Julien Chevallier

Stéphane Goutte
email: stephane.goutte@univ-paris8.fr

Khaled Guesmi

Samir Saadi

Dominique Guegan

On the Bitcoin price dynamics: an augmented Markov-Switching model with Lévy jumps *

Keywords: G22, C58 Bitcoin, Lévy process, Markov-switching model

This study contributes to the existing literature on the empirical characteristics of virtual currency allowing for a dynamic transition between different economic regimes and considering various crashes and rallies over the business cycle, that is captured by jumps. We combine Markov-switching models with Levy jump-diffusion offer a new model that captures the different sub-period of crises over the business cycle, that is captured by jumps. This method also enables to test the relevance of dynamic measures of regime switching concerning the independent pure-jump process, which are not frequently used in the literature. Bitcoin offers something different than a traditional currency; there is potential value of having a network that helps as a secure repository for the common knowledge of all transactions. Besides, the value of Bitcoin fluctuates so wildly that it may be too risky to serve as a credible store of value.

As a peer-to-peer crypto-currency, Bitcoin holds the promise of being free from central banks' and governments' interventions. A decade after its inception following Satoshi [START_REF] Nakamoto | Bitcoin: A peer-to-peer electronic cash system[END_REF] vision, its price movements are far from being tamed nowadays. This study contributes to the existing literature on the empirical characteristics of virtual currency allowing for a dynamic transition between different economic regimes and considering various crashes and rallies over the business cycle, that is captured by jumps. In our model, we combine Markov-switching models with Lévy jump-diffusion. This strategy allows capturing the different sub-period of crises over the business cycle, that is captured by jumps, whereas the trend is modeled under a Gaussian process. This method also enables to test the relevance of dynamic measures of regime switching concerning the independent pure-jump process, which are not frequently used in the literature.

Our study differs from past ones in that we investigate the Bitcoin price fluctuations. Bitcoin offer something different than a traditional currency; there is potential value of having a network that helps as a secure repository for the common knowledge of all transactions. Besides, the value of Bitcoin fluctuates so wildly that it may be too risky to serve as a credible store of value.

Bitcoin hogged the limelight in the cryptocurrency markets in the first week of May 2018 as it fell to one-month lows below $8,000. Nevertheless, while the world's largest cryptocurrency by market capitalization suffered a 3.6 percent week-on-week drop in prices, it still outclassed other significant names like Bitcoin cash (BCH) and EOS, which both reported double-digit losses.

Furthermore, the US CFTC rules that Bitcoin and other crypto-currencies are commodities subject to the CFTC's jurisdiction. Thus, Bitcoin falls within the definition of "commodity" under the Commodity Exchange Act such that The CFTC regulates Bitcoins derivatives contracts.

Consequently, Bitcoin is officially a commodity, according to U.S regulator. Dyhrberg (2015) claim that Bitcoin has liquidity limitations like other commodities. Bitcoin and gold derive most of their value from the fact that they are scarce and costly to extract.

The remainder of the paper is structured as follows. Section 2 details the stochastic model.

Section 3 provides an application to the dynamics of the Bitcoin price. Section 4 concludes.

Stochastic model

Let (ω, F, P ) be a filtered probability space and T be a fixed terminal time horizon. We propose in this paper to model the dynamics of a sequence of historical values of the price using a regimeswitching stochastic jump-diffusion. This model is defined using the class of Lévy processes. 1

Lévy Process

Definition 1 A Lévy process L t is a stochastic process such that 1. L 0 = 0.

2. For all s > 0 and t > 0, we have that the property of stationary increments is satisfied. i.e.

L t+s -L t as the same distribution as L s .

3. The property of independent increments is satisfied. i.e. for all 0 ≤ t 0 < t 1 < • • • < t n , we have that L t i -L t i-1 are independent for all i = 1, . . . , n.

4. L has a Cadlag paths. This means that the sample paths of a Lévy process are right continuous and admit a left limits.

Remark 1 In a Lévy process, the discontinuities occur at random times.

Markov-switching

Definition 2 Let (Z t ) t∈[0,T ] be a continuous time Markov chain on finite space S := {1, 2, . . . , K}.

Denote F Z t := {σ(Z s ); 0 ≤ s ≤ t}, the natural filtration generated by the continuous time Markov chain Z. The generator matrix of Z, denoted by Π Z , is given by

Π Z ij ≥ 0 if i = j for all i, j ∈ S and Π Z ii = - j =i Π Z ij otherwise.
(1)

Remark 2 The quantity Π Z ij represents the switch from state i to state j.

Regime-switching Lévy

Let us define the regime-switching Lévy Model:

Definition 3 For all t ∈ [0, T ], let Z t be a continuous time Markov chain on finite space S := {1, . . . , K} defined as in Definition 2. A regime-switching model is a stochastic process (X t ) which is solution of the stochastic differential equation given by

dX t = κ(Z t ) (θ(Z t ) -X t ) dt + σ(Z t )dY t (2)
where κ(Z t ), θ(Z t ) and σ(Z t ) are functions of the Markov chain Z. Hence, they are constants which take values in κ(S), θ(S) and σ(S) κ(S) := {κ(1), . . . , κ(K)} ∈ R K * , θ(S) := {θ(1), . . . , θ(K)}, σ(S)

:= {σ(1), . . . , σ(K)} ∈ R K + .
where Y is a stochastic process which could be a Brownian motion or a Lévy process.

Remark 3

The following classic notations apply:

• κ denotes the mean-reverting rate;

• θ denotes the long-run mean;

• σ . denotes the volatility of X.

Remark 4

• In this model, there are two sources of randomness: the stochastic process Y appearing in the dynamics of X, and the Markov chain Z. There exists one randomness due to the market information which is the initial continuous filtration F generated by the stochastic process Y ; and another randomness due to the Markov chain Z, F Z .

• In our model, the Markov chain Z infers the unobservable state of the economy, i.e. expansion or recession. The processes Y i estimated in each state, where i ∈ S, capture: a different level of volatility in the case of Brownian motion (i.e., Y i ≡ W i ), or a different jump intensity level of the distribution (and a possible skewness) in the case of Lévy process

(i.e. Y i ≡ L i ).
Remark 5 One could propose to use a regime-switching stochastic volatility model, à la Heston, to better capture the flexibility of the volatility changes and levels. Nevertheless, this kind of model increases the computational burden dramatically during the simulations without improving the empirical fit greatly.

NIG distribution

We recall the main properties of the Normal Inverse Gaussian (NIG) distribution. Indeed, we assume that a Lévy process L follows a Normal Inverse Gaussian (NIG) distribution. Note that the Variance-Gamma could have been an alternative at this stage [START_REF] Kaishev | Dirichlet bridge sampling for the variance gamma process: pricing pathdependent options[END_REF]. The NIG family of distribution was introduced by [START_REF] Barndorff-Nielsen | Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions[END_REF].

The NIG density belongs to the family of normal variance-mean mixtures, i.e., one of the most commonly used parametric densities in financial economics.

Taking δ > 0, α ≥ 0, then the density function of a NIG variable N IG(α, β, δ, µ) is given by

f N IG (x; α, β, δ, µ) = α π exp δ α 2 -β 2 + β(x -µ) K 1 αδ 1 + (x -µ) 2 /δ 2 1 + (x -µ) 2 /δ 2 . ( 3
)
where K ν is the third Bessel kind function with index ν. It can be represented with the following integral

K ν (z) = 1 2 ∞ 0 y ν-1 exp - 1 2 z(y + y -1 ) dy .
For a given real ν, the function K ν satisfies the differential equation given by

x 2 y + xy -(x 2 + ν 2 )y = 0 .
This class of distribution is stable by convolution as the classic normal distribution. i.e.

N IG(α, β, δ 1 , µ 1 ) * N IG(α, β, δ 2 , µ 2 ) = N IG(α, β, δ 1 + δ 2 , µ 1 + µ 2 ) .
Lemma 1 If X ∼ N IG(α, β, δ, µ) then for any a ∈ R + and b ∈ R, we have that

Y = aX + b ∼ α a , β a , aδ, aµ + b .
The log cumulative function of a NIG variable is given by

φ N IG (z) = µz + δ α 2 -β 2 -α 2 -(β + z) 2 , for all |β + z| < α , (4)
The first moments are given by

E[X] = µ + δβ γ , Var[X] = δα 2 γ 3 . (5)
with γ = α 2 -β 2 . And finally the Lévy measure of a NIG(α, β, δ, µ) law is

F N IG (dx) = e βx δα π|x| K 1 (α|x|) dx . ( 6
)
Remark 6 Each parameter in N IG(α, β, δ, µ) distributions can be interpreted as having a different effect on the shape of the distribution:

• α -tail heaviness of steepness.

• β -skewness.

• δ -scale.

• µ -location.

3 Application on Bitcoin

Background on Bitcoin price development

The data is retrieved from Coinbase over the period going from July, 10th 2010 to March, 31th

2018 with a daily frequency, totaling 2814 observations. Figure 1 displays the Bitcoin price development. 3. The Mt. Gox Calamity of 2014: The price of Bitcoin had been making significant gains after the bubble pop of 2013 when, in February, the price fell from $867 to $439 (a 49% drop). This triggered a doldrums period for Bitcoin that lasted until late 2016. The February crash came after the operator of Mt. Gox-long the go-to trading place for longtime Bitcoin owners -announced the exchange had been hacked. On February 7, the exchange halted withdrawals and later revealed thieves had made off with 850,000 Bitcoins (which would be worth around $3.5 billion today). The incident, which created existential doubts about the security of Bitcoin and undercut liquidity in the currency, likely harmed the currency's value for years.

4. Summer Selloff of 2017: In early January, Bitcoin broke $1,000 for the first time in years and started climbing like crazy. By June, the currency nudged $3,000-but then lurched back all of a sudden, falling 36% to $1,869 by mid-July. Why it happened: Even as Bitcoin boomed anew, many worried something was wrong with the code under the hood. Specifically, Bitcoin was slow compared to other crypto-currencies like Litecoin and Ethereum, and its core developers could not agree on how to update the software. This raised the prospect of a "fork" (which would produce two versions of Bitcoin's canonical blockchain) and future schisms, which in turn appeared to give rise to market jitters and the significant fall in price.

Ironically, such a fork did materialize in August in the form of rival Bitcoin Cash-but this seems to have done no longterm harm to Bitcoin.

5. The Great China Chill: After fears over the fork subsided, Bitcoin went on another crazy tear: It climbed close to $5,000 at the start of September before plunging 37% by September 15, shaving off over $30 billion from Bitcoin's total market cap in the process. Recovery is already underway, though, as prices climbed above $4,000 three days later. Why it happened: While Bitcoin price moves can be inscrutable, the prime reason for the latest crash can be summed up in one word: China. After it cracked down on so-called "Initial Coin Offerings," there have been widespread rumors the Communist government is going to ban trading crypto-currency altogether. In response, the most prominent exchange, BTCChina, said it would end trading this month. This crackdown, combined with questions about China's de facto monopoly on Bitcoin mining, explains the recent price swoon.

