M Lazar

S Sejil

L Lalouat

C Raynaud

D Carole

D Planson

G Ferro

F Laariedh

C Brylinski

H Morel

P-type doping of 4H-SiC for integrated bipolar and unipolar devices

Keywords: ion implantation, VLS epitaxial growth, RIE, JFET

P-type 4H-SiC layers formed by ion implantation need high temperature processes, which generate surface roughness, losing and incomplete activation of dopants. Due to dopant redistribution and channeling effect, it is difficult to predict the depth of the formed junctions. Vapor-Liquid-Solid (VLS) selective epitaxy is an alternative method to obtain locally highly doped p-type layers in the 10 20 cm -3 range or more. The depth of this p-type layers or regions is accurately controlled by the initial Reactive-Ion-Etching (RIE) of the SiC. Lateral Junction Field Effect Transistor (JFET) devices are fabricated by integrating p-type layers created by Al ion implantation or VLS growth. The p-type VLS layers improve the access resistances on the electrodes of the fabricated devices.

Introduction

Power discrete SiC devices as Schottky diodes, Bipolar Junction Transistors (BJT), Junction Field Effect Transistors (JFET) and MOSFET are already commercialized. To improve performance and reliability in power electronics, a monolithic integration of this type of devices in SiC single crystal wafers is recommended. Thus the size of converters is reduced, the switching operation is improved and the power loss is decreased. Monolithic integration of SiC devices was already invoked to build vertical JFET and Junction Barrier Schottky diodes [1] and vertical or lateral BJT [2]. Industrial development of the SiC monolithic technology is still in progress, delayed by the need to improve some technological steps, particularly those that can enhance the quality of semi-insulating SiC substrates or that can reduce the resistivity of the p-type layers.

In this paper, SiC p-type doping by aluminum (Al) ion implantation is compared to the alternative VLS epitaxy method to fabricate integrated power devices.

Al ion implantation

P-type localized regions in SiC are in general obtained by ion implantation which is still the unique method of local doping due to the very slow diffusion of dopants in SiC. Moreover, due to the high activation energies of the p-type dopants in SiC (~0.2 eV for aluminum and ~0.3 eV for boron), obtaining a net doping level requires to implant higher doses of ions. That leads to create numerous crystalline defects up to local amorphisation of the material if the ion implantation is processed at room temperature (RT).

To recover a good crystalline quality and to activate the implanted impurities by migration of implanted ions in SiC atomic substitutional sites (for example, Al is a dopant only when it substitutes to a Si site), a high temperature annealing is necessary, typically in the range of 1600-1800°C [3]. Figure 1 presents sheet resistances (R sh ) measured with a four-point probe technique on p-type layers created by Al ion implantation. Post-implantation annealing has been performed during 30 min or one hour at various temperatures from 1600°C to 1800°C. R sh decreases when either the annealing temperature or the annealing time increases. The linearity of the R sh variations with the post-implant annealing temperature, proves a nonsaturation of the electrical dopant activation even after annealing at 1800°C [4]. [4].

Fig.1. Sheet resistance measured at RT by four-point probe technique on p-type SiC layers created by Al ion implantation and annealed at different temperatures and durations

The annealing temperature is limited by the surface deterioration of the implanted layers. A strong degradation of the surface occurs by sublimation of Si from the SiC, resulting in a rough surface. Surface roughness limits device performances by inducing an increase of leakage currents in off-state and a degradation of the breakdown voltages. The roughness increases when either the annealing temperature or the annealing time increases (Fig. 2a) by forming terraces of several tens of nanometers (Fig. 2b). nm By increasing temperature, the sublimation of Si can induce also an etching effect of the SiC surface and thus a significant loss of implanted dopants in the near surface region [5]. Figure 3 presents Al implanted doping profiles measured before and after annealing. Compared to samples just implanted, the Al profiles after annealing seems to be offset toward the surface. The quantity of the dopant loss increases with the annealing temperature. It is worth mentioning that on these samples the ion implantation was performed at 300°C to avoid SiC amorphization [6]. In amorphous layers, a redistribution of the dopants occurs, this phenomenon being discussed thereafter at the end of this section. Surface deterioration and the shift towards the surface of the implanted dopants should induce an increase of the sheet resistance of the p-type layers which is in contradiction with the results we present in figure 1 where the sheet resistance decreases by increasing the postimplantation annealing temperature. The p-type SiC sheet resistances from figure 1 have been measured on samples whose surfaces were protected with a dedicated cap (detailed hereafter). Therefore, in figure 1 the decreasing of the sheet resistances either with the temperature and the duration of the annealing is fundamentally caused by the increase of the dopant activation.

Si sublimation has been reported to be prevented by capping the surface with AlN [7], graphite (C) [8] or by using silane partial overpressure during annealing [9]. In this later case an accurate control of the gas flux is necessary to prevent Si droplet deposition.

However, the efficiency of these protections is shown to be temperature limited except for the graphite layer which resists up to 1800°C. The graphite cap layer is obtained by baking at ~750°C a photoresist layer deposited by spin-coating at the SiC wafer surface. This method is widely used today in SiC device fabrication needing doping by ion implantation.

The C cap showed in general a very good preservation of the sample surface state. Nevertheless, the surface morphology after annealing was shown to depend on implantation parameters (temperature and total doses), even when using a graphite cap protection. The use of the graphite cap influences also the diffusion of dopants and their activation rate [10]. Figure 4 presents AFM images on Al implanted SiC samples measured after annealing with and without a C-cap. After annealing the C-cap has been removed by an oxygen dry plasma etching. For an Al total dose of 1.75×10 15 cm -2 the presence of the C-cap during annealing allowed a very good preservation of the sample surface sate. After 1800°C/30 min annealing, the average roughness is 0.46 nm, slightly higher than for an unimplanted sample (0.31 nm). Preserving the surface state during annealing by using the C-cap allowed us to focus on the study of the crystal recovery of amorphous SiC layers created by Al room temperature (RT) ion implantation. The presence of the amorphous layers, in general, is identified and localized by Rutherford Backscattering Spectrometry in the Channeling mode (RBS/C) measurements. Figure 5 presents the example of the 1.75×10 15 cm -2 Al implanted sample at RT. When ion implantation is performed at RT, dopant redistribution occurs also during the annealing, by a gettering phenomenon localized at the proximity of the end-of-range defects formed at the initial amorphous-crystalline interface [11]. In figure 6, SIMS (Secondary Ion Mass Spectroscopy) dopant profiles are presented for four total implanted doses after 1700°C/30min annealing. After annealing, dopant diffusion increases with the dopant dose. [0001]

[0001]

[1120]

[1100]

[0001]

C Si

(b)

[0001]

[0001]

[1120]

[1120]

[1100]

[0001]

C Si

(c) rotation

[0001]

[0001]

[1120]

[1120]

[1100]

[0001]

C Si

(d) tilt

[0001]

[0001]

[1120]

[1120]

[1100] [0001] C Si

The shape of the dopant profile considerably affects the electrical behavior of devices based on these p-n junctions. Dopant redistribution may have a positive influence on the increase of the p-n junction breakdown voltage. Nevertheless, graduate junctions are in general not compatible with high levels of carrier injection.

Implanted dopants profiles and the depth of the formed junctions are also difficult to be accurately estimated due to the channeling effect. As 4H-SiC presents a hexagonal crystalline structure, a strong channeling effect is observed especially in the case of Al ions. Figure 7 presents the crystalline structure of the 4H-SiC polytype and disorientation of the ion implanted surface (with two defined angles rotation and tilt) in order to avoid or at least minimized the channeling effect. The channeling of the Al ions is quite difficult to control and to eliminate in 4H-SiC even when in addition to the surface disorientation an amorphous layer as SiO 2 is present in the surface. To estimate the doping profile of Al implanted ions, dedicated Monte-Carlo software has been developed considering the crystalline structure of the SiC [12]. Electrical characteristics of the final fabricated SiC devices could be predicted, but with uncertainties due to the difficulty to estimate the depth of the different p-type layers.

Vapor-Liquid-Solid Growth

An alternative method to locally dope SiC is Vapor-Liquid-Solid (VLS) selective epitaxy. Highly p-type doped SiC can be obtained leading to considerable reduction of both the p-type SiC resistivity and the specific resistance of ohmic contacts formed on it.

The technological process is detailed in [13] and is schematically presented in Fig. 8. The depth of the p-type layers is accurately controlled by the initial plasma dry etching of the SiC, performed to create the wells where the Al-Si stacks are deposited and converted in 4H-SiC by VLS epitaxy. Thus, as Al ion implantation, our VLS epitaxy is a local doping method to create p-type 4H-SiC layers.

Al-Si stacks are deposited by electron-beam PVD deposition, thicknesses of the layers and deposition rate are precisely monitored by quartz piezoelectric crystals. Dry etching of the SiC is made by Reactive-Ion-Etching (RIE) or Inductively-Coupled-Plasma (ICP) with SF 6 /O 2 chemistry and using a Ni hard mask defined by photolithography at the surface of the SiC samples. [13].

Fig. 8 -Schematic description of the VLS process: a) deposition of Al-Si stack inside the trenches formed by RIE or ICP in SiC, b) melting of Al-Si and addition of propane for the VLS growth, c) formation of P + -type SiC

During the VLS growth the samples are heated in a vertical cold wall reactor (Figure 9) up to 1100°C. This value is very low compared to temperatures needed for post-implantation annealing (~1700°C) or classical SiC CVD epitaxy (~1400°C-1600°C), reducing thus device cost fabrication if this alternative method is implemented at an industrial scale. Ohmic contacts have been evaluated on these p-type VLS layers with patterned Transfer Length Method (TLM) structures using a specific Ni-Ti-Al metal alloy [15]. The contact is ohmic even without post-metallization annealing. Specific contact resistances as low as 1.3×10 -6 Ω.cm 2 have been obtained after 500-800°C Rapid Thermal Annealing performed under a pure argon atmosphere [16].

