [C2Car]Br. While the autocorrelation function of DTAB is similar to the long-chain carnitine derivatives, that of CTAB is in the range of [C6Car]Br. Usually, a distinct selfassembly could be expected for CTAB with its C16 carbon chain. However, this result can be attributed to the low concentration of the CTAB solution (0.015 M) due to its low water solubility. Please note that as usual, due to the significant ionic strength of the samples, the calculation of micelle hydrodynamic radii from the autocorrelation function via Stokes-Einstein-equation did not yield reasonable results [START_REF] Sutherland | Diffusion in Solutions of Micelles. What Does Dynamic Light Scattering Measure?[END_REF].

Hydrotropic efficiency of short-chain [CnCar]Br compounds (n = 2 to 8)

The hydrotropic efficiency of the carnitine ester bromides [CnCar]Br (n  8), i.e. their ability to enhance the solubility of a hydrophobic, sparingly water-soluble compound [START_REF] Bauduin | Unified concept of solubilization in water by hydrotropes and cosolvents[END_REF],

was evaluated by measuring the solubilization of the hydrophobic dye Disperse Red 13 (i.e. It can be observed that the increase in solubilization starts after a certain threshold concentration. From the Disperse Red 13 solubilization curves, these concentrations cannot be determined precisely but they can be roughly estimated. They are correlated to the hydrotrope efficiency for solubilizing hydrophobic substances. The following order can be observed:

[C8Car]Br > [C6Car]Br > [C4Car]Br > [C2Car]Br. This means that hydrotrope efficiency increases with chain length [START_REF] Williamson | Organic experiments: Macroscale and microscale[END_REF].

Contrary to what could be expected, the critical values in these ternary systems seem to be smaller than for the binary systems (see Figure 3). We assume that the hydrophobic component promotes aggregation and the molecules start to self-assemble earlier in the presence of a third hydrophobic component.

Nevertheless, the mechanism of hydrotropic solubilization is not exactly clarified yet and different theories are reported. One is based on the existence of water-hydrotrope interactions that might drive hydrotropic solubilization [START_REF] Buchecker | The impact of the structuring of hydrotropes in water on the mesoscale solubilisation of a third hydrophobic component[END_REF]. Another one describes the formation of lowstochiometric solute-hydrotrope complexes as the origin of hydrotropy [START_REF] Booth | Mechanism of hydrophobic drug solubilization by small molecule hydrotropes[END_REF]. However, we believe that for carnitine ester bromides, the mechanism resembles a micellar solubilization process, as we observed self-assembly of carnitine esters in aqueous solution for all chain lengths in DLS measurements (see section 3.3). Nevertheless, the addition of a third hydrophobic compound might enhance aggregation and also hydration might probably be an issue. As hydrotropic solubilization is driven by the complex interplay of interactions between water, solute and hydrotrope, further investigations on these systems have to be done to clarify the system structure. (Supporting Information, Fig. S1), no CAC could be determined, whereas from the solubility curve the formation of aggregates can be assumed. This would mean that these compounds form aggregates only in the presence of a hydrophobic compound.

Solubilization of vanillin

Vanillin, or 3-methoxy-4-hydroxy-benzaldehyde, is a biomolecule occurring in green vanilla beans in high amounts. This molecule belongs to one of the most popular flavouring materials and is used in food, beverage, pharmaceutical and cosmetic industry. In addition to that, it exhibits antioxidant, antimicrobial and radical-scavenging properties and is therefore highly interesting as a preservative in all kinds of chemical formulations. Its water-solubility is quite low ( 0.06 M) and extraction processes typically use volatile organic solvents [START_REF] Cláudio | Extraction of vanillin using ionic-liquid-based aqueous two-phase systems[END_REF][START_REF] Sinha | A comprehensive review on vanilla flavor: Extraction, isolation and quantification of vanillin and others constituents[END_REF]. Thus, we have investigated the solubility of vanillin in water in the presence of increasing amounts of the its lower water-solubility. Especially the conventionally used surfactant CTAB was restricted through its own water-solubility, so that the increase of vanillin water-solubility by CTAB was only small. DTAB, in contrast, performed similarly to the longer chain carnitine derivatives (n = 12 and 14), i.e.  0.2 M vanillin for 0.2 M of surfactant.

Cytotoxicity

The cytotoxicity of all [CnCar]Br compounds was evaluated in vitro with human skin keratinocytes (HaCaT cells) and compared to that of SXS, [C4C1Im]Br, DTAB and CTAB.

The resulting EC50 values correspond to the concentrations at which 50% of the examined cells survived. Results are reported on Figure 7. The corresponding cell viability data can be found in the Supporting Information (Fig. S5). with 26 and 21 µM [START_REF] Zhang | Toxicity of nanoparticle surface coating agents: Structurecytotoxicity relationship[END_REF]. This is an expected result because the toxicity is mainly given by the hydrocarbon-chain length, as it is well known for many other compounds.

With EC50 values of 0.1 and 4.5 mM, respectively, the middle-chain compounds [C8Car]Br and [C6Car]Br are less toxic than the long-chain ones. Also, the reference substance

[C4C1Im]Br exhibits low cytotoxicity compared to the long chain carnitine bromides.

However, in comparison to its carbon-chain analogue [C4Car]Br, the carnitine compound shows better cytotoxicity characteristics.

