M Foerster
email: mfoerster@icmab.es

F Rigato

K Bouzehouane

J Fontcuberta

Tunnel transport through CoFe 2 O 4 barriers investigated by conducting atomic force microscopy

Keywords: 75.47Lx, 73.40.Gk, 81.40.Rs, 68.37Ps

Conducting atomic force microscopy has been used to monitor the quality of spin-filtering CoFe 2 O 4 tunnel barriers by mapping current as a function of their thickness. We show that appropriate film annealing leads to a substantial improvement of their tunnelling properties. The contact force between tip and sample was identified to have a determining influence on the width of the distribution P(I) in current maps, thus precluding its reliable use to infer barrier characteristics. Therefore, assessment of tunnel transport should be done by means of the typical current which is a well defined parameter at a given contact force, rather than by the current distribution width.

1.Introduction

Spin filters, formed by a ferromagnetic insulating layer sandwiched between two normal metal electrodes, are expected to constitute spin polarized sources of relevance in future spintronics devices [1]. Spinel ferrites such as NiFe 2 O 4 and CoFe 2 O 4 are being considered for this purpose and recently, spin filtering effects have been demonstrated using these materials [2][3][4][5]. However, as the reported spin filtering efficiency is much below expectations, questions concerning the quality/homogeneity of the spinel barriers and on their effective height arise.

Characterization of tunnel devices is far from simple, as local variations of the tunnel barrier properties may produce large variations in conductance due to its extremely non-linear dependence on the barrier characteristics. Therefore the knowledge of barrier properties at submicronscale is of the highest relevance. Atomic force microscopy with a conducting tip (C-AFM) is a suitable tool to analyze electric transport across nanometric barriers. Current mapping at a given bias voltage (V) has been used to determine the dependence of the conductance of different barriers as a function of their thickness (t), eventually confirming tunnel transport across them [2,3]. Also attempts have been made to infer barrier characteristics from current maps recorded at different voltages, e.g. the altitude of the effective tunnel barrier [6] or the nanoscopic roughness of the barriers [7]. Typically, current maps contain a distribution of current values P(I) whose width reflects the homogeneity of the barrier properties [7][8][9][10]. For instance, it has been predicted that a Gaussian distribution of thicknesses of width σ should produce a log-normal P(I) distribution whose width is directly related to the ratio σ/λ, where λ is the electronic attenuation length in the barrier [8]. This relation is remarkable as it reveals that a tunnel barrier extremely homogeneous in thickness (σ/t <<1) may still be rough in terms of electron tunnelling if σ/λ >>1. Therefore, the analysis of the P(I) distribution may provide a deeper insight into the properties of tunnel barriers. We strengthen that reliable analysis of I(V) characteristics, as measured by C-AFM, requires the use of a good metallic contact between the tip and the probed material, which cannot be commonly achieved unless the film surface is metal-capped [11].

In this work we present a detailed analysis of current maps of CoFe 2 O 4 (CFO) barriers as a function of thickness, growth and measuring conditions, namely the applied voltage and the contact force F m between the tip and the sample. We show that a suitable annealing process allows improvement of their tunnel properties. Using such improved CFO barriers, we critically revise the analysis of the P(I) distribution as a tool to determine barrier properties and demonstrate the crucial influence of the tip-sample contact force F m not only on the absolute current scale of P(I) but also on its width. Therefore, when comparing films of different thicknesses, assessment of tunnel transport should be done by means of the typical current which is a well defined parameter at a given contact force, rather than by the current distribution width.

Experimental details

CoFe 2 O 4 thin films of thicknesses ranging from t = 2 nm to 8 nm were grown on a SrRuO 3 -bottom electrode (25 nm) on SrTiO 3 (111) single crystalline substrates by rf-sputtering. Growth conditions and structural and magnetic characterization of the films will be reported elsewhere [12]. Two series of films, denoted "as-prepared" and "annealed" where grown under the same nominal conditions. While as-prepared films were cooled down to room-temperature after growth, annealed films were kept at growth temperature (450 ºC) in 350 mTorr oxygen for 2 h before cooling down. The C-AFM measurements have been performed using a Nanotec Cervantes microscope and Nanosensors CDT-NCHR conducting tips. All scans were done with the same lateral tip speed of about 3 µm/s under N 2 atmosphere. The SrRuO 3 electrode was positively biased (V) and the tip was grounded. The cantilever force constant was determined from the frequency response of the free oscillation [13]. For each scan we calibrated the cantilever deflection by a deflection-displacement F(z) curve and adjusted the contact force F m . The simultaneously recorded topographic images (see inset of figure 1d for a typical image) indicate a rms roughness below 0.2 nm for all films, slightly smaller than that measured in dynamic mode (σ ~0.3 nm ), reflecting their extreme flatness (the unit cell of CoFe 2 O 4 is ~0.8 nm).

Results and discussion

In figure 1 we show typical current maps obtained for annealed samples (right panels, measured at 1 V and 300 nN) and for the as-prepared samples (left panels, 0.6 V and 600 nN respectively). The colour contrast denotes the changes of conductance. Using a new batch of tips for the characterization of the annealed series, the contact force F m had to be reduced to 300 nN since samples showed signs of indentation after applying 600 nN. In all images two well defined regions can be distinguished: The central square corresponds to the area that had been scanned by the tip in a preliminary scan, while the complete image is the current map measured during a subsequent larger-area scan. We ascribe the difference in conduction between the two areas to a cleaning of the sample surface by the tip during the first scan. Note that in the measurements used for current maps, there are no marks in the corresponding topography image. Further, this difference is independent of the voltage applied during the first scan (also for 0 V) and disappears after 2-3 subsequent scans (see inset in figure 1b), ruling out indentation or electronic modification of the surface as possible origin for such difference. All data Figure 2a shows the probability distribution P(I) of local current values for films of the annealed series (V =1 V). The shift of I typ (I typ is the current at which P(I) is maximum) towards higher values when reducing t corresponds to the exponential dependence of I on t which is a signature of tunnel transport and can be more clearly seen in figure 5 below. In figure 2b we show data corresponding to as-prepared samples measured at various bias voltages, using F m = 600 nN. In addition to the thickness dependence, when increasing the measuring voltage there is the expected shift to higher currents of P(I), but without influencing its shape. As can be better appreciated in the scaling of figure 3, for all samples P(I) displays a tail towards high current values. Turning to a more quantitative description, we consider a Gaussian distribution (σ) of local barrier thicknesses. Then P(I) should be a log-normal distribution [8]:

- - = 2 2 2 ) (ln exp 2 1 ) ( β µ π β I I I P ( 1
)
where µ is a scale parameter; the shape parameter (width) β is given by β = σ/λ where λ is the barrier electronic attenuation length and σ is taken as the roughness of the film. Notice that a narrow P(I) distribution implies a smaller β and thus, for a given roughness, a larger λ. Based on the relation β = σ/λ, and using values of λ estimated from the expected potential barrier height (either from literature or IV curve fitting), remarkable low barrier thickness variations, typically below 0.1 nm and much smaller than the respective surface roughness have been reported for different materials [7].

To extract β and thus λ from our data presented in figure 2, the P(I) distribution was fitted using equation (1) (inset figure 3a). According to Refs. [8], the P(I) distribution can be better fitted using the log-log data representation (inset figure 3b). Both methods lead to excellent fits and similar λ values.

All λ values extracted from the log P(I)-log (I) data of annealed and as-prepared films at different voltages are collected in figure 4a. Inspection of data in figure 4a reveals some important insights. Firstly, for both series of samples the λ values vary only slightly with V. As in the most simple description [8], λ ~(V 0 -V) -(1/2) , the observed weak dependence of λ on V could be an indication that the used bias voltage V (~1V) is much smaller than the barrier height V 0 , which however is in disagreement with the predicted band gap of CFO ( 0.8V) [14]. Secondly, for annealed films λ is roughly constant (~0.6 nm) which indicates that all films have very similar barrier height and roughness. One notices thatλ ~σ, implying that although the films are extremely smooth (t/σ > 6), from the point of view of their electronic transparency they are not so and thus roughness still may play an important role on the tunnel transport. Finally, we notice that for the as-prepared films, λ increases gradually as a function of thickness thus suggesting a change in the electronic properties of these barriers with thickness. CFO thickness (nm)

Figure 5: Thickness dependence of I typ of films of the annealed and as-prepared series, as determined from P (I) at 1 V and 0.6 V respectively. IThe expected dependence for annealed samples with = 0.6 nm is also shown (continuous line). Dashed and dotted lines are guides to the eye.

It is a common practice to use the thickness dependence of I typ , which is expected to be exponential (I typ ~exp[-t/λ] ), to assert the tunnel character of transport across thin barriers. In figure 5 we show the data corresponding to our films, determined from distributions such as those of figure 2. It is clear that annealed films display an exponential decrease of conductance with thickness in agreement with the constancy of the attenuation length shown in figure 4a. The decrease of conductance of asprepared films with increasing t is sub-exponential, which is in agreement with the increase of λ with thickness. We conclude that the process used to obtain annealed samples has improved the electronic properties by homogenizing the tunnel barrier.

Thus we may use the data obtained on the annealed samples (constant λ(t), exponential I typ (t) ) to check the validity of the relation β = σ/λ, which reveals important inconsistencies. First, as mentioned before, a stronger systematic dependence of λ on the applied voltage is expected in terms of electron tunnelling, since the applied voltage will lower the average barrier height. Second, the attenuation length λ extracted from the current map distributions is inconsistent with the experimental thickness dependence I typ ~exp(-t/ ) shown in figure 5 (dashed line), which is characterized by Λ. The solid line in figure 5 indicates the log(V/I typ ) vs. t slope expected from the measured attenuation length values ( λ ~0.6 nm). From the comparison of the experimental and the predicted slope it is clear that the λ value extracted from the current maps does not reproduce the observed thickness dependence of the tunnel barrier conductance. In fact, a much larger value of Λ ~1.8 nm is needed to describe the experimental thickness dependence of I typ . This observation implies that the experimentally measured width of P(I) is larger than expected from electronic roughening (~σ/λ ) and rather determined by other, probably experimental, conditions. We will identify in the following, the contact force F m as a main influence on the distribution width.

The applied contact force is a most critical parameter in C-AFM, since it ultimately determines the contact area and stability and therefore strongly influences experimental noise. We measured the current distribution P(I) of annealed samples at various F m , and indeed observed a shrinking of the width of P(I) when increasing F m . Consequently, the extracted attenuation length λ increases as shown in figure 4b. It has to be concluded that despite of excellent fits of all measured P(I) to log-normal distributions the current distribution width cannot be used to extract the electronic attenuation length.

Summary

In conclusion, we have used C-AFM to monitor the quality of the CoFe 2 O 4 tunnel barriers, demonstrating a substantial improvement of their tunnelling characteristics through an appropriate annealing process. This is of highest significance for further spin-filter development. For improved CoFe 2 O 4 barriers, the analysis of distributions P(I) measured in current maps for different voltages and barrier thicknesses, by applying the commonly used model of statistical thickness variations, revealed inconsistencies. The contact force was identified as having a crucial influence on the current distribution width; it thus follows that the attenuation length λ extracted by using such model is not a characteristic parameter of the barrier but is strongly dependent on the measuring conditions.