Nevertheless, the VLS growth is still an immature technological process. Works remain to identify the steps to be optimized in order to improve the quality of the obtained pn junctions, which still suffer from a poor interface between the p-type VLS layers and the initial substrate. Figure 11 presents current-voltage measurements on two PiN diodes with p-type emitters fabricated by VLS growth. These measurements were made on vertical diode configuration with a "Signatone S1160" probe station and Keithley 2410 or 2636 Source Measure Unit (SMU). Current densities obtained on diodes with varied sizes from 150 µm to 1mm are plotted in forward bias. For the diodes with larger sizes, the constant current plateau obtained at the end of the current densities -voltage characteristic, is a measurement artifact due to the SMU current limit.

In general very high current densities are obtained (several thousand of A/cm 2 ) thanks to the high Al level doping in the p-type emitters that VLS method allows (Figure 10 -SIMS spectra of the Al doping profile). But if for the sample presented in Fig. 11a the current density is independent on the diode sizes and a typical threshold (~3V for 4H-SiC P/N junctions) is obtained, for the sample presented in Fig. 11b lower threshold is obtained in forward regime. Moreover, for these diodes the current density varies with the diameter sizes. However in reverse bias, for the all presented VLS diodes, low values are obtained for the leakage current.

Integrated lateral JFETs

Lateral JFET (LJFET) transistors with n-type and p-type channels have been fabricated, integrated monolithically within the same SiC wafer. The ultimate goal of this study is to realize an integrated inverter leg with complementary LJFET switches on the same 4H-SiC chip (Figure 12). Several batches have been fabricated, the first one (detailed in [17]) with p + wells created by Al ion implantation, and the second one (detailed in [18]) with VLS p + wells. The p + wells were utilized as source and drain contacting layers for the P-JFET and as p + gate for the N-JFET. For both batches, the n + wells have been obtained by nitrogen ion implantation.

For the structure with both p + and n + wells realized by ion implantation, the thickness control of the p-type channel was problematic due to the channeling of the ions implanted to dope the gate (Figure 13). This phenomenon was less marked for the N-JFETs due to the higher project ranges (the average penetration depths during ion implantation) of the nitrogen ions used to dope the n-channel. Whereas the p-type channels of the P-JFETs were unintentionally normally-off or even always blocked. Finally an experimental study based on SIMS measurements and MC simulations was necessary to consider the channeling effect in the 4H-SiC hexagonal crystalline structure. A technological process has been defined to obtain devices working despite the presence of the channeling effect.

The VLS process allowed us to accurately control the thickness of the p-type wells. However, the electrical characteristics of the JFET with p + wells created by VLS are affected by the interface between the VLS p + gate and the n-type channel, which is a CVD epilayer. The structural defects observed by Transmission Electronic Microscopy [19] explain the high reverse leakage current measured between gate and source. Nevertheless, the obtained results have shown that the p + VLS layers could already be used to improve the access resistances on the P-JFET transistors where the VLS p + layers have been utilized as source and drain. Figure 14 compares typical results we obtained for the first p-type JFETs fabricated by Al ion implantation and those made by VLS. The VLS process allows us to multiply drain-source current by more than one order of magnitude.

Conclusion

Industrial development of the SiC monolithic technology needs a significant drop of the resistivity of the p-type layers and of the ohmic contact formed on them. A technological breakthrough will be obtained when an alternative method to ion implantation will be found in order to locally dope SiC, using only low temperature processes, thus preserving the surface state and allowing to precisely predict the junction depths. VLS selective epitaxy is a good candidate for such objectives. If creating reliable P/N junctions by this method seems to be yet premature, highly p-type doped layers and ohmic contacts with specific contact resistances as low as 1.3×10 -6 Ω.cm 2 have already been obtained by VLS. Therefore we demonstrated that VLS layers could already be utilized on the top of p-type CVD epilayers in order to improve the access resistances.

Fig2.

  Fig2. Surface deterioration of SiC: arithmetic average roughness (R a ) measured with Tencor alpha-step stylus after annealing at different temperatures and durations (a), Atomic Force Microscopy (AFM) measurement on a virgin SiC sample annealed at 1700°C during 30 min (b).

Fig. 3 .

 3 Fig.3. Al profiles measured by Secondary Ion Mass Spectrometry (SIMS) before and after postimplantation annealing highlighting the lost of the dopants after high temperature annealing.

Fig. 4 .

 4 Fig.4. 5×5 µm 2 AFM images for 1.75×10 15 Al.cm -2 implanted sample after 1800°C/30 min annealed without graphite cap (a) and with a graphite cap (b). The right AFM image (c) is for 3.5×10 15 Al.cm -2 implanted sample after 1800°C/30 min annealed with a graphite cap.

Fig. 5 .

 5 Fig.5. RBS/C spectra on 4H-SiC samples implanted at RT with 1.75×10 15 cm -2 Al total dose, before and after annealing

Fig. 6 .

 6 Fig.6. SIMS doping profiles obtained after annealing on samples implanted Al with varying total doses

Fig. 7

 7 Fig.7 4H-SiC crystalline structure with cross-section (a), top view (b) and disorientation of the surface with two defined angles, rotation (c) and tilt (d), in order to avoid the channeling effect.

Fig. 9 -

 9 Fig. 9 -Cold wall vertical reactor utilized during VLS growth P + -type SiC layers obtained by VLS have high Al doping concentrations, in the 10 20 cm -3 range. Fig. 10 presents a typical profile obtained by SIMS [14].

Fig10.

  Fig10. Al doping profile from SIMS on VLS SiC p + layers.

Fig. 11 .

 11 Fig. 11. Current-voltage characteristics of PiN 4H-SiC diodes with the emitter fabricated by VLS. Current densities versus applied bias are presented for various diode sizes; a) and b) correspond to two samples with different technological and VLS growth parameters. The constant current plateau obtained for the diodes with larger sizes is due to the Source Measure Unit current limit.

Fig. 12 .

 12 Fig. 12. Simplified inverter diagram with a leg based on complementary lateral JFETs.

Fig. 13 .

 13 Fig. 13. Schematic presentation of the JFET channels hidden by the ion implanted gates.

Fig. 14 .

 14 Fig. 14. I ds (V ds ,V gs ) characteristics for P-type JFETs with p-wells created by Al ion implantation (solid symbols) and VLS process (open symbols)

Acknowledgments

The authors gratefully thank NanoLyon Technological Platform team for technological means and for training our staff. The authors would also like to thank the financial support of ANR-USAR.

P-JFET by VLS process

V gs =40V I ds (mA)

V ds (V)

P-JFET by Al ion implantation


Roland Badeau
email: @telecom-paristech.fr

Common mathematical framework for stochastic reverberation models

Keywords: Reverberation, Diffusion, Room impulse response, Stochastic models a) roland.badeau

In the field of room acoustics, it is well known that reverberation can be characterized statistically in a particular region of the time-frequency domain (after the transition time and above Schroeder's frequency). Since the 1950s, various formulas have been established, focusing on particular aspects of reverberation: exponential decay over time, correlations between frequencies, correlations between sensors at each frequency, and time-frequency distribution. In this paper, we introduce a stochastic reverberation model, that permits us to retrieve all these well-known results within a common mathematical framework. To the best of our knowledge, this is the first time that such a unification work is presented. The benefits are multiple: several formulas generalizing the classical results are established, that jointly characterize the spatial, temporal and spectral properties of late reverberation.

I. INTRODUCTION

When a microphone records a sound produced by an audio source in a room, the received signal is made of several contributions [START_REF] Kuttruff | Room Acoustics, Fifth Edition[END_REF]: firstly, the direct sound, that corresponds to the direct propagation of the sound wave from the source to the microphone, then a few early reflections, that are due to the sound wave reflections on the various room surfaces (walls, floor, ceiling. . . ), and finally the late reverberation: after a time called transition time (Polack, 1992 , 1993), reflections are so frequent that they form a continuum and, because the sound is partially absorbed by the room surfaces at every reflection, the sound level decays exponentially over time. This phenomenon is called reverberation, and it can be modeled as the convolution between the source signal and a causal room impulse response (RIR), made of a few isolated impulses before the transition time, and of a continuous, exponentially decaying, random process in late reverberation. The Fourier transform of the RIR is called room frequency response (RFR), and the modal theory (Balian and Bloch, 1970;[START_REF] Maa | Distribution of eigentones in a rectangular chamber at low frequency range[END_REF][START_REF] Polack | The relationship between eigenfrequency and image source distributions in rectangular rooms[END_REF] shows that its profile is qualitatively similar to that of the RIR: below a frequency called Schroeder's frequency, the RFR is made of a few isolated modes, and above this frequency the modes become so dense that they can be represented as a continuous random process (Schroeder, 1962 , 1987; Schroeder and Kuttruff, 1962).