In fact there are several studies on the cytotoxicity of quaternary ammonium compounds and cationic surfactants [START_REF] Gilbert | Cationic antiseptics: Diversity of action under a common epithet[END_REF][START_REF] Inácio | Mitochondrial dysfunction is the focus of quaternary ammonium surfactant toxicity to mammalian epithelial cells[END_REF]. It seems clarified surfactants tend to interact with phospholipid cell membranes where they affect their physical properties and function. At high surfactant concentration and long exposure, it comes to an association of components of the cell membrane and the surfactants and to cell lysis. However, for cationic surfactants, cytotoxicity was also observed at concentrations that were too low to cause cell lysis. Inácio et al. reported that the reason for this cytotoxicity was a mitochondrial dysfunction induced by quaternary ammonium compounds [START_REF] Inácio | Mitochondrial dysfunction is the focus of quaternary ammonium surfactant toxicity to mammalian epithelial cells[END_REF]. Therefore it is questionable if the cytotoxicity of cationic surfactants, that are quaternary ammonium species in most cases, can be fully avoided. Nevertheless, it can be reduced by decreasing the alkyl chain length. Developing carnitine cationic surfactants that have high surfactant efficiency even with lower alkyl chain length might be an approach towards less toxic cationic surfactants.

Hydrophilic/lipophilic balance of the [CnCar]Br compounds (n = 6 to 14) according to the PIT-slope method

Finally, the hydrophilic/lipophilic balance of the carnitine alkyl ester bromides [CnCar]Br was determined using the PIT-slope method [START_REF] Ontiveros | A simple method to assess the hydrophilic lipophilic balance of food and cosmetic surfactants using the phase inversion temperature of C10E4/n-octane/water emulsions[END_REF]. More robust than the well-known HLB of Griffin, this method is based on monitoring the evolution of the phase inversion temperature (PIT) of the well-defined C10E4/n-octane/water reference system upon addition of a second amphiphile, S2, what results in an increase or decrease of the emulsion PIT. For nonionic surfactants, it has been shown that a linear regression allows a correlation between the PIT and the molar fraction x2 of S2 providing the so-called PIT-slope as a descriptor of the hydrophilic/lipophilic balance of S2. According to Shinoda et al., the PIT corresponds to the temperature where the hydrophilic/lipophilic character of a nonionic ethoxylated surfactant flips over [START_REF] Shinoda | The Correlation between Phase Inversion Temperature In Emulsion and Cloud Point in Solution of Nonionic Emulsifier[END_REF]. The dehydration of the polyoxyethylene units during heating leads to the inversion from an oil-in-water (o/w) to a water-in-oil (w/o) emulsion. This process can be monitored by measuring the emulsion conductivity [START_REF] Anton | Particular conductive behaviors of emulsion phase inverting[END_REF][START_REF] Kunieda | Spontaneous Formation of Highly Concentrated Water-in-Oil Emulsions (Gel-Emulsions)[END_REF]. The C10E4/n-octane/water is used as a reference system as its initial PIT is close to room temperature [START_REF] Pizzino | Relationship between Phase Behavior and Emulsion Inversion for a Well-Defined Surfactant C10E4/n -Octane/Water Ternary System at Different Temperatures and Water/Oil Ratios[END_REF]. By addition of S2, the PIT changes positively or negatively relative to the reference system as follows: positive values for the PIT deviation indicate a more hydrophilic character and negative values a less hydrophilic character of S2 compared to C10E4 [START_REF] Ontiveros | A simple method to assess the hydrophilic lipophilic balance of food and cosmetic surfactants using the phase inversion temperature of C10E4/n-octane/water emulsions[END_REF]. Thus, although this method does not provide absolute values, it delivers a comparative criterion that illustrates the activity of a molecule at the water/oil interface and its relative hydrophilicity. and 14 () from ref [START_REF] Ontiveros | Structureinterfacial properties relationship and quantification of the amphiphilicity of well-defined ionic and non-ionic surfactants using the PIT-slope method[END_REF].

As can be seen on Figure 8, all surfactants have a positive impact on the PIT meaning that, they are more hydrophilic than C10E4. It is noteworthy that the addition of S2 was restricted, as with higher amounts, the emulsion did not invert within the experimentally used temperature range of up to 60 °C. As far as carnitine derivatives are concerned, the linear fits match the obtained data for [C6Car]Br and [C8Car]Br very well. In contrast, longer-chained carnitine compounds with n = 10, 12, 14 do not exhibit a linear behaviour. Indeed, as already reported, the linearity for nonionic surfactants S2 (R 2 > 0.99) is better than for ionic surfactants (R 2 > 0.95) [START_REF] Ontiveros | Structureinterfacial properties relationship and quantification of the amphiphilicity of well-defined ionic and non-ionic surfactants using the PIT-slope method[END_REF]. This can be attributed to the presence of a non-ideal surfactant mixture of ionic and nonionic surfactants due to strong interaction at the interface [START_REF] Lukowicz | Fragrance solubilization in temperature insensitive aqueous microemulsions based on synergistic mixtures of nonionic and anionic surfactants. 458 (2014) 85-95[END_REF][START_REF] Hong | Synergy between bis(dimethyldioctylammonium) molybdate and tetraethylene glycol monooctyl ether, A winning combination for interfacial catalysis in thermo-controlled and switchable microemulsions[END_REF][START_REF] Nardello-Rataj | Aqueous solutions of didecyldimethylammonium chloride and octaethylene glycol monododecyl ether: Toward synergistic formulations against enveloped viruses[END_REF][START_REF] Rauwel | Aqueous mixtures of di-n-decyldimethylammonium chloride/polyoxyethylene alkyl ether: Dramatic influence of tail/tail and head/head interactions on co-micellization and biocidal activity[END_REF]. However, linearity can be considered at low x2 values and, by taking into account only the data for which the PIT is lower than 30 °C, we obtain for the carnitine compounds the following order: In other words, they would have a similar hydrophilicity which tends to show that the carnitine headgroup, though containing ester and hydroxyl groups in its structure, does not bring as much hydrophilicity as expected, probably due to intramolecular hydrogen bonding between the two functions. Hence, with four additional carbon atoms, the carnitine moiety would be equivalent to two CH2 groups. This could be of interest in the use of C6 or C10 alkyl chains instead of C8 or C12 ones respectively for designing surfactants with same functional properties.