MS-Lévy model

In this paper we apply the estimation process of [START_REF] Chevallier | On the estimation of regime-switching Lévy models[END_REF]. This methodology proposes to estimate Eq.( 2) in two steps: (i) model parameters in a regime-switching Brownian process, and (ii) the distribution parameters. We fit a regime-switching Lévy model such as Eq.( 2)

where the stochastic process Y is a Lévy process that follows a Normal Inverse Gaussian (NIG)

distribution. Thus the optimal set of parameters to estimate is Θ := κi , θi , σi , αi , βi , δi , μi , Π , for i ∈ S.

We have the three parameters of the dynamics of X, the four parameters of the density of the Lévy process L, and the transition matrix of the Markov chain Z.

The table ?? reports the results of (i) the set of diffusion parameters, and (ii) the NIG density parameters of the Lévy jump process fitted to each regime. The remaining problem in this work is to specify the number of regimes in the Markov chain. For simplicity, we proceed with two regimes that relate to the 'boom' and 'bust' phases of the business cycle. Standard errors are given in parentheses below parameter estimates.

In Table 1, the parameter α represents the jump intensity. The lower the alpha, the higher the jump intensity in a given regime. We observe that α = 0.0012 during regime 2, which indicates a high jump intensity. On the contrary, α = 0.0801 during regime 1, which points out a low jump intensity.

We also report a plot where each regime is reported with a different color (e.g., blue (red) corresponds to regime 1 (regime 2)). To provide the reader with a clearer picture, we have chosen to plug the regimes identified back into the raw (non-stationary) data. Of course, all the estimates were performed on returns, e.g., stationary data. Below this first plot, the filtered and smoothed probabilities are displayed. They reflect the regime switches at stake.

Inspecting the Figures 1 and2, we observe that the regime 1 is characterized by a rather stable and then switch definitively in regime two of high volatility and high frequency of jump after 2017. So beyond that date, the data enters the regime two which is characterized by ups and downs, and thus a higher jump activity.

β is the skewness parameter: β < 0 (β > 0) implies a density skewed to the left (right). The skewness of the density increases as β increases. In the case where β is equal to 0, the density is symmetric around µ. In both regime the values of β are close to 0; these values reflect a balance between positive and negative jumps, as illustrated in Figure 2. δ is the scale parameter representing a measure of the spread of the returns. We see that the regime two with high frequency and intensity of jumps exhibits a huge value of delta δ = 161.0635 against a vanishing value for regime one. This fact demonstrates the crisis or bubble situation captured by the regime two of our regime-switching Lévy model.

In Table 1, of particular interest is the volatility parameter σ: it is more than sixty times higher during regime 2 than during regime 1. The relatively higher jump intensity during regime 2, therefore, translates into higher volatility levels. The mean-reverting parameters κ are close to zero in both regimes.

When inspecting the filtered and smoothed probabilities in the middle and bottom panels of Figure 3, we notice that the probability of staying in the current regime is very high (e.g., close to unity). This information is also visible in the last column of Table 1: the probability to stay in regime 1 (in regime 2) is equal to 95.52% (85.50%). If there is some general form of persistence in the chain (e.g., high probability of staying in a given regime), then this could have important implications for the computation of the Value-at-Risk and dynamic portfolio allocation, because the benefits of portfolio diversification would be less volatile.

What we learn mainly from these probability graphs is that the stochastic process fitted to each regime does not have the same jumps characteristics during the sample period. Indeed, there are periods with a prominent presence of jumps (recorded during regime 2) in the Bitcoin price, and others without. Hence, this first set of results illustrates the interest of resorting to the regime-switching Lévy model to model the Bitcoin dynamics.

Conclusion

Most of the Bitcoin price downward movements seem to coincide with speculative run-ups coupled with exogenous shocks, such as an exchange hack or government regulation forbidding the use of cryptocurrencies. In most cases, Bitcoin has bounced back from the downward movements within months or even weeks -suggesting Bitcoin buyers hodling for the long run.

Based on this mean-reverting behavior, we investigate a new way to model the dynamics of the Bitcoin price. This paper studies the volatility of the Bitcoin price using a Markov-switching model augmented with jumps under the Lévy form.

Distinct and significant dynamics are captured during crisis and bubbles periods. We identify the persistence of periods of high volatility and jumps. Policy implications unfold in terms of portfolio/risk management for hedging and pricing future derivatives on cryptocurrencies.

Figure 1 :

 1 Figure 1: Bitcoin price

Figure 2 :

 2 Figure 2: Important Dates

Figure 3 :

 3 Figure 3: Probabilities

Table 1 :

 1 Estimated parameters

	Process	κ	θ	σ	P Z ii
	State 1	1.0299	0.3082	23.8652	0.9552
		(0.1055) (0.0074)	(0.7274)
	State 2	0.9130	9.9467	1367.4328 0.8550
		(14.4028) (0.0389) (7529.9928)
	NIG	α	β	δ	µ
	State 1 0.0801	0.0238	0.1603	-0.0499
		(0.0672) (0.0023) (0.7534) (10.0061)
	State 2 0.0012	0.0001 161.0635 -14.5843
		(0.0883) (0.0001) (0.5497) (0.0001)

Regime-switching classification

An ideal model is one that classifies the regimes sharply, with smoothed probabilities close to either zero or one. In order to measure the quality of regime classification, we propose the following two measures:

(1) The regime classification measure (RCM) introduced by [START_REF] Ang | Regime switching in interest rates[END_REF] and generalized for multiple states by [START_REF] Baele | Volatility spillover effects in European equity markets[END_REF].

where the quantity

is the smoothed probability and Θ(n) 1 is the vector of parameters estimated. The constant serves to normalize the statistic to be between 0 and 100. Good regime classification is associated with low RCM statistic value: a value of 0 means perfect regime classification, and a value of 100 implies that no information about the regimes is revealed. Consequently, even if a model has the highest log-likelihood value, its RCM needs to be close to zero.

(2) The smoothed probability indicator, introduced by Goutte and Zou ( 2013) and developed by [START_REF] Goutte | Conditional Markov regime switching model applied to economic modelling[END_REF]. With this indicator, a proper classification for the data can be achieved when the smoothed probability is less than 0.1 or greater than 0.9. This means that the data at time t ∈ [0, T ] is in one of the regimes at the 10% error level.


Selsabil Sejil
email: selsabil.sejil@etu.univ-lyon1.fr

Loïc Lalouat
email: loic.lalouat@ec-lyon.fr

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

Davy Carole
email: davy.carole@univ-lyon1.fr

Christian Brylinski
email: echristian.brylinski@univ-lyon1.fr

François Jomard
email: ffrancois.jomard@uvsq.fr

Dominique Planson
email: dominique.planson@insa-lyon.fr

Gabriel Ferro
email: hgabriel.ferro@univ-lyon1.fr

Christophe Raynaud
email: ichristophe.raynaud@insa-lyon.fr

Very High Sustainable Forward Current Densities on 4H-SiC p-n Junctions Formed by VLS Localized Epitaxy of Heavily Al-Doped p++ Emitters

Keywords: SiC, p ++ doping, VLS, selective epitaxy, roughness, p-n Junction, p-type ohmic contact

Introduction

Continuous improvement of bulk and epilayers of 4H-SiC semiconductor materials, allows currently the marketing of power devices. However, technological difficulties still remain unresolved, like the implementation of a fully satisfactory process to locate p-type doping area into 4H-SiC. Indeed, ion implantation is the only viable technique in industry to perform localized p + doping, however with significant drawbacks [1]. The damages created during ion implantation require a high temperature annealing (~1700°C or higher [2]) to both partially activate the Al elements and restore the crystal quality. Moreover, in practice, with standard ion implanter energy (< 200 keV), the depth of the p-type implanted areas is limited to a few hundreds of nm, which can lead in turn to severe design restrictions. To solve this problem, VLS localized epitaxy has been previously proposed by some of the authors as an alternative to implement deeper localized Al doping at high concentrations [3,4]. Further works have resulted in the improvement of localized VLS process, yielding optimized structural morphology [5], preserving a threshold voltage of ~ 3 V in forward bias, value predicted by the classical theory, on a 4H-SiC p-n junction, and eliminating the need of high temperature post-growth annealing (previously ~1700°C) [6]. To the best of our knowledge, this electrical behaviour has never been obtained before with such low temperature doping process (1100°C). This study was followed with a new one whose aim was to increase the deposit thickness up to 1 µm [7]. Implementing these new p-type VLS layers with an adequate technological process has allowed us to obtain 4H-SiC diodes withstanding spectacular high current densities in forward bias, and providing low leakage current densities in reverse bias. Hereafter are presented elements of the technological process and the J-V characteristics of these diodes which clearly present a bipolar behavior (light emission in forward bias).

Experimental section

The manufacturing process to prepare the sample, and details about the optimization of VLS localized epitaxy have been reported elsewhere [7]. After VLS growth sequence, the amount of Al incorporated into the grown layer (doping profile) was determined by Secondary Ion Mass Spectrometry (SIMS) for different diameters of the wells. The doping uniformity was checked.

Then, to fabricate the PiN diodes (see Fig. 1), classical Ni-based ohmic contacts were deposited on the back side of the n-type substrate by e-beam evaporation, followed by a Rapid Thermal Annealing (RTA) at 900°C in argon atmosphere. Because of the high doping level of the VLS p ++ epilayers, it was possible to get ohmic contacts on these p-type layers just by depositing an layer (by sputtering), i.e. without annealing this metallization. This Al layer was thick enough (800 nm) to get full coverage of the rather rough VLS epilayers surface (see Fig. 2), presenting a pronounced step-bunching which is a good evidence that the growth from the melt occurred in a homo-epitaxial way.

The as-fabricated PiN diodes were electrically characterized in order to validate the quality of the pn junction, using a Keithley 2636 System Measure Unit with a "Signatone S1160" probe station. These measurements were done on vertical diode configuration, at room temperature. Note that neither passivation nor edge termination (such as guard rings, mesa or JTE) were used.

Results and discussions

As a first result, most of the aluminum concentration values measured by SIMS were in the range 1-3×10 20 at.cm -3 (see Fig. 3) with a nearly constant value over the layer thickness, clearly visible for the largest diodes (Ø 920 µm) where the deposit thickness is ~ 900 nm. It should be emphasized that such a high and constant doping level is quite difficult to be reached by standard CVD or by ion implantation without degradation of the layer crystalline quality. The thickness values, measured by SIMS, in the range of 200-900 nm, are in agreement with those measured by Tencor stylus profilometer from the difference between the depth of the etched well, before and after VLS growth [7].