Therefore, assessment of tunnel transport should be done by means of the typical current which is a well defined parameter at a given contact force, rather than by the current distribution width.

Financial support by the Spanish Government (Projects MAT2008-06761-C03 and Nanoselect CSD2007-00041) and the European Union (Projects MaCoMuFi (FP6-03321) and FEDER) is acknowledged.

Figure 1 :

 1 Figure 1: Current maps of films of as-prepared (left) and annealed (right) samples of various thicknesses measured at V = 0.6 V and 1 V respectively. The different contrast observed in each figure corresponds to the changes in current flow after consecutive current mapping scans as explained in the text. Insets: (b) difference of the current distribution maximum between central square and outer area vs. scan sequence; (c) illustrative I-V characteristic indicating tunnel transport; (d) typical topographic image displaying substrate induced steps.

  presented in this work were obtained from only the cleaned, central area. I-V curves recorded in a fixed point (see inset in figure1c) show the characteristic shape of tunnel transport.

Figure 2 :

 2 Figure 2: Normalized probability distribution P(I) vs log (I) of local currents in: (a) annealed films of different thicknesses (F m = 300 nN, V = 1 V) and (b) as-prepared films (F m = 600 nN) of various thicknesses and different measuring bias voltage.

Figure 3 :

 3 Figure 3: Logarithmic probability distributions P(I) of films (annealed, upper panel and as-prepared, lower panel) of Figure 2 emphasizing high current tails. Insets in (a) and (b) are examples of different fits to the same illustrative data (annealed, 4 nm) as described in the text.

Figure 4 :

 4 Figure 4: (a) Attenuation length λ values extracted from the log P(I)-log (I) data of all films of the annealed series (measured at F m = 300nN) and as-prepared series (measured at F m = 600nN). Data extracted from the P(I) distributions measured at different bias voltages are included. (b) Attenuation length (nm) of the t = 4 nm film (annealed series) extracted from current distributions P(I) measured at various forces F m (triangles). The force dependence of I typ for the same film is also shown (squares). In (a) and (b), dashed lines are guides to the eye.


R Gueroult
email: renaud.gueroult@polytechnique.edu

P Q Elias

D Packan

J Bonnet

J M Rax

Particle in cell modelling of the observed modes of a DC wire discharge

Low pressure DC Wire Induced Plasma Sources (WIPS) exhibit two stable modes of discharge -constricted below a threshold pressure and diffuse above. Starting from experimental measurements, we conduct two dimensional particle in cell (PIC) modelling of a DC low pressure (10 -4 -10 -2 mbar), low current (∼ 1 mA) wire discharge in Helium. 2D PIC modelling is required to capture longitudinal non uniformity of the diffuse mode. PIC simulations reproduce the two discharge modes. Voltage versus pressure curve obtained from simulations matches fairly well experimental data, including the transition region. Discharge voltage dependence on pressure is analysed in consideration of electron impact ionization rates evolution with energy. In light of the PIC findings, a model of the discharge mode transition based on a Child Langmuir theory for ions is proposed. Confrontation with simulated data shows good agreement and validate the model for mode transition prediction. Simulations show that the diffuse mode is a space charge dominated regime.

Introduction

A Wire Induced Plasma Source (WIPS) is a type of low pressure gas discharge. Low pressure discharge specificity relies on the fact that the electron mean free path often exceeds the discharge chamber typical length scale, thus requiring a way of confinement. In the case of WIPSs, efficient confinement is achieved electrostatically [1], electrons being captured in the logarithmic potential generated between a cylindric cathode and a positively biased wire anode centered inside the cathode. This technique is the direct transposition -reversed polarity -of the one used in the Kingdon [2] ion trap, and permits low pressure (∼ 10 -5 mbar) gaseous discharge at very low applied voltage (∼ 1 kV) within a chamber a few centimeters in size.

Consequently, WIPSs have been used as an ion sources for secondary emission electron guns [3], and their development have been pushed forward within this framework: versions of WIPS with multiple wires [4] and axial magnetic field [5] to increase their efficiency have been proposed. Nevertheless, those improvements are balanced by the increase weight and complexity of the system induced by the addition of magnets and/or wire tightening systems. Since then, whilst wire sources are widely used in electron guns, efforts to develop an understanding of the WIPS itself had been scarce. Most of the studies have been devoted to pulsed WIPS which can develop current density of about one Ampere per centimeter length, being therefore a good candidate as ion source for high current electron gun [3,6,7,8]. Recently, Makarov undertook both an experimental [9] and a 1D numerical study [10,11] of a high current pulsed WIPS and the temporal evolution of the discharge.

Experiments show that, depending on operating conditions, various modes of discharge are observed. This has been reported in various configurations: single [START_REF] Pigache | Proc. of the XVIII Int. Conf. on Phenomena in Ionized Gases (Swansea)[END_REF] or multiple wires [START_REF] Rushton | [END_REF] at low pressure, single wire at higher pressure [START_REF] Hotta | Proc. of the IEEE Int. Conf. on Plasma Science[END_REF]. Low pressure DC WIPS in the milliAmpere range, used for example in electron beam fluorescence (EBF) optical diagnosis [START_REF] Mohamed | Int. Congress on Instrumentation in Aerospace Simulation Facilities[END_REF] electron guns, exhibits two distinct modes of discharge. Apart Pigache's [START_REF] Pigache | Proc. of the XVIII Int. Conf. on Phenomena in Ionized Gases (Swansea)[END_REF] remark that, in the high pressure mode, the cathode sheath length was of the same order of the planar Child Langmuir length, to our knowledge, no thorough model of the DC low pressure, low current wire source was given. A better understanding of the main physical features of the wire discharge would benefit to EBF electron guns optimization and development of DC WIPS based technologies.