Conclusion

In this study, L-carnitine has been considered as natural cationic building block in the search for novel green amphiphilic compounds. The conversion of L-carnitine into functional molecules (carnitine alkyl ester bromides) was done in a simple, inexpensive one-step synthesis. Some of the resulting esters turned out to be room temperature ionic liquids.

However, this work focused on self-aggregation and solubility behaviour of the corresponding aqueous solutions as well as the hydrophilicity of the resulting amphiphiles.

In a homologous series with alkyl chain length n = 2-14, a progressive transition from hydrotropic to surfactant behaviour was observed via DLS and surface tension measurements, different to what has been found for sodium alkylbenzoates [START_REF] Hatzopoulos | Are hydrotropes distinct from surfactants?[END_REF]. However, according to the PIT slope method used to determine the surfactant hydrophilicity, only carnitine esters with alkyl chain length > 8 showed real surfactant properties, as they were thought to be mainly located in the interfacial layer.

All results were compared to the conventionally known hydrotrope SXS, to [C4C1Im]Br -a typical ionic liquid with hydrotropic properties, and to two well-established cationic surfactants, DTAB and CTAB. The comparison shows that the newly developed compounds can keep up with conventional solubilizers in terms of solubility performance and cytotoxicity, where short chain carntitine esters did not show any cytotoxicity at all.

That the cytotoxicity of cationic surfactants depends mainly on the length and the nature of the alkyl chain and to a much lesser extent on the headgroup, has been known before [START_REF] Leal | On the Toxicity and Biodegradation of Cationic Surfactants[END_REF]. In this respect, no advantage can be respected. However, in contrast to other cationic surfactants, the synthesis is very simple and the carnitine molecule is available industrially through fermentation in biotechnological processes [START_REF] Engel | Pharmaceutical substances[END_REF]. Further, during biodegradation, the cleavage of the ester bond should deliver natural products that are much less toxic than those obtained when classical cationic surfactants are degraded. However, this will be the subject of further research and remains speculation for the moment.

Concerning potential application in colloidal formulations, like cosmetics, food or pharmaceutics, the carnitine ester bromides represent a wide spectrum of properties that can be adjusted to the required properties by changing the alkyl chain length. Especially short chain compounds that did not exhibit cytotoxicity in the measured concentration range, might have a certain potential, especially in mixtures with anionic surfactants. But before, and further to deeper toxicity and biodegradability studies, full phase diagrams must be measured and the underlying structures must be elucidated.

Figure 1 .

 1 Figure 1. Molecular structure of the carnitine alkyl ester bromides; R = CnH2n+1 with n = 2 to

Figure 2 .

 2 Figure 2. One-step synthesis of carnitine alkyl ester bromides; R = CnH2n+1 with n = 2, 4, 6, 8,

Figure 3 .

 3 Figure 3. Surface tension curves of [CnCar]Br compounds with n = 2 (), 4 (), 6 (),8 (),

alkyl- 3 - 3 . 3 .= 2 to 14 )

 33314 methylimidazolium [REF: 2007_LANG_Dong] with the same alkyl chain (n=10 and n=12). Carnitine polar head of a single surfactant molecule had a smaller area at the air-water interface. To summarize, an alkyl chain length dependence of CAC and σCAC values of [CnCar]Br compounds can be observed. CACs of short chain carnitine bromides are in a range typical of hydrotropes, while those of the long chain compounds are in a range typical of surfactants. However, there is no clear distinction between hydrotropes and surfactants, rather a continuous transition when the chain length is increased. Compared to other cationic surfactants as alkyl-3-methylimidazolium , alkylpyridinium bromides or alkyltrimethylammonium bromides with alkyl chain between 10 and 14 [REF: 2007_LANG_Dong] the CMC and the minimum surface tension of alkyl carnitines are lower. Characterization by dynamic light scattering of aggregates formed by the [CnCar]Br compounds (n Dynamic light scattering (DLS) measurements were carried out to further characterize the [CnCar]Br/water binary systems. The concentrations of the DLS solutions were chosen at CMC + 0.1 M. Figure 4 gives the self-correlation functions for the different compounds from n = 2 to 14. The reference solutions of SXS and [C4C1Im]Br had concentrations of 0.5 M and 3.5 M, respectively. DTAB and CTAB solutions were prepared with 0.111 and 0.015 M, respectively.

Figure 4 .