VLS epilayers with a rough surface

Typical current density vs. voltage (J-V) characteristics, measured on diode structures with VLS grown emitters are reported in Fig. 4. For a better readability, only a few J-V characteristics amongst the more representative have been plotted. The total current is limited to 1A in all cases due to instrumental limitations of the K2636 SMU unit. In forward bias, PiN diodes exhibit spectacular high and sustainable current densities: from 150 A.cm -2 for Ø 920 µm (limited by the measurement power compliance) to several kA.cm -2 , in the range 1250-3000 A.cm -2 for the smaller ones. Measured threshold voltages are around ~ 3 V, which is the typical threshold voltage for 4H-SiC p-n junction. Moreover, as an improvement over a previous report [7], the current density curves overlap quite well in forward bias, confirming that the metal deposit of the front side top contact (800 nm thickness) is now thick enough to fully cover the whole VLS layers. Such an optimized front side top contact avoids high electric field spots inducing premature breakdown. In reverse bias, leakage current density is rather low, measured in the range of 10 -6 -10 -7 A.cm -2 for diodes of smaller diameters, and surprisingly higher for Ø 920 µm, measured in the range of 10 -5 -10 -4 A.cm -2 , unlike previously reported value of leakage current density of ~ 10 -8 A.cm -2 [7]. The high leakage current density for Ø 920 µm diodes is probably due to device processing issues. On the border of the anode contact, the metal is probably not completely etched and it slightly overlaps with the n-zone at the well edge, resulting in a parasitic Schottky contact in parallel with the p-n junction.

Then, repetitive current-voltage measurements in reverse regime were carried out. A series of tests at the same reverse voltage was performed on the same diode in order to study the evolution of the leakage current. Compliance has been fixed at 10 or 100 µA, depending on the test, in order to avoid the destruction of the diode. Fig. 5 reports results of measurements performed on Ø 170 µm diodes.

In order to compare our results with those previously presented in [6], breakdown voltage tests were performed under the same conditions, i.e. in air and at room temperature. In the present case, breakdown voltage V BR is estimated at -420 V. Due to the low compliance at 10 µA, the diode is not destroyed when breakdown begins at ~ 420 V. In the second test, with the same compliance, prebreakdown leakage current is a little higher. No degradation is observed between 2 nd and 3 rd test. But for the third test, compliance has been set to 100 µA, and subsequently the diode has been a little more degraded. Indeed, on the 4 th pre-breakdown test, the current increases drastically at ~370 V. For such a doping level and thickness, 420 V represents = 38 % of the theoretical V BR calculated in ideal 1D planar model.

is the "blocking efficiency" of the junction. In a previous study [6], the experimental breakdown voltage was as low as 50 V, which represented only 2 % of the theoretical value. As a result, the net increase of the blocking efficiency confirms the beneficial effects of the crystalline quality optimization of the VLS epilayers.

(Ø 170 µm) without passivation or edge termination

Conclusion

SIMS analysis were performed on p ++ VLS epilayers, and a doping level in the range of 1-3×10 20 at.cm -3 has been measured, confirming deposit thickness in the range of 200-900 nm. Then, PiN structures were implemented including these optimized p ++ VLS layers, together with an optimized technological process, which has provided 4H-SiC diodes presenting a bipolar behaviour, and very high sustainable forward current densities ≥ 3 kA.cm -2 while preserving a low leakage current density in reverse bias. These results were obtained without passivation and edge termination. Reverse I-V measurements were carried out and a net increase in blocking efficiency, up to 38 %, was found, as compared to 2 % in our previous study before optimization of VLS growth process. These improvements in blocking efficiency confirm the beneficial effects of the VLS crystalline quality optimization [7]. Even better performances can be expected in the future from diodes with adequate passivation and edge termination.

Figure 1 :

 1 Figure 1: Schematic cross sectional view of the fabricated PiN diodes, with p ++ emitter formed by VLS epitaxy

Figure 3 :

 3 Figure 3: SIMS depth profiles of Al concentration in SiC layers grown by VLS 0 200 400 600 800 1000 1200 1400

Figure 4 :

 4 Figure 4: J-V behaviors as a function of the diameter of the well (a) Reverse and forward bias in logarithmic scale and (b) forward bias in linear scale.

Figure 5 :

 5 Figure 5: Breakdown voltage test of PiN diodes

Acknowledgements. The authors gratefully thank NanoLyon Technological Platform staff for process expertise. This work has been financially supported by ST Microelectronics Company. LMI and AMPERE labs are members of the I@L Carnot Institute.


Aymen Belghith

Loutfi Nuaymi

Patrick Maillé

Pricing of Real-Time Applications in WiMAX Systems

WiMAX/IEEE 802.16 Broadband Wireless Access (BWA) is a very open technology with different possible services that is likely to be widely used in Metropolitan Access Networks in the near future. While the technical aspects of that technology are now quite well established, the economic implications of the introduction of WiMAX technology have received little attention.

In this paper, we consider the problem of pricing WiMAX sessions: we propose some simple pricing mechanisms aimed at coping with congestion while maintaining a sufficient quality of service and/or yielding a sufficient revenue to the operator. Our mechanisms are evaluated under a specific model of user preferences and WiMAX situations, and the implications of some parameter choices are observed.

I. INTRODUCTION

WiMAX technology, based on IEEE 802.16 standard [START_REF]IEEE Standard for local and metropolitan area networks, Air Interface for Fixed Broadband Wireless Access Systems[END_REF], [START_REF]IEEE Standard for local and metropolitan area networks, Air Interface for Fixed Broadband Wireless Access Systems, Amendment 2: Physical and Medium Access Control Layers for Combined Fixed and Mobile Operation in Licensed Bands and Corrigendum 1[END_REF], has a very rich set of features. Indeed, it is a very promising Broadband Wireless Access (BWA) Technology, whose objective is to have a highly-efficient use of the radio resource while transmitting different types of service (with different constraints).

The IEEE 802.16 standard specifies five physical (PHY) layers: WirelessMAN-SC, WirelessMAN-SCa, WirelessMANOFDM, WirelessMAN-OFDMA, and WirelessHUMAN PHYs. We consider only the WirelessMAN-OFDM PHY which is based on the Orthogonal Frequency Division Multiplexing (OFDM) technology. Yet, our study could be extended to OFDMA access where OFDM symbols allocation is replaced by OFDMA subchannels allocation.

The IEEE 802.16 Medium Access Control (MAC) layer supports two modes of transmission: the point-to-multipoint (PMP) and Mesh modes. It also supports two duplex modes: the Frequency Division Duplex (FDD) and Time Division Duplex (TDD) modes. We consider the PMP transmission mode and the FDD duplex mode. Once again, our study could be extended to the TDD mode, with additional latitude.

The IEEE 802.16 MAC also specifies five types of scheduling service classes or QoS classes: Unsolicited Grant Service (UGS), real-time Polling Service (rtPS), extended real-time Polling Service (ertPS), non real-time Polling Service (nrtPS), and Best Effort (BE) QoS classes. Each QoS class has its QoS parameters and requirements. The MAC Convergence Sublayer (CS) layer makes the classification of the different users applications in those five classes.

Pricing of WiMAX is a new and interesting area. Pricing models are defined for the integration of WiMAX and Wireless Fidelity (WiFi). For example, a strategy game used in economics, called Stackelberg leadership model, is used in [START_REF] Niyato | Integration of WiMAX and WiFi: Optimal Pricing for Bandwidth Sharing[END_REF], in order to propose a pricing model for sharing bandwidth between WiMAX and WiFi users. Other pricing models are defined only for WiMAX context. For example, two market models are defined in [START_REF]Business Case Models for Fixed Broadband Wireless Access based on WiMAX Technology and the 802.16 Standard[END_REF]. The Base Station (BS) allocates resources only for the residential users in the first market model. In the second model, the BS also provides services to the Small and Medium Enterprises (SME) customers. In [START_REF] Lee | Price-based distributed algorithms for rate-reliability tradeoff in network utility maximization[END_REF], a pricing model for wireless networks users is proposed. The utility function of the proposed pricing model depends on the transmission rate and the quality of the channel. In [START_REF] Chang | Adaptive Hierarchical Polling and Cost-based Call Admission Control in[END_REF], a cost-based Call Admission Control (CAC) is proposed for IEEE 802.16 WiMAX. It is based on the required bandwidth and the residual bandwidth. The competitive On-Line (COL) is used in order to have an exponential cost function of the residential bandwidth.

However, there is not yet any published pricing proposal for the different WiMAX QoS classes in the research literature. Evidently, it can be expected that WiMAX operators use different pricing strategies for the QoS classes.

In this paper, we propose pricing schemes for WiMAX. Our pricing model is designed for real-time applications and takes into account the specification of the standard. There are a large number of parameters and other models details that have to be selected for the global WiMAX pricing model. We make some decisions for these parameters in order to obtain a simplified model and some interesting first results.

This paper is organized as follows. In Section II, we present the system model. In Section III, we describe our pricing proposals for UGS and rtPS QoS classes. In Section IV, we present some simulation results. Conclusions and directions for future work are provided in Section V.

II. SYSTEM MODEL

We consider a real-time environment with only the UGS and rtPS QoS classes:

• UGS: supports real-time service flows that have fixed-size data packets on a periodic basis (for circuit-type voice services for instance). • rtPS: supports real-time service flows that generate variable data packets size on a periodic basis (for video services for instance). A minimum transmission rate (R min ) is guaranteed, and the rate can be increased up to a maximum transmission rate (R max ). In IEEE 802.16 standard and for a given system, the frame has a fixed number of OFDM symbols. The number of symbols depends on the frequency bandwidth (BW), the Cyclic Prefix (CP), the sampling factor (n), and the frame duration.

An appropriately-designed pricing mechanism should then properly allocate those symbols among users, so as to satisfy some QoS agreements, or maximize user satisfaction or provider revenue.

However, the number of useful data bits is variable and depends on the Modulation and Coding Scheme (MCS) used by the different subscriber stations (SSs). The MCS to be used is defined by the link adaptation procedure. The choice of the appropriate MCS depends on the value of receiver Signal-to-Noise Ratio (SNR). The transmitting station (the BS or the SS) switches to a more energy-efficient MCS if the SNR is good. It switches to a more robust MCS if the SNR is poor. Once the MCS is defined, the number of bits per OFDM symbol, and then the useful number of bits per frame, can be computed as shown in Table I. Values of some receiver SNR assumptions (see column 3, Table I) are proposed in Table 266 of the IEEE 802.16e amendment of the standard. The BPSK 1/2 MCS is mainly used for management purposes and will not be considered in this work.

III. PRICING MODEL

We propose an utility function for each type of service. The utility is the difference between the price that the user is willing to pay for the service (that we call valuation or willingness to pay) and the price actually paid. The utility, valuation, and price actually paid are per unit of time.

U (r) = V (r) -P (r) (1)
where:

• U: represents the utility function.

• V: represents the valuation function.

• P: represents the price actually paid function.

• r: represents the throughput of the connection (expressed in bit/s).

A. UGS users

The transmission rate of the UGS connections is fixed at the beginning of the session. We consider that each UGS user i is willing to pay a given price v i per transmitted bit per second.

V U GS (r) = v i • r (2)
We introduce heterogeneity among the UGS users by assuming that v i is randomly chosen for each user, according to continuous uniform distribution on the interval [V 1 U GS , V 2 U GS ] (in monetary units (mu) by kbit/s transmitted).

We suggest to use a pricing scheme based on two prices. When the network is not congested (i.e. when the proportion of OFDM symbols used by the UGS and rtPS calls is below a given threshold, called th U GS ), the UGS calls are charged a price P U GS Low per unit of transmission rate. On the other hand, during the congested periods, this price actually paid is changed to P U GS High . We consider that P U GS High = α U GS * P U GS Low .

When an UGS call starts, its price per unit of time does not change; if the call started when the network was not congested, the user continue to pay the low price during its whole call even if afterwards it is served at congestion periods.