In this paper, we seek to examine the WIPS physics, and in particular its mode transition, in the light of an experimental characterization. We numerically investigate the operation of a DC low pressure (10 -4 -10 -2 mbar), low current (∼ 1 mA) wire discharge which could typically be used in EBF electron guns. We limit our study to the single centered wire configuration without applied magnetic field. To begin with, an experimental characterization of the wire discharge is conducted in Section 2. Description of the specific features of our 2D particle in cell code follows in Section 3. PIC modelling validation is then achieved by confronting numerical results with experimental data in Section 4. Finally, based on simulations microscopic outputs we propose macroscopic models of observed phenomenon in Section 5.

Experimental characterization of the low pressure Wire Induced Plasma Source

Experimental Setup

Experimental measurements have been made on the setup sketched in figure 1. The cathode is made of a R c = 2.5 cm inner radius, L = 20 cm length stainless steel cylinder. At one end of the cylinder, a silica window allows observation of the radial expansion of the discharge. Another window is installed laterally to monitor the longitudinal plasma development. The anode is a R a = 0.1 mm radius Tungsten wire placed on the axis of the cylinder. Grounded end plates at each end provide longitudinal confinement. These disks have an inner hole diameter of 6 mm. The plasma source was directly mounted on a Pfeiffer HiCube turbo pump reaching an ultimate pressure of 6.10 -7 mbar in measurement configuration. Measurements presented in this paper were done in Helium, but Ne, Ar, Xe, NO, CO, CO 2 and air have been successfully tested. Incoming gas flow was controlled by a an input valve. In order to have reliable reading of the pressure, several pressure gauges were compared and cross calibrated: two Baratron capacitive gauges, one hot cathode Bayard-Alpert gauge and one Pfeiffer Compact Full Range gauge. Pressure resolution in that configuration was about 10 -5 mbar.

Typical anode voltage (V a ) varies from 500 V to 10 kV depending on the gas type and pressure. Heinzinger LNC stabilized DC power generators were used. A +10 kV, 2 mA generator was used for low pressure measurements, whereas a +1200 V, 10 mA generator was used for higher pressure.

Voltage and current were recorded using a P6015A Tektronix high voltage probe and a Hall current probe respectively, and displayed on a LeCroy 100 MHz oscilloscope. Current was measured at the power generator output.

Discharge Modes

Two discharge modes are observed experimentally. These modes are similar to those described by Pigache [START_REF] Pigache | Proc. of the XVIII Int. Conf. on Phenomena in Ionized Gases (Swansea)[END_REF] at higher currents. Figure 2 presents the discharge evolution with pressure.

At low values of pressure (p ≤ 10 -2 mbar), the plasma glow is limited to a region close to the anode, but fully expanded longitudinally. Therefore we refer to this mode as the constricted mode. When the pressure is decreased (p < 2.10 -3 mbar), the wire becomes red hot, and discharge voltage curve diverges. According to Francis [START_REF] Francis | The glow discharge at low pressure Encyclopedia of Physics[END_REF], the divergence of the discharge voltage with pressure decrease is symptomatic of an obstructed discharge. This behaviour combined with, as it will be shown later, the potential profile near the anode, indicate that the constricted mode might be an obstructed discharge in a wire cylinder configuration.

On the other hand, when pressure is increased above the threshold value, the discharge switches to a diffuse mode. The transition is sudden, and a voltage drop of slightly less than a hundred Volts is observed. The system exhibits hysteresis on mode transition, the transition occurring at slightly lower (resp. higher) pressure when decreasing (resp. increasing) the pressure. In the diffuse mode, the plasma glow fills the whole radial inter electrodes volume but is limited longitudinally to a slice of a few centimeters . For a given pressure, the slice width increase with current, as shown in figure 3. In the diffuse mode, current increase is obtained at almost constant voltage, which is a characteristic feature of a normal glow discharge.

Particle in cell modelling of low pressure discharge

Longitudinal constriction of the discharge at the mode transition underlines the importance of 2D effects. It is therefore necessary to use a bi dimensional model to simulate the wire discharge behaviour. Moreover, Knudsen number well above one due to the low working gas pressure implies a particle in cell (PIC) numerical modelling.

In a classical manner [START_REF] Dawson | [END_REF][START_REF] Birdsall | Plasma Physics via Computer Simulation[END_REF][START_REF] Birdsall | [END_REF], a two dimensional 2d 3v (r, z) particle in cell code has been developed. Particles modelled are: electrons, ions and fast neutrals. Fast neutrals population is included mainly for secondary emission purposes. Indeed due to the high value of the charge exchange cross section in Helium, depending on pressure conditions, fast neutrals number density could be rather important and thus neutrals could play a role on secondary emission. Particles trajectories are computed using a Runge-Kutta 4th-order algorithm. The anode radius R a being order of magnitude smaller than the chamber radius R c , a non-uniform cylindrical mesh has been used to minimize the number of cells. Nevertheless, it is strictly necessary to resolve properly the wire to compute high field gradients in this area and hence to integrate properly the particles trajectories. Despite the low working pressure, collisions are fundamental to model the discharge. Consequently, we implemented a complete Monte Carlo Collision (MCC) module [START_REF] Birdsall | [END_REF]. Collisions taken into account in the simulation are synthesized in table 1. The corresponding cross sections are those gathered by Hartmann [20], exception made of electron impact excitation for which we used Stone's [21] data. Indeed, Hartman [20] cross sections for this process differ significantly from standards [21]. Regarding excitation process, only one excited level is taken into account, but the energy loss during the excitation process is computed randomly between the energy of the first excited level and the ionization energy. For a given collision process, energy distribution between all particles involved is computed by inverting the differential cross section σ(E, χ) for that collision [START_REF] Birdsall | [END_REF]22]. Gas dependent fitting parameters for ionization processes are taken from Opal [23]. Null collision technique [24] has been implemented to speed up calculations.