 4 Figure 4. Time-dependent self-correlation functions as obtained by DLS for the

2 -[ 4 -

 24 (2-chloro-4-nitrophenylazo)N-ethylphenylamino]ethanol), which can be readily followed by UV/visible spectroscopy. In Figure 5, the absorbance of the dye at 503 nm has been plotted against the hydrotrope concentrations of the [CnCar]Br with n = 2 to 8 as well as SXS, [C4C1Im]Br and L-carnitine. The measured absorbance is proportional to the amount of Disperse Red 13 dissolved.

Figure 5 .

 5 Figure 5. Hydrotropic solubilization of Disperse Red 13 in [CnCar]Br (n = 2, 4, 6 and 8)

Likewise, Disperse Red 13

 13 solubilization tests were performed with SXS, [C4C1Im]Br and Lcarnitine. [C4Car]Br seems to have a solubilization efficiency (threshold concentration) similar to the classical hydrotrope SXS. However, the carnitine derivative shows better performance, as the amount of solubilized Disperse Red 13 is higher. All [CnCar]Br solutions were more efficient for Disperse Red 13 solubilization than [C4C1Im]Br and Lcarnitine. It was interesting that in the surface tension curves of [C4C1Im]Br and L-carnitine

  [CnCar]Br compounds in comparison with L-carnitine, SXS, [C4C1Im]Br, DTAB and CTAB solutions as references. Results are depicted on Figure 6.

Figure 6 .

 6 Figure 6. Solubility of vanillin at room temperature in aqueous solutions of [CnCar]Br. For

Figure 7 .

 7 Figure 7. EC50 values in [M] for [CnCar]Br compounds and two common cationic

Figure 8

 8 shows the evolution of the PIT at different mole fractions of the [CnCar]Br compounds from n = 6 to 14 in comparison with already reported alkyltrimethylammonium bromides [Cn]Br[START_REF] Ontiveros | Structureinterfacial properties relationship and quantification of the amphiphilicity of well-defined ionic and non-ionic surfactants using the PIT-slope method[END_REF].

Figure 8 . 2 = 0 . 5

 8205 Figure 8. Phase inversion temperature of the system 3 wt.% C10E4/octane/10 -2 M NaClaq at fw

  This ranking clearly indicates a typical surfactant behaviour for [CnCar]Br with n = 10, 12 and 14 which behave very closely: they are mainly localized in the interfacial layer impacting the PIT according to their hydrophilicity. It is worth noting that when the molar fraction x2 increases, the deviation from linearity is more pronounced for [C10Car]Br compared to [C12Car]Br and [C14Car]Br. This probably reflects a stronger interaction with C10E4 at the interface as already shown in our previous work with C10Ej and [DiC10]Cl [59]. In contrast, the shortchain ester compounds with n = 6, 8 do not show proper surfactant properties, as they are partitioned between the aqueous phase and the interface. Compared to typical cationic surfactants reported in the literature [55], it is interesting to note that [C6Car]Br behaves almost as [C8]Br and that [C10Car]Br behaves almost as [C12]Br.

Table 2

 2

	)

Table 1 .

 1 Degradation temperature Tdeg, melting temperature Tm and glass transition temperature Tg of [CnCar]Br compounds for different chain lengths.

	n	Tdeg [°C]	Tm [°C]	Tg [°C]
	2	190	129	/
	4	189	/	-22
	6	187	/	-29
	8	185	34	/
	10	235	59	/
	12	239	69	/
	14	241	71

/

The results in Table

1

show that [C2Car]Br with a melting temperature of 129 °C cannot be considered as an IL, while [C4Car]Br and [C6Car]Br are room temperature ILs (RTILs) with glass transitions at -22 and -29 °C respectively. Glass transitions typically occur for ILs, as they organize into amorphous states upon cooling. The formation of a crystal lattice is hindered due to steric reasons

Table 2 .

 2 The

surface excess concentration (m) is calculated only for the surfactants which CMC are below 0.01M (See Supplementary Information)

Table 2 .

 2 Aggregation state, water-solubility, critical aggregation concentration (CAC), minimum surface tension σCAC and surface excess concentration of [CnCar]Br compounds.

	[CnCar]Br	2	4	6	8	10	12	14
	State at rt	solid liquid liquid	liquid	solid	solid	solid
	water-solubility [M] > 10 > 10	5.9	5.3	2.4	1.3	0.65
	CAC [M]	1.8	1.0 2.1x10 -1 9.0 x10 -2 1.4 x10 -2 2.7x10 -3 7.1x10 -4
	σCAC [mN/m]	45.5 41.9	37.3	35.2	38.7	36.2	36.2
	m×10 10 [mol/cm 2 ]	-	-	-	-	2.8	3.3	4.3

The CAC values for the [CnCar]Br compounds with n = 2 to 14 range from 1.8 to 7.1 x 10 -4


T A Oproglidis

D H Tassis

A Tsormpatzoglou

T A Karatsori

C G Theodorou

S Barraud

G Ghibaudo

C A Dimitriadis

Threshold voltage of p-type triple-gate junctionless transistors

Keywords:

The threshold voltage of rectangular p-type triple-gate junctionless transistors (JLTs) is studied experimentally using the transconductance derivative (dg m /dV g ) method, after correcting the drain current from the impact of series resistance. The effect of series resistance on the dg m /dV g behavior is highlighted. In the investigated devices, the high series resistance affects the dg m /dV g behavior more than the short-channel effects. The results show that, in addition to the flat-band voltage, for the first time two threshold voltages V th1 and V th2 are observed within the partial depletion region in devices with channel length varying from 95 to 25 nm. Numerical simulations of the holes density distribution reveal the absence of corner effects due to the unique bulk neutral conduction, whereas V th1 and V th2 correspond to the threshold voltages of the side gates and top gate, respectively. The correct extraction of the flat-band voltage has been confirmed with numerical simulations of the holes density distribution. Experimental measurements of p-type JLTs with variable being the fin width indicate that the threshold voltages V th1 and V th2 are due to the different interface states density at the side and top gates.