To sum up, reverberation can be modeled as a stochastic process in a rectangular region of the time-frequency domain [START_REF] Jot | Analysis and synthesis of room reverberation based on a statistical time-frequency model[END_REF], as depicted in Fig. 1. If in addition the source and the microphones are located at least a half-wavelength away from the walls, then in this time-frequency region, the sound field can generally be approximated as diffuse [START_REF] Cremer | Principles and applications of room acoustics, 1, Chap. (ii)[END_REF][START_REF] Joyce | Sabine's reverberation time and ergodic auditoriums[END_REF]Schultz, 1971). Diffusion is a consequence of the reflections on the room surfaces not being specular (i.e. mirror-like), but rather scattered in various directions, as represented in Fig. 2. After many reflections, the sound field can be considered as isotropic: the sound waves come uniformly from all directions. (Jot et al., 1997, p. 30) and (Baskind, 2003, p. 20)).

Historically, the first stochastic reverberation model is due to [START_REF] Schroeder | Frequency-correlation functions of frequency responses in rooms[END_REF] and [START_REF] Moorer | About this reverberation business[END_REF]: the RIR at microphone i is

h i (t) = b i (t)e -αt 1 t≥0 (1)
where α > 0 and b i (t) is a centered white Gaussian process. Parameter α is related to the reverberation time T r in seconds by the equation T r = 3 ln (10) α . The Gaussian distribution of b i (t) arises from the central limit theorem: in late reverberation, h i (t) is the sum of many independent contributions. Schroeder (1962 , 1987); [START_REF] Schroeder | On frequency response curves in rooms. Comparison of experimental, theoretical, and Monte Carlo results for the average frequency spacing between maxima[END_REF] also noticed that the independency of the samples h i (t) implies that the RFR, defined as their Fourier transform 1 F h i (f ), is a stationary random process. From (1), he derived several formulas that can be summarized by expressing the complex autocorrelation function 2 of F h i (f ):

corr [F h i (f 1 ), F h i (f 2 )] = 1 1 + ıπ f 1 -f 2 α .
(2)

Following a similar approach in the spectral domain, under the diffuse field assumption, [START_REF] Cook | Measurement of correlation coefficients in reverberant sound fields[END_REF] computed the correlation at frequency f between two sensors at distance D (with c > 0 the speed of sound in the air):

corr [F h 1 (f ), F h 2 (f )] = sinc 2πf D c . (3)
Equation ( 3) was later generalized to combinations of pressure and velocity sensors [START_REF] Jacobsen | The coherence of reverberant sound fields[END_REF] and to differential microphones [START_REF] Elko | Spatial coherence functions for differential microphones in isotropic noise fields[END_REF].

Finally, [START_REF] Polack | La transmission de l'énergie sonore dans les salles (The transmission of sound energy in rooms)[END_REF] generalized model (1) by assuming that b i (t) is a centered stationary Gaussian process, whose power spectral density (PSD) B(f ) has slow variations 3 . Then he showed that the Wigner distribution 4 [START_REF] Cohen | Time-frequency distributions-a review[END_REF] of the RIR is

W h i ,h i (t, f ) = B(f )e -2αt 1 t≥0 . (4)
In order to account for the fact that the attenuation coefficient α actually depends on the frequency f , he also proposed an empirical generalization of (4):

W h i ,h i (t, f ) = B(f )e -2α(f )t 1 t≥0 . (5)
In other respects, based on the billiard theory, Polack (1992 , 1993) also showed that the durations of the various trajectories in a room, from a given source position to the microphone, are distributed according to a Poisson process [START_REF] Chiu | Stochastic Geometry and Its Applications, Chap. 2 "Point processes I -The Poisson point process[END_REF]. In this paper, we propose a unified stochastic model of reverberation, that will permit us to retrieve all formulas (1) to (4) in a common mathematical framework 5 , to establish a link with the Poisson distribution proposed by Polack, and to show how the probability distribution of the RIR, which is impulsive in early reverberation, converges to the Gaussian distribution in late reverberation. In addition, this model will also permit us to go deeper into the description of the statistical properties of the RIR over the space, time and frequency domains, and to prove several new results.

This paper is structured as follows: Section II presents important mathematical definitions and notation that will be used throughout the paper. Then our general stochastic reverberation model is introduced in Section III. The statistical properties of this model at one sensor are investigated in Section IV. The statistical relationships between two sensors are then analyzed in Section V. The new model is validated experimentally in Section VI, by comparing the predicted second order statistics with their estimates from a database of synthetic RIRs. Finally, some conclusions and perspectives are presented in Section VII. In order to make the main discussion as clear as possible, all mathematical proofs were moved to Appendices A to G.

II. MATHEMATICAL DEFINITIONS

• N: set of whole numbers

• R, C: sets of real and complex numbers, respectively

• R + : set of nonnegative real numbers

• ı = √ -1: imaginary unit • x (bold font), z (
• L p (V )
, where V is a Borel set and p ∈ N\{0}: Lebesgue space of measurable functions

f of support V , such that f p = ( V |f (x)| p dx) 1 p < +∞ • L ∞ (V ), where V is a Borel set: Lebesgue space of essentially bounded functions f of support V (i.e. such that f ∞ = ess sup V |f | < +∞)
• δ: Dirac delta function

• . 2 : Euclidean/Hermitian norm of a vector or a function

• z: complex conjugate of z ∈ C • x : transpose of vector x • S 2 : unit sphere in R 3 (S 2 = {x ∈ R 3 ; x 2 = 1}) • E[X]: expected value of a random variable X • φ X (θ) = E[e ıθX ]: characteristic function of a real random variable X • Covariance of two complex random variables X and Y : cov[X, Y ] = E[(X -E[X])(Y -E[Y ])] (6)
• var[X] = cov[X, X]: variance of a random variable X

• Correlation of two complex random variables X and Y :

corr[X, Y ] = cov[X, Y ] var[X] var[Y ] (7)
• P(λ): Poisson distribution of parameter λ > 0:

N ∼ P(λ) ⇔ P (N = n) = e -λ λ n n! ⇔ φ N (θ) = e λ(e ıθ -1) (8)
• sinc(x) = sin(x)

x : cardinal sine function

• 1 A : indicator function of a set A (1 A (x) is 1 if x ∈ A or 0 if x / ∈ A)
• ψ(t) = ψ(-t): conjugate and time-reverse of ψ : R → C

• Convolution of two functions ψ 1 and ψ 2 : R → C:

(ψ 1 * ψ 2 )(t) = u∈R ψ 1 (u)ψ 2 (t -u)du
• Fourier transform of a function ψ : R → C:

F ψ (f ) = t∈R ψ(t)e -2ıπf t dt (f ∈ R) (9)
• Two-sided Laplace transform of a function ψ : R → C:

L ψ (s) = t∈R ψ(t)e -st dt (s ∈ C) (10)
• Wigner distribution (a.k.a. Wigner-Ville distribution) of two second-order random processes ψ 1 (t) and ψ 2 (t):

W ψ 1 ,ψ 2 (t, f ) = R cov[ψ 1 (t + u 2 ), ψ 2 (t -u 2 )]e -2ıπf u du. (11)

III. DEFINITION OF THE STOCHASTIC MODEL

The model that we present in this section is based on the source image principle (Allen and Berkley, 1979;[START_REF] Kuttruff | Room Acoustics, Fifth Edition[END_REF]. As illustrated in Fig. 3 in the case of specular reflections in a rectangular room 6 , the trajectory inside the room from the real source to the microphone is equivalent to a virtual straight trajectory from a so-called source image which is outside the room. A remarkable property of this principle is that, regardless of the room dimensions, the density of the source images is uniform in the whole space: the number of source images contained in a given disk, of radius sufficiently larger than the room dimensions, is approximately invariant under any translation of this disk.

Since we aim to define a general stochastic reverberation model that holds for any room geometry, based on the billiard theory (Polack, 1992 , 1993), we will consider that the positions of the source images are random and uniformly distributed (note that this assumption The original room walls are drawn with thick lines. A virtual straight trajectory from one source image to the microphone is drawn, along with the real trajectory in the original room.

is a fortiori valid in the case of a diffuse sound field, which is uniform). More precisely, given a Borel set V ⊂ R 3 of finite volume |V |, we will assume that the number N (V ) of source images contained in V follows a Poisson distribution of parameter λ|V |: N (V ) ∼ P(λ|V |).

Mathematically, this is formalized through the concept of Poisson random measures with independent increments: given a non-negative, locally integrable function Λ(x) on R p , the Poisson random increment dN (x) ∼ P(Λ(x)dx) corresponds to an infinitesimal volume |V | = dx. Then for any Borel set V ⊂ R p of finite Lebesgue measure, the number

N (V ) = V dN (x) of points contained in V follows a Poisson distribution of parameter V Λ(x)dx: N (V ) ∼ P( V Λ(x)dx).
In the proposed stochastic reverberation model, we will consider a spatially uniform distribution of the source images in the 3D-space, so that p = 3 and Λ(x) = λ > 0 is constant.

In other respects, we will assume that the sound waves undergo an exponential attenuation along their trajectories, that is due to the multiple reflections on the room surfaces and to the propagation in the air. In this paper we focus on the case of omnidirectional microphones, so we will further assume that this attenuation is isotropic (in accordance with the diffuse field approximation) and independent of the frequency. It will thus only depend on the length of the trajectory, as in [START_REF] Polack | La transmission de l'énergie sonore dans les salles (The transmission of sound energy in rooms)[END_REF].

Finally, we suppose that several microphones 7 indexed by an integer i are placed at arbitrary positions x i in the room.