Due to the low working pressure, collisions between active populations are neglected. Similarly, the low ionization fraction justifies the implementation of a fixed density background gas.

Secondary emission models are used for all three species impacts on the electrodes. The processes involved for heavy particles are kinetic emission (ions, fast neutrals) and potential emission (ions) [START_REF] Kaminsky | Atomic and Ionic Impact Phenomena on Metal Surfaces[END_REF][START_REF] Katz | A three dimensional dynamic study of electrostatic charging in materials[END_REF]. For ion impact secondary electron emission at a few hundreds electron volts, fitting parameters references in the literature are scarce. For example, Bohm [START_REF] Bohm | [END_REF] gives a value γ He + ∼ 0.25 emitted electrons per incident ion at Table 1. Active Species -Background gas collision processes modeled in the simulations. Corresponding cross section are from Hartmann [20], except for electron impact excitation taken from Stone [21].

350 eV whereas Szapiro [28] has γ He + ∼ 0.6 at 500 eV, both for clean stainless steel.

In addition, the secondary emission coefficient depends strongly on surface condition, this parameter being hardly known. Following Bohm [START_REF] Bohm | [END_REF], simulations are made using

γ He + = 0.3, γ He 0 = 0.1, (1)
as default values. Influence of those parameters will be analysed parametrically. Impacting electrons can be elastically or inelastically reflected, or create true secondaries [29,30,31]. Depending on the plasma impedance, ideal voltage sources can induce instabilities. It is thus desirable to drive the plasma with a ideal current source. The anode potential evolves due to the cumulated effect of impacting particles and electrons drawn by the current source. Simulation runs until electrodes current have converged to the specified value.

Code validation: comparison with experiments

In order to validate the particle in cell code described in the previous section, test cases are run and results are confronted to experimental data. Simulations results are compared with experiments on two different levels. First visual aspect and spatial properties of the simulated discharge are compared qualitatively to experimental pictures. Then computed voltage versus pressure curve is examined in light of the experimental one.

Discharge Modes: spatial properties

Three cases matching the experimental modes of discharge are numerically modelled. Working conditions of these cases are summarized in table 2. The first one corresponds to a constricted mode at a 1 mA discharge current (Case A). The others are diffuse modes for 1 and 2 mA (Cases B and C respectively). Figure 4 presents the Abel transform of the volumic excitation rate for the three cases. The Abel transform of the excitation rate is directly related to the emitted light and its comparison with observed glow is a good indicator. Simulations confirm the existence of a constricted mode of discharge where excitation is localized near the anode and uniform longitudinally. At higher pressure, for the same 1 mA discharge current, the excitation area increases radially and collapses longitudinally. The longitudinal expansion with current is well modelled by the simulations.

Case

Longitudinal profiles of the Abel transform of excitation are plotted with profiles obtained from experimental pictures in figure 5. No correction has been made to the raw data, except a normalization of all simulated curves with a single coefficient equal to the ratio of the observed maxima over the simulated one for the 2 mA diffuse mode case. Relative levels of intensity for simulated and experimental data are almost equal, and shapes, although not exactly superimposed, are qualitatively identical. Indeed, simulated emission profiles maxima are more peaked than experimental ones. Reflected light -not modelled in our PIC code -is thought to contribute to the peak broadening. Indeed excitation produces photons with directions distributed equally on a solid sphere angle, and only a small fraction of it hit directly the camera detector. Another fraction of photons is diffused from the cathode surface and then possibly enter the camera detector. Diffusion can take place in areas far from the median plane, increasing in turn the signal on both sides of the excitation maximum.

Discharge voltage variation with pressure.

The simulated discharge voltage versus pressure is plotted along with experimental data in figure 6. The sharp transition observed experimentally between the two modes is found as well on the simulated curve. The voltage jump is well defined, and curves evolutions around the mode transition are very similar: computed curves slopes above and below the mode transition match closely the experimental ones. Discrepancy between simulated and experimental data is mainly present on the high pressure region, where, as it will be shown in the next paragraph, secondary emission coefficient value play a role. For lower pressures, particle in cell results stick closely to the experimental ones, modelling properly the strong rise of discharge voltage with pressure decrease.

Influence of secondary emission coefficients

As mentioned in Section 3, constant secondary electron emission value for ion and fast neutral impact were used due to the lack of data in the literature. A parametric study was made to discriminate the influence of these coefficients on the simulations. Before any computational study, an analysis could be formulated as follows. Increasing secondary emission coefficients would lead to higher electron population number N e -, and in turn to higher ionization rate, yielding eventually higher discharge current (I > I 0 ). On the pressure range considered, the plasma is resistive, meaning that the original I 0 discharge current can be obtained at lower voltage using higher value of γ. Moreover, the fraction of secondary electrons returning to the cathode increases with mean free path, thus dependence on γ should diminish with pressure decrease. Simulations made with γ He + = 0.6 validates this analysis: the transition occurs numerically at the same voltage V ∼ 330 V despite of a drop of the mode transition pressure from p t = 6.10 -2 mbar for default γ He + value to p t = 4.4.10 -2 mbar.

Influence of secondary emission on the voltage versus pressure curve given in figure 6 is hence mainly a shift to the left, the mode transition occurring then at lower pressure. This shift will be explained by the mode transition model proposed in Section 5.

To summarize, secondary emission values tuning could diminish, if not remove, discrepancies between simulations and experimental data in terms of pressure of transition. Considering that secondary emission values do not change the fundamental behaviour -mode transition, spatial properties of the modes -of the discharge on the pressure range considered and that these coefficients are hardly predictable, no further attention has been paid to this phenomena.