Introduction

Triple-gate junctionless transistors (JLTs) have been considered as promising devices to extend the scaling limit to lower channel lengths, avoiding ultra-high doping concentration gradients at the source/drain junctions [START_REF] Colinge | Nanowire Transistors Without Junctions[END_REF]. Compared to the inversion mode triple-gate transistors, JLTs offer lower off-state current, near ideal subthreshold slope and less mobility degradation with gate voltage [START_REF] Colinge | Nanowire Transistors Without Junctions[END_REF]. In a recent work, it has been suggested that the triple-gate architecture in JLTs exhibits corner effects in spite of the bulk neutral conduction mechanism [START_REF] Sandilya | Variations of corner effects in junctionless Fin-Body FETs[END_REF]. The corner effects have been extensively studied by three-dimensional device simulations in inversion mode triple-gate transistors by several groups [START_REF] Fossum | Suppression of corner effects in triple-gate MOSFETs[END_REF][START_REF] Burenkov | Corner effect in double and triple gate FinFETs[END_REF][START_REF] Stadele | A comprehensive .study of corner effects in tri-gate transistors[END_REF][START_REF] Poljak | Suppression of corner effects in triple-gate bulk FinFETs[END_REF][START_REF] Cano De Andrade | Threshold voltages of SOI MuGFETs[END_REF][START_REF] Ruiz | A comprehensive study of the corner effects in pi-gate MOSFETs including quantum effects[END_REF][START_REF] Ruiz | A comprehensive study of the corner effects in Pi-gate MOSFETs including quantum effects[END_REF]. Due to local increase of the electron density at the corners of the fin, independent channels located at the fin corners are activated at different gate voltages. As a result, the transconductance derivative (dg m /dV g ) method revealed that the triple-gate inversion mode transistors can present more than one threshold voltages, which becomes a serious technological problem [START_REF] Cano De Andrade | Threshold voltages of SOI MuGFETs[END_REF]. In inversion mode triple-gate transistors, due to the corner effects the channels located next to the top and bottom corners of the fin have lower threshold voltages compared to the sidewall gates [START_REF] Cano De Andrade | Threshold voltages of SOI MuGFETs[END_REF].

The operation of JLTs present the special feature of involving two conduction regimes, such as bulk neutral channel and surface accumulation layer, separated by the flat-band voltage V fb . In n-type JLTs, the threshold voltage has been investigated experimentally using the dg m /dV g method in devices with a planar structure (fin width W fin = 10 µm) [START_REF] Jeon | Revisited extraction methodology for electrical characterization of junctionless transistors[END_REF][START_REF] Jeon | Separation of surface accumulation and bulk neutral channel in junctionless transistors[END_REF]. Two peaks were observed in the dg m /dV g versus V g plot, with the first (lower) representing the bulk conduction threshold voltage V th and the second (higher) the flat band voltage V fb . A methodology to separate the surface accumulation and bulk conduction in n-type JLTs with single threshold voltage has been proposed by using the standard Y-function method to extract the bulk and accumulation low-field mobility [START_REF] Jeon | Separation of surface accumulation and bulk neutral channel in junctionless transistors[END_REF]. Furthermore, the impact of the source-drain series resistance R sd on the dg m /dV g versus V g plot has been highlighted in planar n-type JLTs, suggesting a method to eliminate the R sd effect for extracting the parameters V th and V fb [START_REF] Jeon | Impact of series resistance on the operation of junctionless transistors[END_REF]. Last years, the electrical properties, the dynamic response, the effect of crystallographic orientations and reliability issues in p-type triple-gate JLTs have been studied extensively [START_REF] Doria | Effective mobility analysis of n-and p-types SOI junctionless nanowire transistors[END_REF][START_REF] Doria | Physical insights on the dynamic response of SOI n-and p-types SOI junctionless nanowire transistors[END_REF][START_REF] Trevisoli | Analysis of p-type junctionless nanowire transistors with different crystallographic orientations[END_REF][START_REF] Reshi | Investigation of NBTI effect in p-type junctionless transistor with uniform and graded doping profiles[END_REF][START_REF] Ribeiro | Analysis of the electrical parameters of SOI junctionless nanowire transistors at high temperatures[END_REF]. Furthermore, using the threshold voltage extracted with the g m /I d method [START_REF] Trevisoli | A physically-based threshold voltage definition, extraction and analytical model for junctionless nanowire transistors[END_REF], a new method for extraction of the n-type JLT parameters (flat-band voltage, fin width and doping concentration) has been proposed [START_REF] Trevisoli | A new method for junctionless traansistors parameters extraction[END_REF].