P U GS (r) = P U GS Low • r, if nb res nb tot ≤ th U GS P U GS High • r, otherwise (3)
where:

• nb res : represents the number of reserved symbols.

• nb tot : represents the total number of symbols.

B. rtPS users

rtPS services offer a flexibility in the transmission rate. We therefore consider that users have elastic demand, i.e. their valuation function is increasing and concaves in the transmission rate r. We propose the following valuation function:

V rtP S (r) = v i • ( -r 2 R i max + 2r), if r ≤ R i max v i • R i max , otherwise (4)
We also introduce heterogeneity among the rtPS users by assuming that v i is randomly chosen for each user, according to continuous uniform distribution on the interval [V 1 rtP S , V 2 rtP S ] (in mu by kbit/s transmitted).

A minimum transmission R min rate is guaranteed to rtPS users upon connection. We suggest that this minimum rate is provided (when there are enough OFDM symbols remaining to serve this call) at a fixed unit price that we denote by P rtP S min , independently of the congestion. On the other hand, the rtPS QoS Class offers the possibility to extend the transmission rate up to R max . We propose that the additional transmission rate is sold P rtP S add that could increase with congestion. This way, during congestion periods, rtPS users will be elicited to reduce their transmission rate close to R min due to high prices. Formally, a user i with transmission rate r that belongs to [R min , R max ] is therefore charged a price:

P rtP S (r) = P rtP S min • R min + P rtP S add • (r -R min ) (5)
The price for extra rate is equal to:

P rtP S add =
C rtP S nb rem min [START_REF] Chang | Adaptive Hierarchical Polling and Cost-based Call Admission Control in[END_REF] Where:

• C rtP S : represents a pricing constant fixed by the operator.

• nb rem min : represents the number of OFDM symbols remaining after all the UGS users are served and the rtPS users get their minimum transmission rate.

IV. SIMULATION RESULTS

A. Simulation model

This section describes the situation we have considered to simulate the behavior of a WiMAX system under the pricing schemes described in the previous section. In this work, we consider the OFDM PHY layer used in the PMP topology and FDD mode. The number of OFDM symbols per frame (N OF DM ) is constant and depends on the frame duration and the OFDM symbol duration. According to [START_REF] Nuaymi | WiMAX: Technology for Broadband Wireless Access[END_REF], the OFDM symbol duration is computed as follows:

OFDM symbol Duration = 1+G n• BW N F T T
where:

• G: represents the ratio of the guard time to the useful symbol time.

• n: represents the the sampling factor.

• BW: represents the channel bandwidth.

• N F T T : represents the total number of subcarriers. For the OFDM PHY, the total number of subcarriers is equal to 256 (N F T T = 256). For the following parameter values: BW = 7 MHz, n = 8/7, G = 1/8, and frame duration = 10 ms obtain an OFDM symbol duration equal to 35.86 µs and N OF DM equal to 278 symbols. The simulation duration is equal to 300 s.

In our simulation framework, we consider Poisson arrivals of each class of customer UGS and rtPS, with respective arrival rate λ U GS = λ rtP S = 1/30s -1 . We assume that the durations of the UGS and rtPS sessions follow exponential distributions with respective mean µ U GS = µ rtP S = 5 minutes.

We consider three types of UGS users: 10 kbit/s, 50 kbit/s, and 200 kbit/s, that represent 25%, 25%, and 50% of the UGS users, respectively. We also consider four types of rtPS users, with respective minimal data rate values (in kbit/s) R min = 30, 50, 100, 200 and maximal data rate values (in kbit/s) R max = 100, 300, 500, 800. We assume that each of these [R min , R max ] couples is chosen by 25% of the rtPS users.

The user SNR values are randomly chosen at the beginning of the session, considering a simple cellular model with three cells per cluster: the SNR distribution can then be obtained using classical results of the SNR repartition in cellular networks [START_REF] Lagrange | CIR cumulative distribution in a regular cellular network[END_REF]. Having the SNR values, the MCS used is directly deduced from Table I. The distribution of SNRs (and then MCSs) is given in Table II.

The valuation function (in mu by kbit/s transmitted) of the UGS and rtPS users are according to continuous uniform

V 1 U GS , V 2 U GS ] and [V 1 rtP S , V 2 rtP S ]
, respectively. In the rest of the paper, we assume that

V 1 U GS = V 1 rtP S = 0.5 and V 2 U GS = V 2 rtP S = 1.
Since the UGS and rtPS QoS classes are designed for realtime applications, an admission control mechanism is required. An UGS (resp. rtPS) customer is admitted only if there are enough OFDM symbols, taking into account its MCS used, for its data rate (resp. for its minimum data rate R min ).

B. Investigation of P ref

In this paper, we assume that P U GS Low = P rtP S min = P ref .

For the UGS pricing model, we adopt α U GS = 2 (P U GS High = 2 * P ref ), th U GS = 0.5. For the rtPS pricing model, we adopt C rtP S = 10.

V 1 U GS , V 2 U GS ]).
We notice that the throughput of the UGS connections decreases when P ref increases and P ref > 0.25. This is due to the high price actually paid and so some UGS users cannot pay to reserve for their data rates. When P ref is upper than V 2 U GS (V 2 U GS = 1), the throughput is equal to 0 because the utility function is always negative.

The throughput of the rtPS connections also constant when P ref increases and P ref < 0.9. It decreases when P ref increases and P ref > 0.9. Indeed, the rtPS users cannot extend their transmissions rate to R max and are be served,when the utility function > 0, at a rate r in [R min , R max ] (depending on the price actually paid).

Fig. 2 shows the mean blocking rate of the UGS and rtPS connections as a function of P ref . A connection is blocked when there are not enough symbols (noted "symbols < 0") or the price actually paid is upper than the price that the user is willing to pay (noted "utility < 0"). The blocking rate (because "utility < 0") of the UGS connections increases when P ref increases and this is due to the high price actually paid. When P ref is upper than V 2 U GS , the blocking rate (because "utility < 0") becomes constant because all the UGS connections are blocked (UGS throughput = 0, see Fig. 1).

When P ref > 0.75, rtPS connections can be blocked because of the high price actually paid. This happens when rtPS connections cannot pay to reserve R min . We observe that the blocking rate (because "symbols < 0") of UGS and rtPS connections are almost equal to 0. Indeed, the throughput decreases when the blocking rate is > 0. Since, when P ref is low, the throughput of the UGS and rtPS connections do not decrease (Fig. 1), the blocking rate (because "utility < 0" and "symbol < 0") of the UGS and rtPS connections have to be equal to 0. Fig. 3 shows the mean number of the UGS and rtPS connections in the system as a function of P ref . The mean number of connections in the system decreases when P ref increases. This is due to the increase of the blocking rate. We observe that P ref has less influence on the rtPS connections. Indeed, an UGS connection is blocked when it cannot pay to reserve its data rate. On the other hand, an rtPS connection is served even if it cannot pay to reserve its R max . It is served when it can pay to reserve r in [R min , R max ]. When P ref increases, the revenue from the rtPS connections increases when P ref < 1 and decreases when P ref > 1. The behavior of the revenue depends on the increase of the price actually paid and the increase of the blocking rate.

We note that the total revenue increases when P ref < 0.9, otherwise, it decreases. If the objective of a pricing scheme is only to maximize the network revenue, we choose 0.9 as a value of P ref . In addition of the maximization of the revenue, we can add some constraints to our pricing model. For example, if we add the minimization of the blocking rate and the maximization of the total throughput, we assign 0.25 to P ref .

C. Investigation of C rtP S

We recall that the price actually paid for rtPS connections (specially the price for extra rate) depends on C rtP S (see [START_REF] Chang | Adaptive Hierarchical Polling and Cost-based Call Admission Control in[END_REF] and ( 5)). In this Section, we adopt P U GS Low = P rtP S min = P ref = 0.25, α U GS = 2, and th U GS = 0.5.

Fig. 5 shows the mean throughput of the UGS and rtPS connections as a function of C rtP S . The throughput of the rtPS connections decreases when C rtP S increases. This is due to The throughput of the UGS connections slightly increases when C rtP S increases. This is because the remaining symbols increases when the symbols used by the rtPS connections decreases.

Fig. 6 shows the mean blockage rate of the UGS and rtPS connections as a function of C rtP S . The blocking rate (because "utility < 0") of the UGS connections is equal to 0 because the minimum price that an UGS user is willing to pay (V 1 U GS = 0.5) is upper or equal than the maximum price that can actually paid (P U GS High = 2 * P ref = 0.5).

The blocking rate (because "utility < 0") of the rtPS connections is equal to 0 because all the rtPS connections can pay to reserve R min (P rtP S min is independent of C rtP S ).

The blocking rate (because "symbols < 0") of the UGS and rtPS becomes equal to 0 when C rtP S exceeds a certain value (C rtP S = 250). This is because of the decrease of the total throughput when C rtP S increases. Then, there are more available symbols in the system. Fig. 7 shows the mean number of the UGS and rtPS connections in the system as a function of C rtP S .

The mean number of the UGS and rtPS connections in the Otherwise, it becomes constant. Indeed, this directly depends on the blocking rate. All the connections are served when C rtP S > 250. Therefore, the total number of the served connections cannot increase even if there are more available symbols in the system. Fig. 8 shows the mean revenue from the UGS and rtPS connections as a function of C rtP S . We observe that the revenue from the UGS connections is almost constant because the price actually paid for this kind of user is independent of C rtP S .

The revenue from the rtPS connections increases when C rtP S increases and C rtP S < 250. Indeed, although the decrease of the throughput of the rtPS connections, the price actually paid for the extra data increases and the rtPS users are still able to pay to reserve an additional rate (its depends on the valuation function).

When C rtP S > 250, the revenue from the rtPS connections decreases when C rtP S increases. This is because the price actually paid for extra data is too high that the rtPS users restricts on a throughput closer than their R min .

Therefore, we can assign 250 to C rtP S if our single aim is to maximize the total revenue. We also choose 250 as a value of C rtP S if we add the minimization of the blocking rate and secondly the maximization of the throughput as constraints.

V. CONCLUSION

In this paper, we have proposed a pricing scheme for WiMAX systems. This scheme is designed for the UGS and rtPS QoS classes that represent the real-time applications in WiMAX. Moreover, our proposed mechanism is easily understandable by users and reasonably fair, and could therefore be implemented in some WiMAX systems. We have set a precise simulation framework to study the implications of our proposals, and exhibited some good properties of our pricing model in terms of revenue maximization, throughput and user satisfaction (via the blocking rate). These proposals include many parameters, which gives way to interesting optimizations. The choice of the system parameters depends on

Fig. 1

 1 shows the mean throughput of the UGS and rtPS connections as a function of P ref . The throughput of the UGS connections is constant when P ref increases and P ref < 0.25. This is because the highest price actually paid (P High = 2 * P ref = 0.5) is lower or equal than the lowest call valuation (belongs to [

Fig. 1 .Fig. 2 .