Discussion

Numerical results on the previous test cases justify the use of the code as a mean to dig out microscopic data, e.g. ionization volumic rates or densities, to propose macroscopic model of the discharge.

Model of the mode transition

Based on experimental measurements, Pigache [START_REF] Pigache | Proc. of the XVIII Int. Conf. on Phenomena in Ionized Gases (Swansea)[END_REF] noted that the high pressure cathode sheath width was of the same order as the Child Langmuir length computed for an Helium ion current between planar electrodes. Makarov [9] used the same approach to describe one state of its pulsed wire discharge. Those analyses rely on the hypothesis that the discharge can be on first approximation considered as an unipolar current of Helium ions. In order to extend Pigache's argument to the mode transition analysis, this hypothesis has to be justified at first.

When pressure is increased, the ion mean velocity decreases because of the lower discharge voltage and higher collision rate with the neutral background gas. Considering that the cathode current is mainly an ion current, in order to sustain a given current, the ion population number should increase to balance the velocity decrease. This way, the ion flux to the cathode remains constant. Due to charge exchange collisions, fast neutrals population follows the ion population increase, until pressure reaches the threshold value. Beyond this limit, the fast neutrals energy is limited as well by a lower discharge voltage and elastic collisions, and hence more and more fast neutrals are transferred back to the background gas population as pressure increases. Indeed neutrals are monitored as fast neutrals as long as their energy exceeds a given value, taken as a threshold for kinetic secondary emission on electrodes.

Simulations results presented in figure 7 confirms these variations. The ion population number is about ten times more important than electron one's in the vicinity of the mode transition. Charged particles are thus mainly ions, justifying the study of the discharge as a space charge limited ion current between electrodes. Nevertheless, ion neutral cross sections for elastic and charge exchange processes in Helium (σ ∼ 2.10 -19 m 2 ) indicate that, at the pressure of transition, a typical ion has approximately five collisions on its way to the cathode. Collisions limit ion speed, increasing space charge and in turn limiting the sustainable current. Consequently, accounting for collisions is important to model the transition. Thus, assuming an ion current between cylindrical electrodes, and using the collision frequency to model ion impacts, the Child Langmuir problem can be stated as

d dr r dV dr = - I 2π 0 Lv , (2)
mv dv dr = -e dV dr -νmv, (3)
(2) and (3) being the Poisson's equation expressed in cylindrical coordinates and the momentum equation for helium ions, respectively. Here V is the electric potential, v is the ion speed, I is the discharge current, L is the discharge length, m is the Helium ion mass, e is the elementary charge and ν = σn g v is the ion collision frequency. Resolving numerically the system formed by ( 2) and (3) for ions a having thermal velocity at the anode with

V (R a ) = V a V (R c ) = 0 E(R a ) = - dV dr (R a ) = 0 (4)
yields the Child Langmuir limiting current I CL for the studied configuration -anode potential and electrodes geometry. I CL is obtained when the space charge cancels the electric field at the anode, meaning that no more ions could be drawn.

For comparison, in the analytic situation of a classical Child Langmuir current between planar electrodes [32], I CL is expressed as

I CL || ∝ V 3/2 d 2 , ( 5
)
where d is the inter electrode gap and V the applied voltage. In our situation, the Child Langmuir limiting current is a function of V , the inter electrode gap δR, as well as the pressure p due to the collisional term in (3):

I CL = I CL (V, δR, p). ( 6
) [.u L6 0PEDU & K L O G 0 / D Q J P X L U 6nRR 4n5e 67nu 9un9 94n4 4un5 47n5 45ne 5HGXFHG03RWHQWLDO09g9 D u un9 . 5HGXFHG05DGLXV u un89 un9 un59 .
Figure 8. Child Langmuir limit for the pressure of mode transition and PIC simulations reduced potential V /V A at z = zm against reduced radius r/Rc for various gas pressures. When increasing pressure, the positive space charge builds up near the anode. Above a given pressure, the voltage profile crosses the one obtain from Child Langmuir model and transition occurs. Potential is then flattened near the anode to sustain higher currents.

Remembering that the discharge voltage decreases with pressure, the discharge mode transition for a given current I 0 can be explained as follows. Figure 8 shows the reduced potential V /V a versus reduced radius (r/R c ) at z m = L/2 on a wide range of pressure values obtained from PIC simulations, as well as the Child Langmuir limit obtained using the pressure p = p t of mode transition in the collisional term. At low pressure, the discharge voltage is high enough compared to the inter electrode gap δR to collect I 0 < I CL (V, δR, p t ). When increasing pressure, thus decreasing voltage, the discharge voltage reaches V t , the computed Child Langmuir potential for I 0 and δR, so that I 0 = I CL (V t , δR, p t ). Above this pressure, it is no longer possible to sustain the discharge current I 0 without reducing the sheath width. Transition occurs and a space charge builds up near the anode. Grounded end plates push electrons away, favouring space charge build up close to the longitudinal mid plane. This space charge flatten the radial potential profile near the anode, in turn decreasing the sheath width and hence permitting stronger current. Flat behaviour of the potential at z ∼ z m near the anode observed in figure 9 creates a longitudinal confinement of electrons near the mid plane. Electrons therefore oscillates longitudinally close to z m , increasing in turn the number density and confinement in that area. The plasma is no longer uniform longitudinally, but centered near the mid plane defined by z = z m . This longitudinal constriction justifies the fact that diffuse mode sheath width has a minimum. Indeed, working at constant current I = 2πrLj(r), a longitudinal collapse induces an increase of j(r). To sustain this current density rise without increasing the potential drop across the sheath, a step decrease of the sheath width is necessary. Using this modified Child Langmuir model, it is possible to predict the voltage of the mode transition for a given electrodes geometry and discharge current. Moreover, knowing the voltage drop across the transition, we can calculate the sheath width of the diffuse mode and reciprocally. Results obtained from experimental measurements, simulations and Child Langmuir model are compared in table 3. Data show fair agreements and confirm the efficiency of the modified Child Langmuir model for mode transition prediction.