In this article, with careful multiple-peak fit analysis of the experimental dg m /dV g versus V g plots derived after eliminating the effect of series resistance, the threshold voltage V th of rectangular crosssection p-type triple-gate JLTs is studied in terms of the channel length. In contrast to planar n-type JLTs of low series resistance exhibiting two peaks corresponding to the conduction threshold voltage and flat-band voltage [START_REF] Jeon | Impact of series resistance on the operation of junctionless transistors[END_REF], in all rectangular p-type triple gate JLTs of high series resistance the intrinsic dg m /dV g versus V g plots exhibit three peaks, highlighting the presence of the flat-band voltage and two conduction threshold voltages within the partial depletion region. The impact of the series resistance and short-channel effects on the dg m /dV g behavior for extracting the threshold voltage and flat-band voltage in p-type JLTs is discussed. Multiple threshold voltages are a technological problem causing difficulties in separating surface accumulation and bulk conduction and complexity in compact modeling. However, multiple threshold voltages in rectangular p-type JLTs has never been observed so far. The origin of the two conduction threshold voltages is clarified by numerical simulation and experimental results of p-type JLTs with rectangular cross-section.

Experimental details

The measured single p-type JLTs were fabricated at CEA-LETI on (100) SOI wafers with 145 nm thick buried oxide (BOX) [START_REF] Barraud | Scaling of trigate junctionless nanowire MOSFET with gate length down to 13nm[END_REF]. After thinning the Si body down to about 10 nm, a full-sheet implantation was carried out before active patterning with a boron doping targeted at about 2 × 10 19 cm -3 . The high-k-metal gate stack is composed of HfSiON/TiN/p + -polysilicon, with equivalent oxide thickness of 1.2 nm, and the TiN metal gate used features a work function close to mid-gap. Using a nitride spacer on the sidewalls of the gate, an additional implantation was performed to source/drain regions in order to improve electrical performance by reducing access resistance, resulting in source/drain doping level about 10 20 cm -3 . The implantation process can give rise to an increased number of interface states at the gate oxide-Si interface, which in turn may cause increase in threshold voltage, reduction in transconductance and increase in leakage current. Previous investigation has shown that for a certain substrate dopant concentration, the Si-SiO 2 interface state density is much higher in boron-implanted silicon than in phosphorus-implanted silicon [START_REF] Peterstrom | Si-SiO 2 interface trap density in boron-and phosphorus-implanted silicon[END_REF]. This difference in interface state density has been attributed to a higher strain in boron-implanted silicon, due to a larger lattice mismatch caused by the smaller boron atoms, explaining the low series resistance in n-type JLTs [START_REF] Jeon | Impact of series resistance on the operation of junctionless transistors[END_REF]. In the investigated devices of rectangular cross-section, the channel length L is varying from 25 to 95 nm, with fixed fin height H fin = 10 nm and fin width W fin = 26 nm. The impact of the fin width on the performance behavior has been investigated in p-type JLTs with rectangular crosssection, composed of an array of 50 parallel nanowires with doping concentration 2 × 10 19 cm -3 , channel length L = 10 m, fin height H fin = 10 nm and fin widths W fin = 20, 40 and 190 nm.

Three-dimensional numerical simulations of a long channel device (L=95nm) were performed, using the Silvaco (Atlas) simulation tool, using the models of band-gap narrowing, Shockley-Read-Hall generation-recombination and field-dependent mobility. Quantum confinement effects were not considered, since these effects become significant when the silicon thickness is less than 10 nm [START_REF] Yesayan | Physics-based compact model for ultra-scaled FinFETs[END_REF].

Results and discussion

Fig. 1(a) shows the transfer characteristics (I d -V g ) at drain voltage V d = -0.03 and -1 V of a p-type JNT with L = 95 nm, H gin = 10 nm and W fin = 26 nm. The drain induced barrier lowering (DIBL), defined as the difference between the gate voltages at constant drain current I d = 10 -7 A/m divided by ΔV d between V d = -0.03 V and V d = -1 V, is -0.02 V/V. The low DIBL parameter indicates negligible short channel effects. For accurate extraction of the series resistance R sd of the triple gate JLTs, an extraction method based on the total resistance R tot = V d /I d,tot , measured at V d = -30 mV in the accumulation regime, has been applied [START_REF] Trevisoli | A new series resistance extraction method for junctionless nanowire transistors[END_REF]. The total resistance, defined as the sum of the channel resistance and series resistance, is given by [START_REF] Trevisoli | A new series resistance extraction method for junctionless nanowire transistors[END_REF]:

  1 m d tot g fb sd ox d g VL R V V R CI         , ( 1
)
where  is the mobility degradation factor,  is the carrier mobility and C ox is the gate capacitance per unit of length. Considering that the inverse Y-function g m 1/2 /I d is R sd independent, the curve of R tot as a function of g m 1/2 /I d can be approximated by a straight line, the intercept of which with the vertical axis gives the series resistance. The accuracy of the series resistance extraction method has been demonstrated through numerical simulations and experimental results [START_REF] Trevisoli | A new series resistance extraction method for junctionless nanowire transistors[END_REF]. The total resistance R tot measured at V d = -30 mV is plotted as a function of the inverse Y-function g m 1/2 /I d in Fig. 1(b). This curve is approximated by a straight line and the intercept with the vertical axis gives the series resistance R sd = 18.15 k. It is noticed that the commonly used method of exponential fitting in the R tot versus gate voltage curve [START_REF] Dixit | Analysis of the parasitic S/D resistance in multiple-gate FETs[END_REF], the value of the obtained R sd is strongly dependent on the analyzed voltage range. Furthermore, the transfer length method (TLM) for extracting the series resistance is not used, since non-ideal ohmic contacts at the source and drain sides can result in severe device-todevice variability in series resistance.