We end up with the following model:

h i (t) = x∈R 3 h i (t, x) e -α c x-x i 2 dN (x), (12)
where h i (t) is the RIR at microphone i, α > 0 is the attenuation coefficient (in Hz), and c > 0 is the speed of sound in the air (approximately 340 m/s in usual conditions). The impulse h i (t, x), propagated from the source image at position x, is modeled as a coherent sum of monochromatic spherical waves:

h i (t, x) = f ∈R A(f ) e 2ıπf t- x-x i 2 c x -x i 2 df, (13)
where A(f ) is a linear-phase frequency response (in order to ensure coherence). In Appendix A, we show that ( 12) and ( 13) are equivalent to the following model:

Definition 1 (Unified stochastic reverberation model).

Let λ > 0, T > 0, α > 0, and c > 0. Let dN (x) be a uniform Poisson random measure on R 3 with independent increments:

dN (x) ∼ P(λdx). ( 14
) Let g(t) ∈ L ∞ ([-T, T ]), such that F g (0) = dF g df (0) = 0, (15) ∀f ∈ R, L g (α + 2ıπf ) ≥ 0. ( 16
)
At any sensor position

x i ∈ R 3 , h i (t) is defined as ∀t ∈ R, h i (t) = e -α(t-T ) b i (t), ( 17
)
where

b i (t) = x∈R 3 g t -T - x -x i 2 c dN (x) x -x i 2 . ( 18
)
Equivalently, the Fourier transform of h i (t) is

F h i (f ) = L g (α + 2ıπf )e -2ıπf T R 3 e -α+2ıπf c x-x i 2 x -x i 2 dN (x). ( 19
)
This definition calls for comments. Firstly, the linear-phase frequency response A(f ) in ( 13) was parameterized as

A(f ) = L g (α + 2ıπf )e -2ıπf T . (20)
This technical definition aims to simplify the mathematical developments in the next sec-

tions. Secondly, any function g ∈ L ∞ ([-T, T ]) is such that F g (f ) and f → L g (α + 2ıπf ) are
smooth, so F g (0), dFg df (0) and L g (α + 2ıπf ) are well-defined. The constraints ( 15) and ( 16) imposed to g are required to prove Lemma 1 and Proposition 2 in Section IV 8 . Moreover, the support of g is chosen so that h i (t) in ( 17) and b i (t) in ( 18) are causal. Thirdly, the existence of functions g that satisfy these constraints is guaranteed by Lemma 3 in Appendix A.

Now it is time to investigate the properties of this model. In Section IV, we will focus on one sensor at spatial position x i . Then in Section V, we will analyze the spatial relationships between two sensors at different positions x i and x j .

IV. STATISTICAL PROPERTIES AT ONE SENSOR

Let us first introduce an equivalent model definition:

Proposition 1 (Equivalent model definition at one sensor). With the same notation as in Definition 1, we have:

b i (t) = r∈R + g t -T - r c dN (r) r , (21)
F h i (f ) = L g (α + 2ıπf )e -2ıπf T r∈R + e -α+2ıπf c r dN (r) r (22)
where dN (r) are independent Poisson increments on R + : dN (r) ∼ P(4πλr 2 dr).

(

) 23
Proposition 1 is proved in Appendix A. Let us now investigate the statistical properties of this model:

Lemma 1. The random process b i (t) introduced in Definition 1 is such that ∀t ≥ 2T , 1. b i (t) is a centered wide sense stationary (WSS) process 9 , of autocovariance function ∀τ ∈ R, Γ(τ ) = cov[b i (t + τ ), b i (t)] = 4πλc ( g * g)(τ ), ( 24
)
autocorrelation function ∀τ ∈ R, γ(τ ) = corr[b i (t + τ ), b i (t)] = ( g * g)(τ ) g 2 2 , ( 25
)
and power spectral density

∀f ∈ R, B(f ) = F Γ (f ) = 4πλc |F g (f )| 2 . (26)
2. When t → +∞ (i.e. t T ), b i (t) converges in law to a stationary Gaussian process.

Lemma 1 is proved in Appendix C.

Proposition 2 (Statistical properties at one sensor position). With the notation introduced in Lemma 1, the reverberation model in Definition 1 has the following properties:

1. First order moments:

• in the spectral domain:

∀f ∈ R, E[F h i (f )] = 4πλc 2 L g (α + 2ıπf ) e -2ıπf T (α + 2ıπf ) 2 (27)
• in the time domain:

∀t ≥ 2T, E[h i (t)] = 0. ( 28
)
2. Second order moments:

• in the spectral domain:

∀f ∈ R, var[F h i (f )] = 2πλc L g (α + 2ıπf ) 2 /α (29) ∀f 1 , f 2 ∈ R, corr[F h i (f 1 ), F h i (f 2 ]) = e -2ıπ(f 1 -f 2 )T 1 + ıπ f 1 -f 2 α (30)
• in time domain:

∀t ≥ 2T, var[h i (t)] = 4πλc g 2 2 e -2α(t-T ) (31) ∀t 1 , t 2 ≥ 2T, corr[h i (t 1 ), h i (t 2 )] = γ(t 1 -t 2 ) (32)
• in the time-frequency domain:

∀f ∈ R, ∀t ≥ 2T, W h i ,h i (t, f ) = B(f ) e -2α(t-T ) . ( 33
)
3. Asymptotic normality: when t → +∞, h i (t) converges in law to a Gaussian process.

Proposition 2 is proved in Appendix D. Note that ( 26) and ( 15) show that B(f ) is very

flat at f = 0: B(0) = dB df (0) = d 2 B df 2 (0) = d 3 B df 3 (0) = 0.
The asymptotic normality is related to the central limit theorem: when r becomes large, the volume contained between the spheres of radius r and r + dr increases as r 2 dr, and so does the number of source images included in this volume as shown in ( 23), which leads to the addition of an increasing number of independent and identically distributed (i.i.d.) random increments dN (x).

This proposition permits us to retrieve most of the classical results listed in the introduction. Firstly, h i (t) is centered for t ≥ 2T (the fact that it is not centered for t ∈ [0, 2T ] explains why the expected value of the frequency response

E[F h i (f )] in (27) is not zero).
Secondly, (17) corresponds to Schroeder and Moorer's model defined in (1) when T → 0 (in this case the process b i (t) becomes white, and it is Gaussian when t T ), and in the general case it is equivalent to Polack's model [START_REF] Polack | La transmission de l'énergie sonore dans les salles (The transmission of sound energy in rooms)[END_REF]

V. STATISTICAL PROPERTIES BETWEEN TWO SENSORS

Let us now focus on the relationships between two sensors:

Lemma 2. Let us consider the model in Definition 1 at two positions x i and x j ∈ R 3 . Let us define the rectangular window

∀t ∈ R, w(t) = c 2D 1 [-D c , D c ] (t), ( 34
)
where D = x ix j 2 . Then, in addition to the properties listed in Lemma 1, the random

process b(t) = [b i (t), b j (t)
] is also such that

• ∀t ≥ 2T + D c , b(t) is a centered WSS process 9 , of cross-autocovariance function ∀τ ∈ R, Γ i,j (τ ) = cov[b i (t + τ ), b j (t)] = w * Γ(τ ), (35)
cross-autocorrelation function

∀τ ∈ R, γ i,j (τ ) = corr[b i (t + τ ), b j (t)] = w * γ(τ ), ( 36
)
and cross-power spectral density

∀f ∈ R, B i,j (f ) = F Γ i,j (f ) = B(f ) sinc( 2πf D c ). ( 37
)
• When t → +∞ (i.e. t T ), b(t) converges in law to a stationary Gaussian process.

Lemma 2 is proved in Appendix F.

Proposition 3 (Statistical properties between two sensors).

Let us consider the reverberation model in Definition 1 at two positions x i and x j ∈ R 3 , and the rectangular window w(t) defined in (34) with D = x ix j 2 . Then, with the notation introduced in Lemma 1 and in addition to the properties listed in Proposition 2, we also have:

• in the spectral domain:

∀f 1 , f 2 ∈ R, corr[F h i (f 1 ), F h j (f 2 )] = e -αD c -2ıπ(f 1 -f 2 )(T + D 2c ) sinc( π(f 1 +f 2 )D c ) 1 + ıπ f 1 -f 2 α (38)
• in the time domain:

∀t 1 , t 2 ≥ 2T + D c , corr[h i (t 1 ), h j (t 2 )] = w * γ (t 1 -t 2 ) (39)
• in the time-frequency domain:

∀f ∈ R, ∀t ≥ 2T + D 2c , W h i ,h j (t, f ) = B(f )e -2α(t-T ) sinc( 2πf D c ). (40)
• Asymptotic normality: when t → +∞, h(t) converges in law to a Gaussian process.

Proposition 3 is proved in Appendix G. Applying equation (38

) to f 1 = f 2 = f , we get
Cook's formula (3) when α → 0 (no exponential decay). This formula was indeed originally proved by considering plane waves under a far-field assumption [START_REF] Cook | Measurement of correlation coefficients in reverberant sound fields[END_REF]. Besides, equation (39) shows that h i (t) and h j (t) are correlated on a time interval that corresponds to the wave propagation from one sensor to the other. To the best of our knowledge, all formulas in Proposition 3 are novel.

VI. SIMULATION RESULTS

In Propositions 2 and 3, we have listed several statistical properties of the reverberation model introduced in Definition 1. Some of these properties are already-known experimental facts, such as the zero mean and asymptotic Gaussianity of late reverberation. In this section, we aim to experimentally validate the second-order statistics specific to this new stochastic reverberation model, that were not predicted by the existing models listed in Section I. Firstly, we note that at the second order, the spectral domain, time domain, and time-frequency domain formulas in both Propositions 2 and 3 are mutually equivalent; we can thus focus on the time-frequency domain only. Secondly, from ( 33) and ( 40), we get the closed-form expression of the time-frequency correlation:

∀f ∈ R, ∀t ≥ 2T + D 2c , W h i ,h j (t, f ) W h i ,h i (t, f ) = sinc( 2πf D c ). ( 41
)
Since ( 33) was already known [START_REF] Polack | La transmission de l'énergie sonore dans les salles (The transmission of sound energy in rooms)[END_REF] and since (40) can actually be deduced from ( 33) and ( 41), the novelty regarding the second order statistics of late reverberation in Propositions 2 and 3 lies in the single equation ( 41). Therefore we just need to experimentally validate (41).