 12 Fig. 1. Mean throughput versus P ref

Fig. 4

 4 shows the mean revenue from the UGS and rtPS connections as a function of P ref . The revenue from the UGS connections increases when P ref increases without exceeding 0.25. This is because no UGS connections is blocked (blocking rate = 0 when P ref < 0.25, see Fig. 2) and the price actually paid increases when P ref increases. When P ref increases and P ref > 0.25, the revenue from the UGS connections

Fig. 3 .Fig. 4 .

 34 Fig. 3. Mean number of the connections in the system versus P ref

Fig. 5 .Fig. 6 .

 56 Fig. 5. Mean throughput versus C rtP S

Fig. 7 .

 7 Fig. 7. Mean number of connections in the system versus C rtP S

Fig. 8 .

 8 Fig. 8. Mean revenue from the connections versus C rtP S

ACKNOWLEDGMENT

This work is supported by the Britanny (Bretagne) Region in the framework of Op WiMAX Project.


Julien Pezard
email: julien.pezard@ec-lyon.fr

Véronique Soulière
email: bveronique.souliere@univ-lyon1.fr

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

Naoufel Haddour
email: naoufel.haddour@ec-lyon.fr

François Buret
email: efrancois.buret@ec-lyon.fr

Christophe Raynaud
email: fchristophe.raynaud@insa-lyon.fr

Dominique Planson
email: dominique.planson@insa-lyon.fr

Realization and Characterization of Carbonic Layers on 4H-SiC for Electrochemical Detections

Keywords: electrochemical sensors, carbonic layers, PPF, BDD, graphene

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

Introduction

For electrochemical use, carbon materials exhibit better behaviors than metallic gold or platinum electrodes, in terms of detection limit, potential range and sensitivity [1]. Another advantage of these materials is that they can be prepared at the surface of different substrates. In this work, we aim at preparing diamond, graphene and pyrolyzed photoresist films (PPF) at the surface of silicon carbide (4H-SiC), which is considered to be a biocompatible material. These layers are more attractive than glassy carbon, a broadly used carbon material in electrochemistry. Our layers present the advantage to be able to be patterned, to resist to high temperature and also give the possibility to include all them in SiC technological steps for the preparation of more elaborate structures.

Discovered in 1984, fourteen years later, PPF has been used as a working electrode [2], showing interesting properties in terms of durability and reliability. Nowadays, PPF is used in research as a biocompatible working electrode for electrochemical analysis [3], or as a sensitive and selective electrode for heavy metals detections [4]. Del Campo et al. also showed the possibility to include PPF in CMOS technology [5]. PPF can be obtained by the pyrolysis of photoresist layers, by a plethora of conditions (temperature, duration, pressure, presence of different gas). In SiC technology, PPF is used as a protective cap during post-implantation annealing [6].

Graphene has shoved the world of electronics forward by its marvelous theoretical properties. In electrochemistry, its exceptional behavior in sensitivity and limit of detection [7] has attracted the attention of researchers. In this study we focused on graphene grown on SiC by silicon sublimation at its surface. This material has promising characteristics, in particular, a good signal to noise ratio.

Diamond is well known in electrochemistry for its very large potential range, but also a far higher signal to noise ratio than for glassy carbon [8]. Diamond in electrochemistry needs to be conductive enough and in this sense it is heavily doped with boron. The common name used is Boron doped diamond (BDD). These electrodes show a reproductive behavior through time and show no damage of the layer with long term use.

In our work, performances of these electrodes have been evaluated and compared. Furthermore, an application of these layers as transducers in the detection of a biological molecule (Acetylthiocholine) is shown. The results obtained give real confidence to our team that these layers show a real interest in the field in electrochemistry.

Experimental

Pyrolyzed photoresist films were obtained by the annealing of AZ5214E photoresist at temperatures above 1000 °C, which is higher than what can be generally found in literature [2][3][4][5]. PPF layers have been synthetized by spin coating depositing on SiC of AZ5214E photoresist film with a thickness of approximately 1 µm. Then, solvents are evaporated and the resist is pyrolyzed under primary vacuum at 750 °C for 30 minutes. A second annealing is performed with an induction RTA furnace during 30 minutes, under Ar, with a temperature plateau we varied from 1150 to 1650 °C. Graphene has been synthesized by sublimation of Si from SiC at 1550 °C under Ar atmosphere with the same induction RTA furnace. Details upon this method can be found elsewhere [START_REF] Pezard | Thin Solid Films[END_REF]. Boron Doped Diamond has been deposited on SiC by plasma-assisted chemical vapour deposition (PACVD) [START_REF] Tallaire | [END_REF]. All the carbonic layers have been formed on 4H-SiC commercial substrates with low doped n-type epilayers provided from CREE TM . The presence of the low-doped epilayer was necessary to select the response of the carbonic layers from that of the SiC substrate.

For this study, the layers' conductivities have been evaluated through I-V four-point probe method measurements at room temperature. Their electrochemical properties in terms of kinetics, potential range, detection limit, sensitivity and signal to noise ratio were studied by electrochemical characterizations with an Origalys OGS 100 potentiostat. Acetylcholinesterase, chitosan, and Dulbecco's Phosphate Buffered Saline (PBS) 10X were purchased from Sigma Aldrich. µ-Raman measurements were performed using the 633 nm wavelength of an HeNe laser beam (LabRAM ARAMIS HORIDA 20 mW).

Electrode

PPF annealed 1250°C -1.2 -1.35 V -1.2 -1.15 V PPF annealed 1450°C -1.4 -1.37 V -1.25 -1.2 V PPF annealed 1650°C -1.6 -1.39 V -1.4 -1.25 V Graphene -1.4 -1.39 V -1.4 -1.25 V Diamond -1 -1.37 V -1.1 -1.25 V

Results and discussion

Figure 1 and Table 1 present the electroactivity potential domains of our electrodes in KCl 0.1 M (Molarity=moles/volume) and in PBS 10X. For the PPF, the electroactivity range increases with the annealing temperature. This is in agreement with the increasing of the electrical conductivity of the PPF layers with the annealing temperature, as we obtained by I-V measurements. Figure 2 presents linear I-V characteristics obtained by four-point method and the extracted resistivity of the PPF layers versus annealing temperature. The thicknesses of the PPF layers have been accurately measured by alpha-step stylus profilometer. Both the resistivity and the thicknesses of the PPF layers decrease by increasing the annealing temperature. The thickness variation is related to a disorder evolution which is also observed by structural Raman analyses presented in Fig. 3 and Table 2. The relative intensity variation between the peaks D and G indicates a change in the structure of the PPF layers. The optimal annealing temperature can be considered at 1650°C which is close to the temperature of our graphene growth on SiC. The PPF and graphene layers could be generated on SiC during the same technological step.

Fig2. Four-probe I-V measurements on PPF layers and the extracted resistivities versus the temperature annealing. Table 3 resumes the biosensors characteristics obtained with our carbonic electrodes: reaction speed (imax), Michaelis Menten constant (KM), limit of detection, and linearity range.

For the all three materials the KM constant is in the same range of values, the fixed enzyme exhibiting the same activity. On a biosensor based on carbon nanotube films functionalized with the same enzyme and employing a redox mediator Chen et al. [11] obtained a Michaelis constant of 0.25 mM, which is relatively close to our results. Li et al. showed a constant of 0.77 mM for the detection of acetylthiocholine on graphene electrode [12]. In our study, we aim at having the lowest Michaelis constant, as a low value stands for a maximum enzyme activity reached with low substrate concentrations. In terms of sensitivity, compared to graphene, PPF and diamond exhibit superior values. Our results are relatively close to the literature where the sensitivity is around 10 µA/mM [13]. The detection limit observed on our three materials is between 7 to 30 µM. The best threshold values were obtained with graphene. These values correspond to the detection limits described in the literature for electrodes unmodified with mediators. For example, functionalized graphene has allowed obtaining a detection limit of 3 µM [14]. Kesik et al. obtained a limit of 0.09 mM for carbon nanotubes, and a linearity range from 0.05 mM to 8 mM [15].

Conclusion

The layers grown in this study have shown electrochemical behavior comparable with literature, with wide potential range, and the possibility to use them for biodetection was outlined through the detection of acetylthiocholine in chronoamperometry in a concentration range that fits with biological detection needs. The materials used for the development of our electrodes can be easily patterned, one asset that enables us to predict multiple aspects that could be investigated in the future. One of them would be to reduce the size of these electrodes from macrometric to micrometric dimensions in order to use them in all-inclusive devices with compatible technological steps as PPF and graphene generated on SiC during the same annealing.

Fig. 1

 1 Fig.1 Cyclic voltammetry corresponding to the electroactivity of our electrodes in PBS 10X

Fig. 3

 3 Fig.3 Raman analyses of PPF layers formed at varied annealing temperatures from 750°C to 1650°C Figures 4 presents the Michaelis-Menten and Lineweaver Burk calibration curves from our electrodes (1 cm 2 samples), used as biosensors for detection of acetylthiocholine in PBS 10X. Prior to electrochemical detection, the electrodes have been functionalized by adsorption using a solution of chitosan 0.5 % in acetic acid adjusted to pH 5 with NaOH.

Fig 4

 4 Fig 4 Michaelis-Menten (left) and Lineweaver Burk (right) calibration curves extracted from chronoamperometry for detection of acetylthiocholine

Table 1 .

 1 The electroactivity potential range values for our electrodes in KCl 0.1M and PBS 10X

Table 2

 2

	T (°C)	ID/IG
	750	0.946
	1050	1.235
	1150	1.25
	1250	1.265
	1450	1.275
	1650	1.289

. Relative intensity variation between the peaks D and G extracted from Raman analyses on PPF layers.

Table 3 .

 3 The electrochemical characteristics values for our electrodes used as transducers in a biosensor

Acknowledgements

The authors gratefully thank Dr Alexandre Tallaire from LSPM for the diamond films deposition by plasma-assisted CVD on our SiC samples.


T Abi-Tannous
email: tony.abi-tannous@insa-lyon.fr

M Soueidan
email: bmaher.soueidan@insa-lyon.fr

G Ferro

M Lazar

B Toury

M F Beaufort

J F Barbot

J Penuelas
email: hjose.penuelas@ec-lyon.fr

D Planson

A study on the chemistry of epitaxial Ti3SiC2 formation on 4H-SiC using Al-Ti annealing

Keywords: Ti-Al alloy, MAX phase, Ti3SiC2, ohmic-contact, p-type 4H-SiC

In order to form Ti3SiC2 on 4H-SiC(0001) 8°-off, 200 nm of Ti30Al70 was deposited onto SiC substrates by magnetron sputtering from pure Ti30Al70 targets. The samples were then annealed at 1000°C for 10 min under Ar atmosphere in a Rapid Thermal Annealing (RTA) furnace. Structural analyses reveal the formation of epitaxial hexagonal Ti3SiC2 (0001) oriented. Elemental analyses show that high amount of Al and O elements are present inside the deposit. Obviously, the formation of Ti3SiC2 is accompanied by parasitic Al oxide, probably due to some unwanted oxygen residual in the RTA chamber. By using proper backing steps before the annealing, the deposit is not anymore composed of only Ti3SiC2 but accompanied with other compounds (Al3Ti, and Al). On the oxide-free sample, the specific contact resistance ρc of the Ti3SiC2 based contact on p-type 4H-SiC (having Na= 2´10 19 cm -3 ) was measured to be as low as 6´10 -5 Ω.cm 2 .