It is worth noting that the measured series resistance of the p-type JNT is high in spite of the additional implantation in the extensions performed to reduce the access resistance. To understand the reason why the series resistance is high, we studied the effect of fin width on the device performance. measured at drain voltage V d = -30 mV. For the narrow fin device (W fin = 20 nm) the subthreshold slope approaches the ideal value of 64 mV/dec, whereas for the wide fin device (W fin = 190 nm) the subthreshold slope is strongly degraded to 362 mV/dec. Furthermore, the leakage current is increased dramatically with increasing W fin from 20 to 190 nm These findings clearly indicate that the top gate interface is dominated by interface states of density much higher compared to that of the side gates interfaces. The high density of interface states can explain the high series resistance at the source and drain contacts. Moreover, the asymmetric distribution of the density of interface states is expected to cause different threshold voltages for the top and side gates, which can be represented by two peaks in the dg m /dV g versus V g plots. The threshold voltage V th of the triple-gate JLTs was obtained using the transconductance derivative method at drain bias V d = -30 mV [START_REF] Trevisoli | Threshold voltage in junctionless nanowire transistors[END_REF]. For the JLT of Fig. 1, the as-measured derivative dg m /dV g as a function of V g is plotted with dashed line in Fig. 2(a). Two peaks are distinguished in the as-measured dg m /dV g versus V g characteristic of the JLT with L = 95 nm. The drain current corrected from the impact of R sd was calculated from the relation [START_REF] Jeon | Impact of series resistance on the operation of junctionless transistors[END_REF]:

1 , d d sd d d tot V I R V I         , ( 2
)
where I d,tot denotes the measured drain current. The intrinsic dg m /dV g versus V g characteristic, obtained from the R sd -corrected I d of Fig. 1(a) at V d = -30 mV is plotted with symbols in Fig. 3(a), together with its decomposition into three Gaussian curves, indicating that two independent channels located in different regions are activated at different gate voltages. The first and second peaks from right correspond to the conduction threshold voltages V th1 = -0.06 V and V th2 = -0.56 V separating the partial depletion from full depletion region, while the third peak corresponds to the flat-band voltage V fb = -1.13 V, i.e. the threshold of the accumulation conduction channel. The observed two threshold voltages within the partial depletion region of the ptype JLT can be attributed to the asymmetric distribution of the interface states density of the side and top gates. The effect of R sd on the dg m /dV g behavior is shown in Fig. 3(b). As the series resistance is reduced from 18 to 8 k, the peak corresponding to V th2 is more reduced compared to that of V th1 with simultaneous shift to higher values, whereas the peak of V fb is completely vanished. Therefore, as the series resistance seriously affects the dg m /dV g versus V g plots, the value of R sd in Eq. ( 2) is critical to reveal the peak corresponding to V fb and for accurate extraction of the parameters V th1 and V th2 .

To clarify further the physical origin of the conduction threshold voltages V th1 and V th2 within the partial depletion region and verify the obtained flat band-band voltage V fb , the distribution of the holes density has been determined with numerical simulations. The simulated transfer characteristic of the p-type JLT with L = 95 nm , H fin = 10 nm and W fin = 26 nm is presented in Fig. 4, reproducing with good accuracy the experimental transfer characteristic using the extracted series resistance of R sd = 18.15 k, doping concentration N A = 1.4510 19 cm -3 and low field mobility  0 = 30 cm 2 /V•s. After calibration of the simulation tool, the distributions of the hole density in the middle of the channel when the devices is operated at drain bias voltage V d = -30 mV and gate voltages V g = V th1 = -0.06 V, V g = V th2 = -0.56 V and V g = V fb = -1.13 V were obtained as presented in Fig. 5. In Fig. 5(a), a depletion of the holes density in the upper corners of the fin is observed, indicating the absence of the corner effects due to the unique volume conduction in JLTs, in contrast with the inversion mode triple gate MOSFETs [START_REF] Sandilya | Variations of corner effects in junctionless Fin-Body FETs[END_REF][START_REF] Fossum | Suppression of corner effects in triple-gate MOSFETs[END_REF][START_REF] Burenkov | Corner effect in double and triple gate FinFETs[END_REF][START_REF] Stadele | A comprehensive .study of corner effects in tri-gate transistors[END_REF][START_REF] Poljak | Suppression of corner effects in triple-gate bulk FinFETs[END_REF][START_REF] Cano De Andrade | Threshold voltages of SOI MuGFETs[END_REF][START_REF] Ruiz | A comprehensive study of the corner effects in pi-gate MOSFETs including quantum effects[END_REF].

As shown in Fig. 5(a), at V g = V th1 = -0.06 V the holes conduction path is located in the middle of the channel and close to the interface with BOX as in double gate (DG) JLTs [START_REF] Dehzangi | Electrical comparison and charge transmission in p-type double gate and single gate junctionless transistor fabricated by AFM nanolithography[END_REF], indicating that the threshold voltage V th1 is due to the side gates. For gate voltage above V th1 , the holes conduction path is extended closer to the top of the channel, indicating that the threshold voltage V th2 is due to the top gate. The conclusion for the origin of V th1 and V th2 derived from the hole density distribution at different gate voltages is consistent with the finding of the asymmetric distribution of interface states at the side and top gates. At gate voltage V g = -1.13 V, the whole channel becomes neutral with hole density equal to the doping density N A and the current flows through the entire channel [Fig. 5(c)]. Therefore, numerical simulations show that the gate voltage of V g = -1.13 V corresponds to the flatband voltage V fb , which is in agreement with the flat-band voltage extracted experimentally in Fig. 3(a) after correcting the drain current from the impact of series resistance R sd .