For this purpose, we need an experimental setup with the greatest spatial diversity possible: a high number (typically a thousand) of RIRs, with random source position x, random middle position of the sensors

x i +x j
2 , random orientation of the sensor axis

h i ,h i (t, f ) and W h i ,h j (t, f ) were esti- mated as W h i ,h i (t, f ) = 1 M M m=1 |S (m) h i (t, f )| 2 and W h i ,h j (t, f ) = 1 M M m=1 S (m) h i (t, f )S (m) h j (t, f ),
where S (m) an overlap of 64 samples in the time domain. The distributions

h i (t, f ) (resp. S (m) h j (t,
W h i ,h i (t, f ) and W h i ,h j (t, f )
obtained in this way are smoothed estimates of W h i ,h i (t, f ) and W h i ,h j (t, f ) in the timefrequency domain [START_REF] Cohen | Time-frequency distributions-a review[END_REF]. We observe in Fig. 4 that the estimated time-frequency correlation is noisy but, as expected, it seems approximately constant as a function of time, and looks like a cardinal sine function as a function of frequency.

In order to accurately compare the frequency variation with Cook's formula (3), we computed the ratio ρi,j (f ) of the projections of the estimated distributions onto the frequency version of this noisy signal. We can observe that the average time-frequency correlation is indeed approximately independent of time.

axis, i.e. ρi,j (f ) = R W h i ,h j (t, f )dt R W h i ,h i (t, f )dt (

VII. CONCLUSION AND PERSPECTIVES

In this paper, we proposed a new stochastic model of reverberation, that permitted us to retrieve various well-known results within a common framework. This unification work resulted in several new results, that jointly characterize the properties of late reverberation in the space, time, and frequency domains. The most noticeable result in our opinion is (40), which very simply makes the connection between Polack's time-frequency model ( 4) and

Cook's formula (3).

Although this model was motivated by physical assumptions that only hold in a particular region of the time-frequency domain (after the transition time and above Schroeder's frequency), from a signal processing perspective however, one of its most interesting features is that it is also applicable before the transition time and below Schroeder's frequency. Indeed, since the parameter of the Poisson distribution dN (r) in ( 23) increases quadratically with the distance r, the model permits us to describe both the impulsiveness of the RIR before the transition time, and its asymptotic normality in late reverberation. However, because it does not take the exact geometry of the acoustic scene into account (i.e. the room geometry and the source and sensors positions), it is able to capture neither the accurate temporal localization in the RIR of the isolated peaks related to early reflections before the transition time, nor the spectral localization of the isolated room modes below Schroeder's frequency. In practice, this blindness to such fine details does not prevent us to estimate the model parameters from the full RIRs, in the whole time-frequency domain. The missing geometrical information can still be recovered afterwards in an ad-hoc manner, for instance by estimating the Poisson random measure dN (x) given the observed RIRs. Indeed, the knowledge of dN (x) permits us to identify the spatial positions of the source images, which carry the geometrical information related to both the early reflections and the room modes.

Finally, we end up with a stochastic model involving very few parameters (α, λ, filter g, and the distances between microphones), that is able to describe reverberation in the whole time-frequency domain. We thus believe that this model has an interesting potential in a variety of signal processing applications.

However this reverberation model, as it is presented in this paper, is not yet suitable for modeling real RIRs. Indeed, one assumption has to be relaxed: the attenuation coefficient α is not constant but rather depends on frequency in practice, as in Polack's generalized time-frequency model ( 5). In a future paper, we will thus present a generalization of the proposed model where we will introduce a frequency-varying attenuation coefficient. A second generalization of this model would be to represent acoustic fields that are not perfectly diffuse. Finally, the generalization to directional microphones is straightforward, by using the same approach as presented in [START_REF] Elko | Spatial coherence functions for differential microphones in isotropic noise fields[END_REF].

Our future contributions will also focus on the signal processing aspects of this work: we will propose a fast algorithm for estimating the parameters of the model in discrete time.

Finally, our ultimate goal is to investigate the potential of this model in applications such as source separation, dereverberation, and synthetic reverberation.

We can now derive equations ( 17), ( 18) and ( 19) in Definition 1. By substituting ( 13) and ( 20) into (12), we get

h i (t) = f ∈R L g (α + 2ıπf )e 2ıπf (t-T ) x∈R 3 e -α+2ıπf c
x-x i 2

xx i 2 dN (x) df which shows that h i (t) as defined in ( 12) is the inverse Fourier transform of ( 19) and therefore proves (19). Besides, applying the Fourier transform ( 9) to ( 17) also leads to ( 19), which proves the equivalence between ( 17)-( 18), and ( 12)-( 13)-( 20).

Finally, let us prove Proposition 1.

Proof of Proposition 1. Let dN (r) = u∈S 2 dN (ru). Then ( 14) implies ( 23). The change of variables x = x i + ru with r ∈ R + and u ∈ S 2 in ( 18) and ( 19) leads to ( 21) and ( 22).

APPENDIX B: PROPERTIES OF THE POISSON RANDOM MEASURE

The following Proposition 4 characterizes the distribution of the Poisson random measure introduced in Section III, by providing the closed-form expression of the characteristic functional of this random measure:

Proposition 4. Let dN (x) be a Poisson random measure on R p with independent increments: dN (x) ∼ P(Λ(x)dx), where Λ is a non-negative, locally integrable function. We consider the stochastic integral I = ψ(x)dN (x), where ψ is an essentially bounded realvalued function with compact support, and its characteristic function φ

I (θ) = E[e ıθI ]. Then we have ∀θ ∈ R, φ I (θ) = exp (e ıθψ(x) -1)Λ(x)dx , (B1) which yields ∀θ ∈ R, ln φ I (θ) = +∞ n=1 ı n n! θ n κ n (B2)
where ∀n ∈ N\{0},

κ n = ψ(x) n Λ(x)dx (B3)
is the n-th order cumulant of I. In particular, we get

E[I] = ψ(x)Λ(x)dx. (B4)
Proof. In the particular case where ψ is the indicator function of a Borel set V of finite

Lebesgue measure, we have I = N (V ) ∼ P( V Λ(x)dx). Therefore ( 8) yields (B1), because both members of (B1) are equal to exp((e ıθ -1) V Λ(x)dx). Consequently, since (B1) holds for any indicator function and since the increments dN (x) are assumed independent, it is easy to show that (B1) also holds for simple functions 12 . Following the construction of the Lebesgue integral, we conclude that (B1) also holds for any essentially bounded function ψ with compact support.

Then the Taylor series expansion of the second exponential function in (B1) yields ∀θ ∈ R,

ln φ I (θ) = +∞ n=1 ı n n! θ n ψ(x) n Λ(x)dx, (B5)
which proves (B2). Note that the permutation of the integral and the sum in (B5) is allowed

thanks to Fubini's theorem, because +∞ n=1 1 n! |θ| n |ψ(x)| n Λ(x)dx ≤ e |θ| ψ ∞ V Λ(x)dx <
+∞, since ψ is essentially bounded, V = support(ψ) is compact, and Λ is locally integrable.

In particular, for n = 1, we get E[I] = κ 1 thus (B3) yields (B4).

The following Proposition 5 provides the closed-form expression of the joint cumulants of two stochastic integrals involving the Poisson random measure:

Proposition 5. Let dN (x) be a Poisson random measure on R p with independent increments: dN (x) ∼ P(Λ(x)dx), where Λ is a non-negative, locally integrable function. We consider two stochastic integrals I 1 = ψ 1 (x)dN (x) and I 2 = ψ 2 (x)dN (x), where ψ 1 and ψ 2 are essentially bounded real-valued functions with compact support, and their joint

characteristic function φ I 1 ,I 2 (θ 1 , θ 2 ) = E[e ı(θ 1 I 1 +θ 2 I 2 ]. Then we have ∀θ 1 , θ 2 ∈ R, ln(φ I 1 ,I 2 (θ 1 , θ 2 )) = +∞ n=1 ı n n! n k=0 n k θ k 1 θ n-k 2 κ k,n-k , (B6)
where

∀n 1 , n 2 ∈ N such that n 1 + n 2 ≥ 1, κ n 1 ,n 2 = ψ 1 (x) n 1 ψ 2 (x) n 2 Λ(x)dx (B7)
is the (n 1 , n 2 )-th order joint cumulant of (I 1 , I 2 ). In particular, we get

cov[I 1 , I 2 ] = ψ 1 (x)ψ 2 (x)Λ(x)dx. (B8)
Proof. By applying (B1) to θ = 1 and to ψ(x) = θ 1 ψ 1 (x) + θ 2 ψ 2 (x), we get ∀θ 1 , θ 2 ∈ R,

ln φ I 1 ,I 2 (θ 1 , θ 2 ) = (e ı(θ 1 ψ 1 (x)+θ 2 ψ 2 (x)) -1)Λ(x)dx = +∞ n=1 ı n n! (θ 1 ψ 1 (x) + θ 2 ψ 2 (x)) n Λ(x)dx = +∞ n=1 ı n n! n k=0 n k θ k 1 θ n-k 2 ψ 1 (x) k ψ 2 (x) n-k Λ(x)dx, ( B9
)
which proves (B6). Note that the permutation of the integral and the two sums in (B9) is allowed thanks to Fubini's theorem, because