Introduction

To achieve reliable, high quality 4H-SiC devices, low resistance ohmic contacts on SiC (n-or ptype) are necessary. This is known to be especially difficult on p-type SiC. Rather good results have been obtained from high-temperature annealed Ti and Al-containing alloys [1][2][3][4][5]. The reason why these alloys display low contact resistance is still under discussion though the beneficial effect of the resulting MAX phase Ti 3 SiC 2 is often suggested [6,7]. This material has a very interesting combination of metallic properties such as high thermal and electrical conductivity and ceramic properties such as being resistant to oxidation and thermal shock [8], with an excellent chemical compatibility with SiC. Epitaxial Ti 3 SiC 2 on on-axis and 8°-off 4H-SiC after Al-Ti annealing was already reported [9,7] but the chemical paths leading to its single formation are not well understood. For instance, where do Al and Si excesses go? Is there any amorphous oxide formation?

The goal of this study is to try answering these questions while finding conditions for forming Ti 3 SiC 2 on 4H-SiC(0001) 8°off after annealing of Ti-Al based stacking.

Experimental section

The samples used for this study are Si face 4H-SiC(0001) substrates 8° disoriented. Before metallization, the samples are chemically cleaned to remove any surface pollution. This implies acetone and ethanol ultrasonic degreasing for 5 min each, followed by H2SO4: H2O2 (75: 25) for 10 minutes and finally HF acid diluted at 5% for 4 min, before rinsing with deionized water. After cleaning, 200 nm of Ti30Al70 was deposited onto SiC substrates by magnetron sputtering from pure Ti30Al70 targets in a high vacuum system. The deposition was carried out with an Ar constant pressure (5×10 -3 mbars) at room temperature. The samples were then annealed at 1000°C for 10 min under Ar atmosphere in a rapid thermal annealing furnace (heating rate of about 20°C/s) [9]. Some experiments were performed on p-type 4H-SiC epitaxial layers, with doping concentration of 2´10 19 at/cm -3 , in order to allow determination of the Specific Contact Resistance (SCR) using the Transfer Length Method (TLM) structure.

Identification of reaction products

Figure 1-a shows X-ray diffraction (XRD) θ-2θ scan of a typical sample annealed in these conditions. As seen, the diffraction peaks unambiguously reveal the formation of the hexagonal Ti3SiC2 structure which is in epitaxial relation with the substrate. The rocking curves obtained on the (0008) diffraction planes of the Ti 3 SiC 2 phase (Figure 1-b) present a full width at half-maximum (FWHM) of 0.4° showing the good crystallinity of the film. Using X-ray photoelectron spectroscopy (XPS) analyses, the presence of Al was also detected in high amount on the surface, but its depth profile was associated with the presence of O element (Figure 2). Such surface oxide layer was not seen from XRD measurements probably because under amorphous state.

In order to confirm these results, transmission electron microscopy (TEM) investigations were carried out. Figure 3 displays a low magnification bright field (BF) image of a typical region of an as-annealed sample. This general view shows that the 4H-SiC substrate is entirely covered by a Ti3SiC2 layer with varying thickness from ~95 to 140 nm. On top of this is found an almost continuous layer which was attributed to amorphous to polycrystalline Al oxide (probably Al2O3). As a matter of fact, the formation of Ti3SiC2 is accompanied by parasitic Al oxide, probably due to some oxygen residual in the RTA chamber. For ohmic contact, such oxide layer is obviously unwanted and conditions shall be found for avoiding its formation. That is why advanced baking steps were implemented before the annealing. The effects of such baking were directly seen from the XRD measurements (Figure 4). Ti3SiC2 is not the only phase detected and its formation is now accompanied by Al3Ti, and pure Al. As a result, oxygen seems to play a key role during the Ti-Al/SiC interaction for obtaining only one Ti-based phase (Ti3SiC2). This is probably what was happening in the work of A. Drevin-Bazin [9]. The following "globally" chemical reactions are proposed to summarize the results obtained:

Without oxygen: 20Al + 9Ti + 2SiC → Ti3SiC2 + 6Al3Ti + 2Al + Si With oxygen: 14Al + 6Ti + 4SiC + 25/2O2 → 2Ti3SiC2 + 7Al2O3 + 2SiO2
Apparently, oxygen is "pumping" preferentially the Al atoms from the Al-Ti alloy so that the system moves towards the single Ti3SiC2 phase instead of forming a mixture with Al3Ti.

Electrical characterization

The geometric patterns for the TLM (Transfer Length Method) measurements were obtained through a photolithographic procedure. The tested layer was laterally isolated by RIE etching of the SiC forming mesa structures. The p-type layers are also insulated vertically by the p/n junction. The commercially Al-etch at 60°C was used for etching the metal patterns. After such procedure, the TLM pattern consists of multiple rectangular electrodes with increasing spacing of 6, 10, 20, 40, 80 and 120 µm. Using a 4-point probe method, the I-V characteristics are plotted as function of the contact spacing for a 200 nm-thick Ti30Al70 layer annealed at 1000°C during 10 min (see Figure 5).

As seen for all the spacing, the I-V curves are perfectly linear showing again the ohmic character of the Ti3SiC2-based contact for p-type 4H-SiC. The total resistances, deduced from the curves of figure 5 are plotted as function of spacing between the contact pads (Figure 6). In the configuration used, the curve showing the total resistance according to the distance between the metal pads is linear. From figure 6 it can be inferred that the specific contact resistance is about 6×10 -5 Ω.cm 2 . This value is among the best compared to the state of the art [1][2][3][4][5].

A l2 O 3 T i3 S i C 2

Figure 1 .Figure 2 .

 12 Figure 1. (a) XRD θ-2θ scan after annealing at 1000°C of Ti30Al70, (b) The rocking curves at 2θ = 40.9° corresponding to the (0008) Ti3SiC2 peak.

Figure 4 .

 4 Figure 4. XRD pattern of the TiAl contacts onto 4H-SiC annealed at 1000°C for 10 min with proper baking of the system.

Figure 3 . 2 A l 2 O 3 Figure 5 .

 32235 Figure 3. Cross-sectional TEM image of the same sample as in figure 1-a. Ti3SiC2 covers completely the surface of the SiC substrate and is itself covered by an Al oxide layer.

Figure 6 .

 6 Figure 6. Evolution of resistance as a function of the distance between the contact pads for the same sample as in figure 5.

Summary

We have shown that it is possible to form epitaxial Ti3SiC2 onto 4H-SiC (0001) 8°-off from 200 nm of Ti30Al70 alloy, deposited by sputtering and annealed in a rapid thermal annealing at 1000°C during 10 min. TEM images show that Ti3SiC2 covers completely the surface of the SiC substrate and is itself covered by an Al oxide layer. By using proper baking steps before the annealing, the results are clearly different. Several compounds (Ti3SiC2, Al3Ti, and Al) are then detected. On the oxide-free sample, the specific contact resistance of the Ti3SiC2 based contact on p-type 4H-SiC (having Na= 2´10 19 cm -3 ) was found to be ρc = 6´10 -5 Ω.cm 2 . Our results are among the best compared to the state of the art.


Tony Abi-Tannous

Maher Soueidan

Gabriel Ferro

Mihai Lazar

Christophe Raynaud

Berangere Toury

Marie-France Beaufort

Jean François Barbot

Olivier Dezellus

Dominique Planson

A Study on the Temperature of Ohmic Contact to p-Type SiC Based on Ti 3 SiC 2 Phase

Keywords: Ohmic contact, silicon carbide, Ti-Al alloy, Ti3SiC2, TFE

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

I. INTRODUCTION

ue to its wide bandgap, high electric field strength and high thermal conductivity, but also to its technological maturity, Silicon Carbide (SiC) is the most likely of all the wide band-gap semiconductors to succeed Silicon (Si) for high power density and high temperature device applications [START_REF] Neudeck | Stable electrical operation of 6H SiC JFETs and ICs for thousands of hours at 500°C[END_REF][START_REF] Okojie | Failure mechanisms in MEMS based silicon carbide high temperature pressure sensors[END_REF][START_REF] Shenai | Optimum semiconductors for high-power electronics[END_REF][START_REF] Chow | SiC and GaN bipolar power devices[END_REF]. But, the high-temperature functionality of SiC devices is useless without ohmic contacts that are also able to operate under the same conditions. The stability of metalsemiconductor contacts is one of the factors which limit the high-temperature operation of SiC electronic devices and circuits. This is generally associated with the evolution of the nature and/or structure of the metal-SiC interface through continuing reactions forming silicides, carbides, free carbon, and/or oxides. Further complications are introduced by the requirements for high temperature ohmic contacts such as i) high resistance to oxidation, ii) stable electrical conductivity and contact resistance, iii) low resistivity, iv) the ability to be connected to external circuitry and v) the reliability at high temperature.

P-type 4H-SiC has a high work function (~7 eV) and it's difficult to found a conventional metal that leads to low Schottky barrier height (SBH) when deposited on p-type 4H-SiC [START_REF] Roccaforte | Critical issues for interfaces to p-type SiC and GaN in power devices[END_REF]. Such contacts are generally employed containing aluminium [START_REF] Luo | Searching for device processing compatible ohmic contacts to implanted p-type 4HSiC[END_REF], and while many different alloys have been investigated, a great deal of attention has been focused on Al-Ti [START_REF] Roccaforte | Critical issues for interfaces to p-type SiC and GaN in power devices[END_REF][START_REF] Wang | Ohmic contacts on silicon carbide: The first monolayer and its electronic effect[END_REF][START_REF] Mohammad | Comparison of Pt-Based Ohmic Contacts with Ti-Al Ohmic Contacts for p-Type SiC[END_REF][START_REF] Konishi | Development of Ni/Al and Ni/Ti/Al ohmic contact materials for p-type 4H-SiC[END_REF][START_REF] Mysliwiec | TiAl-Based Ohmic Contacts on p-Type SiC[END_REF], which has demonstrated specific contact resistances on the order of 10 -3 -10 -5 Ω.cm 2 on p-type SiC. Other investigations have focused on Ni/Ti/Al contact fabricated on highly p-type 4H-SiC [START_REF] Thierry-Jebali | Very low specific contact resistance measurements made on a highly p-type doped 4H-SiC layer selectively grown by vapor-liquid-solid transport[END_REF] leading to low specific contact resistance (2.8×10 -6 Ω.cm 2 ), but this result was not reproducible.