For p-type JLTs with channel length varying from 75 to 25 nm, the as-measured and the intrinsic dg m /dV g versus V g characteristics obtained after correcting the drain current from the impact of R sd , together with their decomposition into Gaussian curves are presented in Fig. 6. The insets of Figs. 6(a)-6(d) present the series resistance R sd and DIBL parameter of the JLTs. The short channel effects become stronger with decreasing the channel length, whereas the series resistance varies lying in the range of about 11-27 k. In all cases, after careful multiple-peak fit analysis of the intrinsic dg m /dV g versus V g curves, three clear peaks are observed corresponding to the two threshold voltages V th1 and V th2 and the flat-band voltage V fb . These results suggest that in the investigated p-type JLTs, the high series resistance affects the dg m /dV g behavior more than short-channel effects. In contrast, in n-type JLTs exhibiting low series resistance, in short-channel devices the short-channel effects affect the dg m /dV g behavior more than the R sd effects [START_REF] Jeon | Impact of series resistance on the operation of junctionless transistors[END_REF]. The extracted device parameters V th1 , V th2 and V fb as a function of the channel length are presented in Fig. 7. In the JLTs of the studied technology, high Fig. 6 Measured plots of dg m /dV g versus V g in p-type triple gate JLTs with fin height H fin = 10 nm, fin width W fin = 26 nm and channel lengths L = 75, 55, 35 and 25 nm. The dashed lines represent the as-measured dg m /dV g versus V g curves and the solid lines their decomposition into three Gaussian curves. Fig. 7. Extracted experimental parameters V th1 , V th2 and V fb as a function of channel length L of isolated p-type triple gate JLTs with fin height H fin = 10 nm and fin width W fin = 26 nm. For comparison, the extracted parameters V th1 , V th2 and V fb of a typical array of 50 parallel p-type triple gate JLTs with fin height H fin = 10 nm and fin width W fin = 20 nm and channel length L = 10 m are also presented. series resistance effects affect the dg m /dV g behavior more than short-channel effects. Indeed, the threshold voltages V th1 and V th2 remain almost invariable with the channel length varying from 25 nm to 10 m as shown in Fig. 7.

Conclusion

Detailed experimental study of the threshold voltage in rectangular p-type triple-gate JLTs was carried out using the transconductance derivative method. In devices with channel length varying from 95 to 25 nm, after correcting the drain current from the impact of series resistance, multiple-peak fit analysis of the intrinsic dg m /dV g versus V g curves present three peaks corresponding to the two conduction threshold voltages V th1 and V th2 and flat-band voltage V fb . The effect of series resistance on the dg m /dV g behavior is highlighted. In contrast to n-type JLTs of low series resistance, in the investigated p-type JLTs the high series resistance affects the dg m /dV g behavior stronger than the short-channel effects. Numerical simulations of the holes density distribution reveal the absence of corner effects in p-type triple-gate JLTs due to the unique bulk neutral conduction, whereas V th1 and V th2 correspond to the threshold voltages of the top gate and side gates, respectively. The correct extraction of flat-band voltage is verified with numerical simulation of the holes density distribution at different gates voltages. Numerical simulations and experimental results of p-type JLTs with variable being the fin width indicate that the observed threshold voltages V th1 and V th2 are correlated with the different density of interface states at the side gates and top gate, respectively. The overall results show that the two threshold voltages in p-type JLTs can be avoided by reducing the implantation process-induced density of top and side gates interface states.

Fig. 2

 2 presents the transfer characteristics of p-type JLTs composed by 50 parallel nanowires with fixed channel length L = 10 m, fin height H fin = 10 nm and fin widths W fin= 20, 40 and 190 nm,

Fig. 1 .

 1 Fig. 1. (a) Measured drain current as a function of gate voltage of p-type triple gate JLT with fin height H fin = 10 nm, fin width W fin = 26 nm and channel lengths L = 95 and 75 nm. (b) Total resistance R tot as a function of the inverse Y-function 1/ 2 / m d g

Fig. 2 .

 2 Fig. 2. Drain current as a function of gate voltage of p-type triple gate JLTs composed of an array of 50 parallel nanowires with channel length L = 10 m, fin height H fin = 10 nm and fin width W fin = 20, 40 and 190 nm, measured at drain voltage V d = -30 mV.

Fig. 3 .Fig. 4 .

 34 Fig. 3. (a) Plots of dg m /dV g versus V g of the p-type triple gate JLT of Fig. 1(a). The dashed line represents the asmeasured dg m /dV g versus V g curve, the symbols the R sd -corrected experimental dg m /dV g versus V g curve and the solid lines its decomposition into three Gaussian curves. (b) Effect of the series resistance R sd on the behavior of dg m /dV g .

Fig. 5 .

 5 Fig. 5. Hole density distributions at drain voltage V d = -30 mV and gate voltages (a) V g = V th1 = -0.06 V, (b) V g = V th2 = -0.56 V and (c) V g = V fb = -1.13 V in the middle of the channel of the rectangular p-type triple gate JLT with dimensions: L = 95 nm, H fin = 10 nm and W fin = 26 nm. The insets show the logarithm of the holes concentration distributions.