+∞ n=1 1 n! n k=0 n k |θ 1 | k |θ 2 | n-k |ψ 1 (x)| k |ψ 2 (x)| n-k Λ(x)dx ≤ e |θ 1 | ψ 1 ∞+|θ2| ψ 1 ∞ V Λ(x)dx < +∞ b. Second order moments
Considering the Poisson random measure dN (r) defined in (23), equation (B8) in Proposition 5, applied to p = 1 and to function Λ(r) = 4πλr 2 defined on R + , shows that for any essentially bounded (possibly complex) functions r → ψ 1 (r) and r → ψ 2 (r) with compact supports in R + , cov r 1 ∈R + ψ 1 (r 1 )dN (r 1 ), r 2 ∈R + ψ 2 (r 2 )dN (r 2 ) = 4πλ r∈R + ψ 1 (r)ψ 2 (r)r 2 dr, which we will rewrite in the symbolic, more compact form:

cov[dN (r 1 ), dN (r 2 )] = 4πλ δ(r 2 -r 1 )r 1 r 2 dr 1 dr 2 . (C3)
From (C3), since u∈R |g(u + τ ) g(u)|du < +∞ (with the change of variable r → u = t -T -r c ), Fubini's theorem shows that we can permute the mathematical expectation with the integral over r in ( 21), leading to (24), which jointly proves both ( 25) and ( 26).

Asymptotic normality

From ( 21), b i (t) can be written as the stochastic integral I in Proposition 4, applied to p = 1, to function Λ(r) = 4πλr 2 defined on R + , and to ψ(r) =

g(t-T -r c ) r . Then (B2) yields ∀θ ∈ R, ln(φ b i (t) (θ)) = +∞ n=1 (ıθ) n n! κ n (t) (C4)
where κ n (t) is the n-th order cumulant of b i (t), whose expression is given by (B3):

κ n (t) = 4πλ r∈R + g t -T - r c n r 2-n dr. (C5)
The change of variables r = c(t

-T -u) in (C5) implies ∀t ≥ 2T, κ n (t) = 4πλc 3-n T u=-T g(u) n (t -T -u) 2-n du.
Therefore, ∀n ≥ 2, ∀t > 2T ,

|κ n (t)| ≤ 4πλc 3-n T u=-T |g(u)| n du (t -2T ) n-2 . ( C6
)
Let ε > 0. Substituting (C6) into (C4), we get:

∀t ≥ (2 + ε)T , ln(φ b i (t) (θ)) -ıθκ 1 (t) - θ 2 2 κ 2 (t) ≤ ψ(θ) t -2T -→ t→+∞ 0,
where

ψ(θ) = 4πλ(εcT ) 3 T u=-T e | θg(u) εcT | - 2 n=0 | θg(u) εcT | n n! du.
Therefore the characteristic function of b i (t) converges pointwise to that of the normal distribution when t → +∞, which proves that b i (t) is asymptotically normally distributed.

In the same way, it can be proved 13 that the random variables b i (t + t 1 ) . . . b i (t + t K ) for all K ∈ N and t, t 1 . . . t K ∈ R are jointly normally distributed when t → +∞ and t 1 . . . t K are fixed, which shows that b i (t) converges in law to a stationary Gaussian process. 17) implies (28).

2. Second order moments a. In the spectral domain From (C3), since

r∈R + e -2α c r dr < +∞, Fubini's theorem shows that we can permute the mathematical expectation with the integral over r in ( 22), leading to:

cov[F h i (f 1 ), F h i (f 2 )] = 2πλc L g (α + 2ıπf 1 )L g (α + 2ıπf 2 ) e -2ıπ(f 1 -f 2 )T α+ıπ(f 1 -f 2 ) . (D1)
Equation (D1) implies both ( 29) and (30).

b. In the time domain

Equations ( 24) and ( 17) imply both ( 31) and (32).

c. In the time-frequency domain

Finally, from (C3) and ( 17), since

u∈R r∈R + g t + u 2 -T - r c g t -u 2 -T - r c drdu < +∞,
Fubini's theorem shows that we can permute the mathematical expectation with the integrals over r in ( 21) and over u in (11) (with ψ 1 = ψ 2 = h i ), leading to:

W h i ,h i (t, f ) = 4πλe -2α(t-T ) r∈R + I(t -T - r c , f )dr (D2)
where

I(t, f ) = u∈R g t + u 2 g t - u 2 e -2ıπf u du = W g,g (t, f ). (D3)
In order to conclude, we will use two properties of the Wigner distribution. Firstly, the time support of W g,g is the same as that of g. Secondly, the projection property shows that t∈R W g,g (t, f )dt = |F g (f )| 2 . Therefore substituting (D3) and ( 26) into (D2) finally implies (33).

Asymptotic normality

Since b i (t) converges in law to a stationary Gaussian process as shown in Lemma 1, equation ( 17) shows that h i (t) converges in law to a Gaussian process.

APPENDIX E: GEOMETRY WITH TWO MICROPHONES Let x i , x j ∈ R 3 . Let ξ 1 , ξ 2 : R + → C.
In the next sections, we will have to compute several integrals of the form:

J ξ 1 ,ξ 2 = x∈R 3 ξ 1 ( x -x i 2 ) x -x i 2 ξ 2 ( x -x j 2 ) x -x j 2 dx. (E1) x j x i x θ r 0 D 2 ρ + v 2 ρ -v 2 D 2
FIG. 6. Geometry with two microphones at x i , x j and a source image at x.

To compute such an integral, we will use the spherical coordinates (r, θ, ϕ), as illustrated in Fig. 6, where θ = 0 corresponds to the direction of vector x jx i , and the origin of the essentially bounded (possibly complex) functions x → ψ 1 (x) and x → ψ 2 (x) with compact supports in R 3 , cov x 1 ∈R 3 ψ 1 (x 1 )dN (x 1 ), x 2 ∈R 3 ψ 2 (x 2 )dN (x 2 ) = λ x∈R 3 ψ 1 (x)ψ 2 (x)dx, which we will rewrite in the symbolic, more compact form: cov[dN (x 1 ), dN (x 2 )] = λ δ(x 2x 1 )dx 1 dx 2 .

(F1)

From (F1), since

x∈R 3 g t 1 -T -x-x i 2 c g t 2 -T - x-x j 2 c x -x i 2 x -x j 2 dx < +∞,
Fubini's theorem shows that we can permute the mathematical expectation with the integral over x in (18), leading to:

cov[b i (t 1 ), b j (t 2 )] = λJ ξ 1 ,ξ 2 (F2)
where ξ 1 (r) = g t 1 -T -r c , ξ 2 (r) = g t 2 -T -r c , and J ξ 1 ,ξ 2 was defined in (E1). Then

substituting (E3) into (F2) shows that cov[b i (t 1 ), b j (t 2 )] = 4πλ +∞ ρ= D 2 I(t 1 -T -ρ c , t 2 -T -ρ c )dρ (F3)
where

I(t 1 , t 2 ) = 1 2D D v=-D g t 1 -v 2c g t 2 + v 2c dv. ( F4
)
With v = cu, substituting (34) into (F4) proves that

I(t 1 , t 2 ) = u∈R g t 1 - u 2 g t 2 + u 2 w(u)du. (F5)
Substituting (F5) into (F3) proves ( 35), which with (25) implies (36). Moreover, applying the Fourier transform (9) to (35) and substituting (26) leads to (37).

(x) = g(t-T - x-x i 2 c ) x-x i 2 and ψ 2 (x) = g(t-T - x-x j 2 c ) x-x j 2 . Then (B6) yields ∀θ 1 , θ 2 ∈ R, ln(φ b(t) (θ 1 , θ 2 )) = +∞ n=1 ı n n! n k=0 n k θ k 1 θ n-k 2 κ k,n-k (t) (F6)
where κ n 1 ,n 2 (t) is the (n 1 , n 2 )-th order cumulant of b(t), whose expression is given by (B7):

κ n 1 ,n 2 (t) = λJ ξ 1 ,ξ 2 , ( F7
)
where

ξ 1 (r) = (g(t-T-r c )) n 1 r n 1 -1 , ξ 2 (r) = (g(t-T-r c )) n 2 r n 2 -1
, and J ξ 1 ,ξ 2 was defined in (E1).

Then substituting (E3) into (F7) shows that Therefore the characteristic function of b(t) converges pointwise to that of the normal distribution when t → +∞, which proves that b(t) is asymptotically normally distributed.

In the same way, it can be proved 13 that the random variables b(t xx i 2 e -α c x-x j 2

+
xx j 2 dx < +∞, Fubini's theorem shows that we can permute the mathematical expectation with the integral over x in (19), leading to: cov[F h i (f 1 ), F h j (f 2 )] = λL g (α + 2ıπf 1 )L g (α + 2ıπf 2 )e -2ıπ(f 1 -f 2 )T J ξ 1 ,ξ 2 , (G1)

where ξ 1 (r) = e - (G2)

By substituting (G2) and ( 29) into (G1), we finally get (38).

In the time domain

Equations ( 17) and ( 35) prove that cov[h i (t 1 ), h j (t 2 )] = 4πλce -2α( t 1 +t 2 2 -T ) w * g * g (t 1t 2 ) . (G3)

Finally, substituting ( 31) and ( 25) into (G3) implies (39).