To date, Al-Ti alloy remains the only material that has been shown to result in stable low-resistance ohmic contacts to ptype SiC due to the formation of Ti3SiC2 after annealing at high temperature [START_REF] Roccaforte | Critical issues for interfaces to p-type SiC and GaN in power devices[END_REF][START_REF] Tsukimoto | Growth and Microstructure of Epitaxial Ti3SiC2 Contact Layers on SiC[END_REF][START_REF] Buchholt | Ohmic contact properties of magnetron sputtered Ti3SiC2 on n-and p-type 4H-silicon carbid[END_REF]. Ti3SiC2 is layered ternary carbide which belongs to the Mn+1AXn (MAX) compounds [START_REF] Barsoum | Kinking Nonlinear Elastic Solids, Nanoindentations, and Geology[END_REF][START_REF] Barsoum | The MN+1AXN phases: A new class of solids: Thermodynamically stable nanolaminates[END_REF]. It is thought to be one of the best candidate materials for high temperature ohmic contact issues to SiC because of the following reasons : i) it advantageously combines metallic and ceramic properties, ii) has good thermal and electrical conductivity (37 Wm -1 .K -1 and 4.6×10 6 Ω -1 m -1 respectively), iii) has relatively low thermal expansion coefficients (8.10 -6 /°C) which value is closed to SiC (4.10 -6 /°C), iv) is a very stable compound in air up to 1400°C and v) Ti3SiC2 and SiC are in thermodynamic equilibrium so that the interface should remain stable even at high temperature (higher than 1200 °C) [START_REF] Eklund | The Mn+1AXn phases: Materials science and thin-film processing[END_REF][START_REF] Utili | Ti3SiC2 as a candidate material for lead cooled fast reactor[END_REF].

This paper reports on the electrical characterization of Ti-Al ohmic contact to p-type 4H-SiC. We study the effect of Ti-Al composition and annealing temperature for the formation of ohmic contact, which can be used in various electronic devices (bipolar diodes, BJT, Thyristors, JFET…). Finally, the temperature dependence of the electrical properties up to 600°C and after 400 h of ageing at 600°C are discussed.

II. EXPERIMENTAL SECTION

A. Sample Preparation

P-type epilayers (~1.3 µm thick) doped with Al in the [1-4] ×10 19 cm -3 range on 4H-SiC (0001) 4°off substrates were used in this study. The samples are chemically cleaned to remove any surface pollution. This implies acetone and ethanol ultrasonic degreasing for 5 min each, followed by two times H2SO4: H2O2 (2: 1) for After RIE, the samples underwent a second chemical cleaning as mentioned above. And then two sets of samples were prepared on such substrates:

(1) Ti-Al stacking: 200 nm of TixAl100-x (20 at% ≤ x ≤ 50 at%) deposited by magnetron sputtering from TixAl100-x target in a high vacuum system. The deposition was carried out at room temperature with an Ar constant pressure (5×10 -3 mbar).

(2) Pure Ti: 200 nm of pure Ti was deposited using e-beam evaporator in a high vacuum chamber. After metal deposition, Transfer Length Method (TLM) structures were fabricated on the mesa in order to characterize electrically the metal/SiC p-type contact. The geometric patterns for the TLM measurements were obtained through a photolithographic procedure. The unwanted metal was removed by wet etching using commercial Al-etch at 60°C. The TLM contact consists of seven rectangular electrodes (500 × 100 µm 2 ) with increasing spacing of 3, 6, 10, 20, 40, 80 and 120 µm. For ohmic contact formation, high temperature annealing was performed in a Rapid Thermal Annealing (RTA) furnace under argon at atmospheric pressure with advanced backing steps (3 times pumping followed by argon filling) before the annealing in order to decrease the residual oxygen concentration in the RTA chamber. The beneficial effect of such baking steps is described in references [START_REF] Abi-Tannous | Parametric investigation of the formation of epitaxial Ti3SiC2 on 4H-SiC from Al-Ti annealing[END_REF][START_REF] Abi-Tannous | A study on the chemistry of epitaxial Ti3SiC2 formation on 4H-SiC using Al-Ti annealing[END_REF]. The annealing temperature was varied from 900 to 1200°C for 10 min plateau and a heating rate of 20°C/s. After annealing, the dimensions of the TLM patterns and the interpad distances remained unchanged. This was confirmed by measurements using optical microscope. So, no correction has been performed during the TLM measurements. Note that the "current crowding effect" is negligible since the length of the rectangular pads (500 µm) is very big compared to the distance between the pads and the distance between the pads and the edge of the mesa (5 µm).

B. Sample characterizations

The structural characterizations were mainly performed using X-Ray Diffraction (XRD), and Transmission Electron Microscopy (TEM). The XRD measurements were conducted on a D8 BRUKER with Cu Kα radiation. The Cross-sectional TEM (XTEM) samples were thinned down to 10 µm by using the tripod polisher. The electron transparency was achieved by ion milling.

For electrical characterization the samples are mounted directly on the heating chuck of a probe station. The samples are characterized up to 600°C and the temperature is monitored using a K-Type thermocouple. Our tests showed a temperature accuracy value of ±1%. Above 300°C, tests are performed under vacuum to prevent oxidation of the TLM structures. The electrical measurements are performed using a Keithley Source Measurement Unit K2602A. A four-probes setup is used (Kelvin configuration) to eliminate cable and tips resistances, which results in a measurement accuracy value better than 1%. I-V characteristics are plotted as function of the contact spacing of different temperatures.

III. RESULTS AND DISCUSSION

A. Identification of reaction products

TABLE I lists the reaction products obtained by XRD for all samples (Ti 20 Al 80, Ti 30 Al 70, Ti 50 Al 50 and Ti) after annealing at various temperatures. From the Ti3SiC2 peaks all belonging to the (0 0 L) family (L = from 2 to 12), we can deduce that the Ti3SiC2 material formed has an epitaxial relationship with the SiC substrate. Epitaxial Ti3SiC2 on SiC, obtained by Ti-Al annealing, was already reported by Parisini and al. [START_REF] Parisini | Structural characterization of alloyed Al/Ti and Ti contacts on SiC[END_REF] and Tsukimoto and al. [START_REF] Tsukimoto | Growth and microstructure of epitaxial Ti3SiC2 contact layers on SiC[END_REF].

Transmission Electron Microscopy (TEM) investigations were carried on Ti30Al70 annealed at 1000°C and Ti annealed at 1200°C. Fig. 1-a displays a bright field (BF) filtered image of a typical region of the as-obtained layer on 4H-SiC from Ti30Al70 annealed at 1000°C. This general view shows that the 4H-SiC substrate is entirely covered by a Ti3SiC2 layer, which is itself covered by an Al rich layer. On the other hand, for Ti annealed at 1200°C (Fig. 1-b), the 4H-SiC substrate is again entirely covered by a Ti3SiC2 layer, but there is a strongly modified SiC region on the side of the interface coming from the solid-solid reaction between Ti and SiC. When comparing with the results obtained using Al based contacts, Al play the role of liquid phase promoter, leading to a less damaged interface. For more details about the interfacial reaction check ref. [START_REF] Abi-Tannous | Parametric investigation of the formation of epitaxial Ti3SiC2 on 4H-SiC from Al-Ti annealing[END_REF].

B. Electrical characterization at room temperature

Using TLM structures, the electrical characterization of the contact resistance was performed by means of current-voltage (I-V) measurements. Fig. 2 shows the corresponding roomtemperature I-V characteristics measured between two adjacent TLM pads taken from Ti50Al50 contacts, as-deposited and after thermal annealing at 900, 1000, 1100, 1200°C.

A non-ohmic behavior was observed on as-prepared samples and on samples annealed at 900°C. But for the samples annealed at 1000°C or above, I-V behaviors are linear and prove that the contact is ohmic. Fig. 3 shows a typical set of curves obtained from TLM measurements from those contacts (i.e. Ti50Al50 annealed at 1000, 1100 or 1200°C) that displayed ohmic characteristics for the purpose of extracting the Specific Contact Resistance (SCR) ρc. As seen, the TLM plots have a different slope at different annealing temperatures; it means that the sheet resistance of the p-layer is different between the samples. This is due to the fact that the annealing was achieved on different samples having a doping concentration varied between 1 and 4 × 10 19 cm -3 . The Ti50Al50 annealed at 1000°C yielded the lowest SCR of 1.1 × 10 -4 Ω.cm 2 (and also the lowest Ron).

This SCR increases with increasing annealing temperature as can be seen from TABLE II and Fig. 4. From this Table and Figure, one can also follow the electrical properties of the others types of contacts (Ti30Al70, Ti20Al80 and Ti) annealed at different temperatures and evaluated by TLM analysis at room temperature. As can be seen, the trends are rather similar for these contacts except for the case of pure Ti for which a nonohmic behavior was always observed. This is probably due to the modified SiC region at the interface.

However, when we add Al, the electrical behavior of the contact become ohmic and ρc increases when the Al content increases. That means the addition of Al to the Ti is necessary for the formation of ohmic contact with low resistivity. The main reason for this variation is still unknown and more physical and chemical characterizations to the interface should be done in order to elucidate this behavior. Moreover, for all compositions containing Al, the ρc increases when the annealing temperature increases from 1000 to 1200°C and this is probably due to the formation of TiC with Ti3SiC2.

We note in spite of a high roughness observed on our contacts (~45nm root mean squared, measured by Atomic Force Microscopy), excellent edge acuity is obtained. This is independent on the Al concentration, on all TLM pads and we carefully measured the width and the distances between the TLM pads.

According to the results presented in Fig. 4 and TABLE II, the samples annealed at 1000°C resulted in the lowest contact resistivities and hence were selected for further electrical measurements at high temperature.

C. Electrical characterization at high temperature

The I-V characteristics of Ti50Al50, Ti30Al70 and Ti20Al80 annealed at 1000°C, measured at different temperatures (25, 50, 100, 150, 200, 250, 300, 400, 500, 600°C) display an increase of the current with increasing analysis temperature. Fig. 5 shows a typical set of curves obtained from TLM measurements performed at different measurement temperatures (between 25 and 600°C) for Ti50Al50/4H-SiC annealed at 1000°C. A similar behavior was observed for Ti30Al70/4H-SiC and Ti20Al80/4H-SiC annealed at 1000°C. From these curves, we can extract the evolution of the sheet resistance (Rsh) of the p-layer (Fig. 6) and the SCR (Fig. 7) as a function of the measurement temperature for each sample. Besides, after these tests were completed and the samples are cooled down to 25°C, Rsh and SCR were measured again and it were found not to differ from the values measured before the high temperature tests for each contact type.

For Ti50Al50, Rsh decreases with increasing measurement temperature, from ~2 kΩ/sq at 300 K to 0.2 kΩ/sq at 875 K. On the other hand, for Ti30Al70 and Ti20Al80, Rsh decreases from ~2.5 kΩ/sq at 300 K to 0.4 kΩ/sq at 875 K, in a similar

To determine exactly the doping concentration NA of the player, the Rsh experimental data, reported in Fig. 6, were fitted using the expression of the sheet resistance defined by: [START_REF] Neudeck | Stable electrical operation of 6H SiC JFETs and ICs for thousands of hours at 500°C[END_REF] where "q" is the elementary charge; "µp(T)" is the holes mobility, "p(T)" is the free hole concentration and "e" is thickness of the p-layer (in our case t~1.3 µm).

The temperature dependence of the holes mobility µp(T) is giving by the following relation [START_REF] Matsuura | Dependence of acceptor levels and hole mobility on acceptor density and temperature in Al-doped p-type 4H-SiC epilayers[END_REF]: [START_REF] Okojie | Failure mechanisms in MEMS based silicon carbide high temperature pressure sensors[END_REF] where "β" is an empirical parameter depending on the doping concentration NA and "µp(300)" is the mobility at 300 K. The values of β and µp(300) results from theoretical calculations (resolution of the electroneutrality equation, calculation of the electron drift mobility as the sum of contribution of the different scattering mechanisms).