To summarize, the mode transition is the result of a positive space charge growth beyond the threshold value, inducing a space charge limited regime. Transition is observed when current exceeds the Child Langmuir current computed for the given configuration. The decrease of the mode transition pressure p t noted in Section 4 is the direct consequence of the accelerated build up of positive space charge, higher value of secondary emission coefficient inducing higher electron number density n e .

Experimental Child Langmuir model Simulation Transition voltage (V)

415 320 330 Sheath width (mm) -12 14 Indeed, n e being many times lower than the ion number density n i near the mode transition, a small increase of n e will induce a larger increase of n i , and consequently the positive space charge limit for mode transition will be obtained at lower pressure. Phenomenologically speaking, potential profiles given in Figure 8 provide a new feature differencing the constricted and diffuse modes: the existence of an electric field at the anode. Following Donko's [33] proposition of definition of an obstructed discharge -the electric field does not vanish at the anode, the constricted mode could be an obstructed discharge mode between coaxial cylinders.

Discharge voltage dependence on pressure

For WIPS based technologies, working conditions should exclude regions of V -p curves divergence. Indeed divergence of the V -p curve at a given current requires higher power input, which is to be avoided as much as possible. It is therefore important to get insights into the discharge voltage behaviour.

The wire cylinder configuration permits low pressure gaseous discharge at low discharge voltage. Indeed, although V -p curves look like typical Paschen's curves, discharge can be operated at pressure orders of magnitude lower that what would predict Paschen's law for a parallel plates configuration and a given potential. Even though the generalisation of Paschen's law to non uniform electric field between coaxial cylinders [34,[START_REF] Engel | Ionized Gases[END_REF] induces slight modifications, it can not explain the low voltage ignition. Actually, due to the small anode radius used in our configuration, a large fraction of electrons misses the anode and orbits around it. The effective electron path for ionization is therefore much longer than the distance an electron would travel if it falls directly on the anode (hypothesis implicitly used for the derivation of Paschen's law between coaxial cylinders). The direct consequence of this longer electron path is the capability to satisfy the breakdown criterion at much lower pressure, making it possible to operate discharges at 1 kV for pressure orders of magnitude lower that what would predict Paschen's law [34]. To summarize, the observed deviation from Paschen's law results from the use of a very small anode.

The study of the V -p curve divergence relies on the electron population number N e balance in the discharge. Considering that the anode current I is an electron current, the electron population number conservation can be expressed as:

dN e dt = n g Ne j=1 ζ j e --I/e (7)
where ζ j e -= σ i e -(v j e -)v e -j is the reduced ionization frequency -ionization collision rate over number density -for the j th electron, N e is the total number of electrons in the discharge, v j e -is the speed of the j th electron, σ i e -is the ionization cross section for electron impact and n g is the neutral gas density. Secondary emission of electrons from ions and fast neutrals impact on cathode should have been included in (7). Nevertheless, our interest here is in the low pressure range where secondary electrons are mainly impacting back on the cathode after crossing the discharge chamber without collisions. We thus neglect both secondary emission of electrons and electron current at the cathode, which are supposed to balance each other. Ion and fast neutral impact ionization phenomenon, which are found to be of second order, have been neglected as well.

Introducing the mean reduced electron ionization rate ζ m and the anode electron collection rate ν a as

ζ m = 1 N e Ne j=1 ζ j e - (8)
and

I = eN e ν a , (9)
equation ( 7) becomes =P f rV a7 As shown in figure 10, dζ e -/dE e decreases with electron energy E e , becoming negative above threshold value E e T -energy for which ζ e -is maximum. Therefore, for an individual electron, increasing ζ e -by ∆ζ e -requires a larger ∆E e as E e increases, until it gets impossible (E e ≥ E e T ). Generalizing to the whole electron population, ζ m has a qualitatively similar behaviour, exhibiting a maximum as the electron energy distribution functions (EEDF) peaks around E e T . As featured in figure 11, electron mean energy varies almost linearly with the discharge voltage in the constricted mode. Consequently, behaviour of ζ m as function of electron mean energy or discharge voltage is identical. This behaviour is well illustrated by the EEDF obtained from PIC simulations (see figure 10). At higher pressure, electrons temperature is low and a voltage increase results in the increase of the ionization cross section of all electrons, leading to a large increase of ζ m . When pressure decreases, due to the combined effect of a lower slope of ζ e -(E e ) and an increase of the fraction of electrons having an energy above E e T , ∂ζ m /dV decreases. Decreasing further the pressure will lead ζ m to reach a maximum and then to decrease. On the other hand, ν a0 can be seen as a measure of the electron confinement. The lower ν a0 , the better is the confinement. On the lower voltage range, ν a0 remains almost constant as low primary electron energy leads mainly to isotropic scattering. As voltage increases, higher electron mean temperature leads to a smaller angle of scattering and higher energy for secondary electrons. Smaller scattering angles allow primaries to keep on their path and miss the anode. Higher energy for secondaries increases their chance to miss the anode. Hence ν a0 is a decreasing function of the electron mean energy and, in turn, of voltage. Scattering models [START_REF] Birdsall | [END_REF] shows that the higher the electron energy E e , the higher is the ∆E e required to decrease the angle of scattering or increase the energy of secondaries by a given value. Therefore, after passing through a maximum, |∂ν a0 /∂V | decreases with electron energy or anode potential.