3. In the time-frequency domain

Finally, from (F1) and ( 17), since

u∈R x∈R 3 g t + u 2 -T -x-x i 2 c g t -u 2 -T - x-x j 2 c
xx i 2 xx j 2 dxdu < +∞, Fubini's theorem shows that we can permute the mathematical expectation with the integrals over x in (18) and over u in (11) (with ψ 1 = h i and ψ 2 = h j ), leading to:

W h i ,h j (t, f ) = λe -2α(t-T ) R J ξ 1 ,ξ 2 e -2ıπf u du. (G4)
where ξ 1 (r) = g t + u 2 -T -r c , ξ 2 (r) = g t -u 2 -T -r c , and J ξ 1 ,ξ 2 was defined in (E1).

Then substituting (E3) into (G4) shows that

W h i ,h j (t, f ) = 4πλe -2α(t-T ) 2D +∞ ρ= D 2 D v=-D I(t -T -ρ c , f, v)dρdv (G5)
with

I(t, f, v) = u∈R g t + u 2 -v 2c g t -u 2 + v 2c e -2ıπf u du = e -2ıπf v c W g,g (t, f ) , ( G6
)
where W g,g (t, f ) was expressed in (D3), and we have used the change of variable u = u -v c .

  FIG.1. Time-frequency profile of reverberation (adapted from(Jot et al., 1997, p. 30) and (Baskind,

  regular): vector and scalar, respectively • [a, b]: closed interval, including a and b ∈ R • ]a, b[: open interval, excluding a and b ∈ R

FIG. 3 .

 3 FIG.3. Positions of microphone (plus sign), source (thick point) and source images (other points).

  FIG. 4. Time-frequency correlation between sensors of the late RIR

j

  FIG. 5. Correlation between sensors of the late RFR over frequency (a) and over time (b)

  42)(represented by the dashed curve in Fig.5-(a)), which is an estimate of (3) (the cardinal sine function is represented by the solid curve in Fig.5-(a)). We can observe a good match between the two curves 10 . Finally, the dash-dotted curve in Fig.5-(b) represents, as a function of time, the same noisy estimate of the time-frequency correlation as in Fig. 4, at a fixed frequency f = 3860 Hz (which corresponds to a local maximum of the cardinal sine function in Fig. 5-(a)), and the solid curve represents a smoothed (i.e. low-pass filtered)

  < +∞, Fubini's theorem shows that we can permute the mathematical expectation with the integral over r in (22), leading to (27). Besides, since E[b i (t)] = 0 as shown in Lemma 1, equation (

  Let b(t) = [b i (t), b j (t)] . From (18), b i (t) (resp. b j (t)) can be written as the stochastic integral I 1 (resp. I 2 ) in Proposition 5, applied to p = 3, to the constant function Λ(x) = λ, and to ψ 1

  variables ρ = c(t -Tu) and v = cw in (F8) implies that ∀t ≥ 2T + D c , κ n 1 ,n 2 (t) = 2πλc 4-n 1 -n 2 D Tu -w 2 ) n 2 -1 dudw.Therefore∀n 1 + n 2 ≥ 2, ∀t ≥ 2T + D c , |κ n 1 ,n 2 (t)| ≤ 2πλc 4-n 1 -n 2 D into (F6), we get ∀t ≥ 2T + D c , ln(φ b(t) (θ))n-k (t) ≤ 1 (t -2T ) ψ(θ 1 , θ 2 ) -→

α+2ıπf 1 cr

 1 , ξ 2 (r) = e -α+2ıπf 2 c r , and J ξ 1 ,ξ 2 was defined in (E1). Then equa-

  t 1 ) . . . b(t + t K ) for all K ∈ N and t, t 1 . . . t K ∈ R are jointly normally distributed when t → +∞ and t 1 . . . t K are fixed, which proves that b(t) converges in law to a stationary Gaussian process.

ACKNOWLEDGMENTS

Ahead of this work, I had some very interesting and inspiring discussions about reverberation with several persons that I would like to thank here, including my former PhD students Simon Leglaive and Arthur Belhomme, and my colleagues Gaël Richard, Yves Grenier, Laurent Girin, Antoine Liutkus, Philippe Depalle and Xavier Boutillon.

APPENDIX A: PROOFS FOR THE DEFINITION OF THE STOCHASTIC MODEL

The following lemma aims to prove the existence of functions g that satisfy conditions (15) and ( 16) in Definition 1.

Lemma 3. Let ψ(t) ∈ L 2 ([0, T ]) with T > 0, such that T t=0 ψ(t)e αt dt = T t=0 ψ(t)e -αt dt = 0, where α ∈ R 11 . Let g(t) = ( ψ * ψ)(t) e α(t-T ) . Then function g satisfies conditions (15) and (16) in Definition 1.

Proof of Lemma 3. Since ψ(t) ∈ L 2 ([0, T ]), function g is continuous and bounded. Firstly, F g (f ) = e -αT L ψ (α + 2ıπf )L ψ (-α + 2ıπf ). Moreover, since ψ(t) has finite support, both functions f → L ψ (-α+2ıπf ) and f → L ψ (α+2ıπf ) are infinitely differentiable. In addition,

since ψ 1 and ψ 2 are essentially bounded, V = support(ψ 1 ) ∪ support(ψ 2 ) is compact, and Λ is locally integrable. In particular, for n 1 = 1 and n 2 = 1, we get cov[I 1 , I 2 ] = κ 1,1 , thus (B7) yields (B8). Considering the Poisson random measure dN (r) defined in (23), equation (B4) in Proposition 4, applied to p = 1 and to function Λ(r) = 4πλr 2 defined on R + , shows that for any essentially bounded (possibly complex) function r → ψ(r) with compact support in R + , E[ r∈R + ψ(r)dN (r)] = 4πλ r∈R + ψ(r)r 2 dr, which we will rewrite in the symbolic, more compact form:

From (C1), since

), Fubini's theorem shows that we can permute the mathematical expectation with the integral over r in ( 21), leading to:

coordinates is the middle of the line segment [x i , x j ] (so that

x i +x j 2 = 0 as represented in Fig. 6). We thus get x = [r sin(θ) cos(ϕ), r sin(θ) sin(ϕ), r cos(θ)] and dx = r 2 dr sin(θ)dθdϕ, with r ∈ R + , θ ∈ [0, π] and ϕ ∈ [0, 2π]. Moreover, as can be seen in Fig. 6, we have

By substitution into (E1), we get

Finally, we make a last change of variables, that is also illustrated in Fig. 6:

and r 2 dr sin(θ)dθ

Equation (E2) thus becomes In order to conclude, we will use the two properties of the Wigner distribution that we already used in Appendix D 2 c. Firstly, the time support of W g,g is the same as that of g. Secondly, t∈R W g,g (t, f )dt = |F g (f )| 2 . Substituting (G6) and ( 26) into (G5) finally implies (40).

Asymptotic normality

Since b(t) converges in law to a stationary Gaussian process as shown in Lemma 2, equation ( 17) shows that h(t) converges in law to a Gaussian process.

1 The Fourier transform is defined in equation ( 9).

2 The correlation of two complex random variables is defined in equation ( 7).

3 Note that in this case, F hi (f ) is no longer a stationary process.

4 The Wigner distribution is defined in equation ( 11).

5 A straightforward generalization of this model will also permit us to prove (5), that will be presented in a future paper .

6 Fig. 3 illustrates the source image principle in 2D-space for convenience, but of course our model will be defined in the 3D-space.

7 For the sake of simplicity, we will focus on the case of two microphones; the generalization to an arbitrary number of microphones is straightforward.

8 More precisely, ( 15) is required to prove that the random processes b i (t) and h i (t) are centered after a given time (cf. Appendix C 1 a), and ( 16) enforces the nonnegativity of L g (α + 2ıπf ) in ( 20), which makes the filter A(f ) linear-phase, and which is required to get an expression (30

Proposition 2 that simplifies to Schroeder's equation (2) when T → 0.

9 A WSS process is a random process with finite second order moments, whose mean and covariances are invariant with respect to any translation of time.

10 Actually, the reader may notice that the magnitudes of the extrema of the dashed curve provided by (42) are slightly lower than those of the solid curve given by Cook's formula (3). This difference is due to the way Wigner distributions are estimated: the STFT produces estimated time-frequency distributions W hi,hi (t, f ) and W hi,hj (t, f ) that are smoothed both in time and frequency. Therefore the estimate ( 42) is the ratio between two spectra that are both smoothed in frequency, which explains the deviations between the two curves in Fig. 5-(a).

11 An example of function ψ(t) satisfying the conditions in Lemma 3 is

12 A simple function is defined as a linear combination of indicator functions of measurable sets.

13 The proof is the same and it is omitted here for the sake of conciseness.

Allen


Mihai Lazar
email: mihai.lazar@insa-lyon.fr

Davy Carole

Christophe Raynaud

Gabriel Ferro

Selsabil Sejil

Farah Laariedh

Christian Brylinski

Dominique Planson

Hervé Morel

Classic and alternative methods of p-type doping 4H-SiC for integrated lateral devices

Keywords: ion implantation, VLS epitaxial growth, RIE, JFET

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

Introduction

Power discrete SiC devices as Schottky diodes, Bipolar Junction Transistors (BJT), Junction Field Effect Transistors (JFET) and MOSFET are already commercialized. To improve reliability in power electronics, a monolithic integration of this type of devices in SiC single crystal wafers is recommended. Thus the size of converters is reduced, the switching operation is improved and the power loss is decreased. Monolithic integration of SiC devices was already invoked to build vertical JFET and Junction Barrier Schottky diodes [1] and vertical or lateral BJT [2]. Industrial development of the SiC monolithic technology is still in progress, delayed by the need to improve technological steps, particularly those that can enhance the quality of semi-insulating SiC substrates or that can reduce the resistivity of the p-type layers.