The temperature dependence of free hole concentration p(T) is giving by the following relation [START_REF] Schroder | Carrier and doping density[END_REF]:

(3)
Where "k" is the Boltzmann constant, "ND" the concentration of compensating donor, "ni" the intrinsic carrier density, "NV" the effective density of states in the valence band, "g" the degeneracy factor for acceptors and "EA" the ionization energy of acceptors referred to the top of the valence band. This ionization energy is given by [START_REF] Koizumi | Compensation-Dependent Carrier Transport of Al-Doped p-Type 4H-SiC[END_REF]:

Where (𝑁 & 𝑁 ' ⁄ ) is the compensation ratio. Note that in equation ( 3) and ( 4) 𝑁 ' and 𝑁 & are the total doping concentration and not the ionized dopant concentration (𝑁 ' * and 𝑁 & + ). In these fittings, the doping concentration NA and the compensation ratio (𝑁 & 𝑁 ' ⁄ ) were considered as fitting parameters.

From the best fit in Fig. 6, we found NA=1.5×10 19 cm -3 (with 2% compensation, µp(300) = 75 cm 2 V -1 s -1 and b = 1.5) for Ti30Al70 and Ti20Al80 annealed at 1000°C. In the case of Ti50Al50 annealed at 1000°C, we found NA=3×10 19 cm -3 (with 3% compensation, µp(300) = 50 cm 2 V -1 s -1 and b = 1.3).

As seen, the NA value found for Ti 50 Al 50 from the temperature dependence fit is higher than those found for Ti 20 Al 80 and Ti 30 Al 70. This is due to the fact that the doping concentration of the p-layer varies between 1 and 4×10 19 cm -3 . These values of NA will be used to fit the experimental SCR.

( )

1 ( ) ( ) sh p R T qµ T p T e = ( ) ( ) β 300 300 p p T µ T µ - ae ö = ç ÷ è ø ( ) ( ) ( ) ( ) 2 2 exp /
D i V A A D i p T p T N n N E N N p T n p g kT ´+ - ae ö = - ç ÷ - - + è ø 1 3 1 8 3
( ) 0.265 3.6 10 1 The SCR values of Ti20Al80, Ti30Al70 and Ti50Al50 annealed at 1000°C, as a function of the measuring temperatures, are shown in Fig. 7. As seen, for all cases, with increasing measurement temperatures (between 300 and 875 k) the SCR decreases, from 4.1×10 -4 to 1.6×10 -5 Ω.cm 2 for Ti 20 Al 80 , from 2.7×10 -4 to 1.15×10 -5 Ω.cm 2 for Ti 30 Al 70 and from 1.1×10 -4 to 9.8 × 10 -6 Ω.cm 2 for Ti 50 Al 50 . In general, this behavior due to the increase of Al doping ionization in the p-layer with increasing temperature, a typical behavior of wide band-gap semiconductors with incomplete ionization of the dopants at room temperature [START_REF] Moscatelli | Improved electrical characterization of Al-Ti ohmic contacts on p-type ion implanted 6H-SiC[END_REF].

D A A A N E eV N N -ae ö = - ´-ç ÷ è ø
According to the classical theory, there are three transports mechanisms in the metal/semiconductor interface: the thermionic emission (TE), the thermionic field emission (TFE) and the field emission (FE). The ratio kT/E00 allowed us to estimate the dominant carrier transport mechanism. E00 is a characteristics energy related to the doping density NA of the p-layer and it can be described by the following relation [START_REF] Padovani | Field and thermionic-field emission in Schottky barriers[END_REF]:

𝐸 --= 𝑞ℎ 4𝜋 3 𝑁 ' 𝑚 * 𝜀 7 8 9
(5)

where "h" is Planck's constant, "m * " the effective mass of the tunneling hole and ε the dielectric permittivity. in the case of SiC (ε = 9.7ε0, ε0 being the permittivity of free space). In this formula (5) we do not take into account the phenomenon of partial ionization but we used the same value 𝑁 ' defined as the doping concentration.

For our doping levels and temperature range (300 -875 k), the ratio kT/E00 is found to be ≈ [0.4 -1.1] for Ti50Al50 and ≈ [0.54 -1.5] for Ti30Al70 and Ti20Al70. So, TFE can be assumed to be the dominant carrier transport mechanism in the Ti 3 SiC 2 /SiC interface. Because if kT/E00 ≫1 TE is the dominate mechanism, if kT/E00 ≪1 FE is the dominate mechanism and if kT/E00 is close to 1 TFE is the dominate mechanism.

To determine the Schottky barrier height of the contact, the SCR experimental data, reported in Fig. 7, were fitted using the TFE model. In this model, the SCR can be expressed as [START_REF] Padovani | Field and thermionic-field emission in Schottky barriers[END_REF][START_REF] Frazzetto | Structural and transport properties in alloyed Ti/Al Ohmic contacts formed on p-type Al-implanted 4H-SiC annealed at high temperature[END_REF][START_REF] Yu | Electron tunneling and contact resistance of metal-silicon contact barriers[END_REF]: [START_REF] Luo | Searching for device processing compatible ohmic contacts to implanted p-type 4HSiC[END_REF] Where [START_REF] Wang | Ohmic contacts on silicon carbide: The first monolayer and its electronic effect[END_REF] and A* Richardson constant (in the case of SiC, A* = 146 Acm -2 K -1 [START_REF] Roccaforte | Richardson's constant in inhomogeneous silicon carbide Schottky contacts[END_REF]) and Vp is the energy difference between the valence-band and the Fermi level.

In these fittings, the Schottky barrier height ϕB was considered as fit parameter. The extracted values of ϕB from curves in Fig. 7 are reported in Table III. They are found to vary from 0.71 to 0.85 eV. Scorzoni et al. found a value of ϕB = 0.82 eV for Ti-Al deposited on p-type ion implanted 4H-SiC [START_REF] Scorzoni | Contact resistivity and barrier height of Al/Ti ohmic contacts on p-type ion implanted 4H-and 6H-SiC[END_REF]. This ϕB value is comparable to that found for Ti 50 Al 50 . On the other hand, a value of ϕB = 0.69 eV was fond by Vivona et al. for Ti/Al/W deposited on p-type implanted layer 4H-SiC [START_REF] Vivona | Ti/Al/W Ohmic contacts to p-type implanted 4H-SiC[END_REF]. This ϕB value is close from that found for Ti 30 Al 70.

Ageing tests were performed on Ti30Al70 at a constant temperature of 600°C in an Ar atmosphere. The contact resistivity was monitored before and after the tests. In fixed time intervals (24, 48, 100, 200 and 400 hours), the contact was cooled down to room temperature, and the contact resistivity was measured. As seen in Fig. 8, the SCR (ρc) has not changed even after 400 hours of aging. So we can deduce that the ohmic contact based on Ti3SiC2 is relabel at 600°C even after 400 h of aging.

IV. CONCLUSION

In this paper, the formation of Ti3SiC2 based ohmic contact on 4H-SiC from Ti-Al alloys annealed at high temperature was studied. The role of Ti-Al composition in the initial metallic stacking and the role of the annealing temperature were investigated. XRD analyses show that at very high annealing temperature of Ti-Al, epitaxial Ti3SiC2 was replaced by epitaxial TiC. The appearance of TiC at higher annealing temperature seems to be to the detriment of Ti3SiC2 while maintaining the epitaxial character. TEM investigation shows that Ti3SiC2 cover directly the SiC substrate, this means that Ti3SiC2 can be used as an ohmic contact on SiC. For pure Ti annealed at high temperature, Ti3SiC2 always co-exists with TiC, but this coexistence was accompanied by some perturbations in the SiC substrate. According to the TLM measurements, all samples annealed at 1000°C (except pure Ti) resulted in the lowest SCR and hence were selected for the high temperature experiments. The temperature dependence of the SCR indicated that TFE is the dominant transport mechanism and the extracted values of ϕB vary from 0.71 to 0.85 eV. Finally, when the high temperature tests are completed and the samples are cooled down to 25°C, the SCR were measured again and it were found not to differ from the values measured before the high temperature tests for each contact type. Finally, the ageing tests performed at 600°C under Ar atmosphere showed that Ti3SiC2 based contacts were stable and relabel up to 400 h.

Fig. 1 .Fig. 3 .

 13 Fig.1. XTEM-BF image taken along the [11-20] SiC zone axis showing that (a) after annealing of Ti30Al70 at 1000°C, Ti3SiC2 covers completely the surface of the SiC and is itself covered by an Al layer. And (b) after annealing of Ti at 1200 °C, modified SiC covers the surface of the SiC and is itself covered by a Ti3SiC2 layer

Fig. 5 .

 5 Fig. 5. Variation of the total resistance versus spacing between pads extracted from I-V measurements for Ti50Al50/4H-SiC annealed at 1000°C, at different measure temperatures (between 25°C and 600°C).

Fig. 6 .

 6 Fig. 6. Temperature dependence of the Rsh of the p-layer for Ti20Al80, Ti30Al70 and Ti50Al50 annealed at 1000°C.

Fig. 4 .

 4 Fig. 4. Evolution of the SCR (ρc) measured at room temperature as a function of the annealing temperature of Ti20Al80, Ti30Al70 and Ti50Al50.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Temperature dependence of the SCR for Ti20Al80, Ti30Al70 and Ti50Al50 annealed at 1000°C.

  10 minutes and finally HF acid A study in temperature of ohmic contact on ptype SiC based on Ti 3 SiC 2 phase

	diluted at 5% for 4 min, before rinsing with deionized water
	and blown dry in N2 gas. The tested SiC layer was isolated by
	RIE etching forming mesa structures. The SiC etching was
	performed in an Alcatel Nextral NE110 reactor during 7 min,
	with a pressure of 60 mTorr and a power (RF) of 250 W. The
	reactive gas is composed by SF6 (25 sccm) and O2 (7 sccm).
	Tony Abi-Tannous, Maher Soueidan, Gabriel Ferro, Mihai Lazar, Christophe Raynaud, Bérangère
	Toury, Marie-France Beaufort, Jean-Francois Barbot, Olivier Dezellus, and Dominique Planson
	D

TABLE II SPECIFIC

 II CONTACT RESISTANCE OF ALL SAMPLES AFTER ANNEALING AT

		VARIOUS TEMPERATURES.
		900°C	1000°C	1100°C	1200°C
	Ti 20 Al 80	1.1 × 10 -3 Ω.cm 2	4.1 × 10 -4 Ω.cm 2	8 × 10 -4 Ω.cm 2	2 × 10 -3 Ω.cm 2
	Ti 30 Al 70	Non-ohmic	2.7 × 10 -4 Ω.cm 2	3 × 10 -4 Ω.cm 2	3.4 × 10 -4 Ω.cm 2
	Ti 50 Al 50	Non-ohmic	1.1 × 10 -4 Ω.cm 2	1.8 × 10 -4 Ω.cm 2	4.8 × 10 -4 Ω.cm 2
	Ti	Non-ohmic	Non-ohmic	Non-ohmic	Non-ohmic

TABLE III SCR

 III

measured at