In this paper, for the first time, SiC p-type doping by aluminum (Al) ion implantation is compared to the alternative VLS epitaxy method.

Al ion implantation

P-type localized regions in SiC are in general obtained by ion implantation which is still a unique method of local doping due to the very slow diffusion of dopants in SiC. Moreover, due to the high energies of ionization of the p-type dopants in SiC (~0.2 eV for aluminum and ~0.3 eV for boron), obtaining a net doping level requires to implant higher doses of ions. That leads to create numerous crystalline defects up to local amorphisation of the material if the ion implantation is processed at room temperature (RT).

To recover a good crystalline quality and to activate the implanted impurities by migration of implanted ions in SiC atomic substitutional sites (for example, Al is a dopant only when it substitutes to a Si site), a high temperature annealing is necessary, typically in the range of 1600-1800°C [3]. Figure 1a presents sheet resistances (R sh ) measured with a four-point probe technique on p-type layers created by Al ion implantation. R sh decreases when either the annealing temperature or the annealing time increases, the linearity of the R sh variations proving a non-saturation of the electrical dopant activation even after annealing at 1800°C [4].

The annealing temperature is limited by the surface deterioration of the implanted layers. A strong degradation of the surface occurs by sublimation of Si from the SiC, resulting in a rough surface. Surface roughness limits device performances by inducing an increase of leakage currents in off-state and a degradation of the breakdown voltages. The roughness increases when either the annealing temperature or the annealing time increases (Fig. 1b) by forming terraces of several tens of nanometers (Fig. 2). By increasing temperature, the sublimation of Si can induce also an etching effect of the SiC surface and thus a significant loss of implanted dopants in the near surface region [5]. Figure 3 presents Al implanted doping profiles measured before and after annealing. Compared to samples just implanted, the Al profiles after annealing seems to be offset toward the surface. The quantity of the dopant loss increases with the annealing temperature. It is worth mentioning that on these samples the ion implantation was performed at 300°C to avoid SiC amorphization. [6]. Moreover, losing the implanted dopants should induce an increase of the sheet resistance of the p-type layers which is in contradiction with the results we present in figure 1a where the sheet resistance decreases by increasing the postimplantation annealing temperature. The p-type SiC sheet resistances from figure 1 have been measured on samples whose surfaces were protected with a dedicated cap (detailed hereafter). Therefore, in figure 1a the decreasing of the sheet resistances either with the temperature and the duration of the annealing is fundamentally caused by the increase of the dopants activation. Si sublimation has been reported to be prevented by capping the surface with AlN [7], graphite (C) [8] or by using silane partial overpressure during annealing [9]. In this later case an accurate control of the gas flux is necessary to prevent Si droplet deposition. However, the efficiency of these protections is shown to be temperature limited except for the graphite layer which resists up to 1800°C. The graphite cap layer is obtained by baking at ~750°C a photoresist layer deposited by spincoating at the SiC wafer surface. This method is widely used today in SiC device fabrication needing doping by ion implantation. The C cap showed a very good preservation of the sample surface state. After 1800°C/30 min annealing, the average roughness is 0.46 nm, slightly higher than for an unimplanted sample (0.31 nm) [10]. Nevertheless, the surface morphology after annealing was shown to depend on implantation parameters (temperature and total doses), even when using a graphite cap protection. The use of the graphite cap influences the diffusion of dopants and their activation rate. When ion implantation is performed at RT, dopant redistribution occurs also during the annealing, by a gettering phenomenon localized at the proximity of the end-of-range defects formed at the initial amorphous-crystalline interface [11]. The presence of the amorphous layers has been identified by Rutherford Backscattering Spectrometry in the channeling mode (RBS/C) measurements. Figure 4 presents the example of the 1.75×10 15 cm -2 Al implanted sample at RT.

After annealing, dopant diffusion increases with the dopant dose. In figure 5 the SIMS (Secondary Ion Mass Spectroscopy) dopants profiles are presented for four total implanted doses after 1700°C/30min annealing. The shape of the dopant profile considerably affects the electrical behavior of devices based on these p-n junctions. Dopant redistribution may have a positive influence on the increase of the p-n junction breakdown voltage. Nevertheless, graduate junctions are in general not compatible with high levels of carrier injection. ) Fig. 5. SIMS doping profiles obtained after annealing on samples implanted Al with varying total doses [11] Implanted dopants profiles and the depth of the formed junctions are also difficult to be accurately estimated due to the channeling effect. As 4H-SiC presents a hexagonal crystalline structure, a strong channeling effect is observed especially in the case of Al ions. The channeling of the Al ions is quite difficult to control and to eliminate even when an amorphous layer as SiO 2 is present in the surface. Dedicated Monte-Carlo software has been developed considering the crystalline structure of the SiC [12]. Electrical characteristics of the final fabricated SiC devices could be predicted, but with uncertainties due to the difficulty to estimate the depth of the different p-type layers.

Vapor-Liquid-Solid Growth

An alternative method to locally dope SiC is Vapor-Liquid-Solid (VLS) selective epitaxy. Highly p-type doped SiC can be obtained leading to considerable reduction of both the p-type SiC resistivity and of the specific resistance of the ohmic contacts formed on it. The technological process is detailed in [13] and is schematically presented in Fig. 6. The depth of the p-type layers is accurately controlled by the initial Reactive-Ion-Etching (RIE) of the SiC, performed to create the wells where the Al-Si stacks are deposited and converted in 4H-SiC by VLS epitaxy. The RIE is made with SF 6 /O 2 chemistry and using a Ni hard mask defined by photolithography at the surface of the SiC samples.

Al

During the VLS growth the samples are heated in a vertical cold wall reactor up to 1100°C. This value is very low compared to the temperatures needed for post-implantation annealing (~1700°C) or classical CVD epitaxy (~1400°C). P + -type SiC layers are obtained by VLS with high Al doping concentrations, in the 10 20 cm -3 ranges. Fig. 7 presents typical profiles obtained by SIMS [14].

Ohmic contacts have been evaluated on these P-type VLS layers with patterned Transfer Length Method (TLM) structures using a specific Ni-Ti-Al metal alloy [15]. The contact is already ohmic without post-metallization annealing. Specific contact resistances as low as 1.3×10 -6 Ω.cm 2 have been obtained after 500-800°C Rapid Thermal Annealing [16].

Integrated lateral JFETs

Lateral JFET transistors with N-type and Ptype channels have been fabricated, integrated monolithically within the same SiC wafer. Several batches have been fabricated, the first one (detailed in [START_REF] Lazar | IBMM[END_REF]) with P + wells created by Al ion implantation, and the second one (detailed in [START_REF] Sejil | [END_REF]) with VLS P + wells. The P + wells were utilized as source and drain contacting layers for the P-JFET and as P + gate for the N-JFET. The N + wells have been obtained by nitrogen ion implantation.

For the structure with both P + and N + wells realized by ion implantation, an experimental study based on SIMS measurements and MC simulations was necessary to consider the channeling effect in the 4H-SiC hexagonal crystalline structure. A technological process has been defined to obtain devices working despite the presence of the channeling effect.

The electrical characteristics of the JFET with P + wells created by VLS are affected by the interface between the VLS P + gate and the Ntype channel, which is a CVD epilayer. The structural defects observed by Transmission Electronic Microscopy explain the high reverse gate/source leakage current measured. Nevertheless, the obtained results have shown that the P + VLS layers could already be used to improve the access resistances on the P-JFET transistors where the VLS P + layers have been utilized as source and drain.

Conclusion

Industrial development of the SiC monolithic technology needs a significant drop of the resistivity of the p-type layers and of the ohmic contact formed on them. A technological breakthrough will be obtained when an alternative method to ion implantation will be found in order to locally dope SiC, using only low-temperature processes, thus preserving the surface state and allowing to precisely predict the junction depths. VLS selective epitaxy is a good candidate for such objectives. If creating reliable P/N junctions by this method seems to be yet premature, highly p-type doped layers and ohmic contacts with specific contact resistances as low as 1.3×10 -6 Ω.cm 2 have already been obtained by VLS. Therefore we demonstrated that VLS layers could already be utilized on the top of ptype CVD epilayers in order to improve the access resistances.

Fig. 1 .

 1 Fig. 1. Sheet resistance measured at RT by four-point probe technique (a) and arithmetic average roughness (R a ) measured with Tencor alpha-step stylus (b) on p-type SiC layers created by Al ion implantation and annealed at different temperatures and durations.

Fig. 2 .

 2 Fig. 2. Atomic Force Microscopy (AFM) image on non implanted SiC sample annealed at 1700°C during 30 min.

Fig. 3 .

 3 Fig. 3. Al profiles measured by Secondary Ion Mass Spectrometry (SIMS) before and after post-implantation annealing highlighting the loss of the dopants after high temperature annealing.

Fig. 4 .

 4 Fig. 4. RBS/C spectra on 4H-SiC samples implanted at RT with 1.75×10 15 cm -2 Al total dose, before and after annealing

Fig. 6 .

 6 Fig. 6. Schematic description of the VLS process: a) deposition of Al-Si stack inside the trenches formed by RIE in SiC, b) melting of Al-Si and addition of propane for the VLS growth, c) formation of P + -type SiC [13].

Fig. 7 .

 7 Fig. 7. Al doping profile from SIMS on VLS SiC P + layers.