Evolution of ζ m , ν a and N e with a diminution of pressure is obtained from time constants of the various processes involved. As pressure is decreased, the anode current I and thus ν a diminish in a fraction of τ = (n g ζ m ) -1 -the inverse of the ionization rate. Then, as I is lower than set point value I 0 , a positive charge builds up on the anode and, in turn, the anode potential rises as (I 0 -I)t until I return to I 0 . As anode potential increases, ζ m evolves in about the time it takes for an electron to make one orbit around the wire. This period is about 10 -8 s for an electron of a few hundreds of eV, being smaller than τ which varies from 8.10 -8 to 7.10 -7 s depending on pressure. Equation (7) shows that N e evolution is slower, with a time constant of a few τ .

The evolution of ∂V /∂p can thus be summarized as follows. At higher pressure, the voltage increase ∆V needed to get a sufficient growth of ζ m to balance the n g decrease is small, and thus obtained quickly. Therefore (n g ζ m -ν a ) returns to zero before N e had time to evolve. Besides, small ∆V at low discharge voltage leaves ν a0 unchanged, so during the short transient regime ν a decreases with I and then returns to almost the same ν a0 . As pressure diminishes, ∂ζ m /∂V decreases, so a larger ∆V is needed. Charging of the anode to get this larger ∆V takes longer, leaving enough time for a growth of N e . In the meantime, ν a0 starts decreasing as discharge voltage rises. Decreasing further the pressure, ζ m is less and less capable of balancing alone a decrease of n g , which is then compensated by a proportional decrease of ν a0 . |∂ν a0 /∂V | being decreasing, V -p curve divergence initiated by the ζ m saturation is extended at low pressure. Since ν a0 is proportional to the neutral number density at low pressure, N e = I 0 (eν a0 ) -1 has to increase as 1/n g . This is confirmed by simulated data shown in figure 7. As pressure increases, n g decrease is mainly compensated by an increase of ζ m , and N e decreasing rate tends to zero. To prevent divergence, and thus extend the optimal range of working conditions, McClure [1] proved experimentally that a solution was to decrease the anode radius or end plates holes radii. As shown in table 4, simulations confirm efficiency of both methods. We already explained how smaller wire radii yield better ionization efficiency (or similarly lower ν a0 ) and thus lower discharge voltage. On the other hand, smaller end apertures lead to better confinement of electrons. This improved confinement allows higher electron density without voltage increase. Looking at figure 7, this indeed means translating the point of the V -p curve divergence to lower pressures.

Summary

In this paper, a 2D3v particle in cell code was used to study a wire discharge and its mode transition. Simulations reproduce fairly well the two discharge modes and their distinct properties -e.g. glow intensity and discharge voltage. Longitudinal constriction and voltage drop at the mode transition match closely the experimental data.

The simulation results justify to model the discharge near the mode transition as a unipolar ion current between cylindrical electrodes and allow us to conclude that the diffuse mode is a space charge dominated regime. A modified Child Langmuir law accounting for collisional effects is proposed and its ability to predict mode transition is successfully validated against experimental and PIC results. Discharge voltage dependence on pressure is satisfactorily explained in light of electrons ionization frequency variations.

Despite a close match in terms of relative intensity levels, glow longitudinal profiles of the diffuse mode obtained from experiments and simulations for various operating conditions differ slightly, experimental profiles being a little wider. Diffusion of light on the electrodes was mentioned as a plausible contributor to that profile broadening, deexcitation of the helium metastable states being possibly another one. To go further, the addition of a collisional radiative module to the existing code would permit to distinguish between the various excited states and thus allow, among others things, to discriminate the influence of metastable states on emission profiles.

Figure 1 .

 1 Figure 1. Experimental setup. The anode is a Tungsten wire 0.1 mm in radius, the cathode is a stainless steel cylinder 2.5 cm in radius and 20 cm long. The vessel is filled with Helium. Anode potential is Va = 0.5 -10 kV DC. End plates assure longitudinal electrostatic confinement.

Figure 2 .

 2 Figure2. Voltage vs pressure curve for a 1 mA current. The diffuse mode is observed at high pressure, transition (red line) occurs when decreasing pressure and the constricted mode takes place. At lower pressure, the anode becomes red hot.

Figure 3 .

 3 Figure 3. Discharge profiles views trough the lateral window -3(a): Constricted mode p = 1.10 -2 mbar, I = 1 mA -3(b),3(c),3(d) Diffuse mode (p = 2.8.10 -2 mbar) longitudinal expansion with current for I = 1, 1.5 and 2 mA.

Figure 4 .

 4 Figure 4. Simulated discharge modes: 4(a) Constricted mode at 1 mA discharge current, p = 1.36 10 -2 mbar (case A)-4(b) and 4(c) Diffuse mode at p = 6.07 10 -2 mbar for 1 and 2 mA discharge current respectively (cases B and C)

Figure 5 .

 5 Figure 5. Simulated and experimental plasma longitudinal expansion profiles. Simulations conditions are summarized in table 2.

Figure 6 .

 6 Figure 6. Experimental and simulated discharge voltage versus pressure for 1 mA current. Voltage jump of slightly less than a hundred volts is observed on both curves.

Figure 7 .

 7 Figure 7. Evolution of electrons, ions and fast neutrals population numbers with pressure. Plasma is mainly made of ions at high pressure, whereas electrons number increase strongly at lower pressures.

Figure 9 .

 9 Figure 9. Evolution of the normalized electric potential distribution in the (r, z) plane with pressure. The formation of the electron confinement area existing in the diffuse mode is clearly seen.

  dN e dt = N e (n g ζ m -ν a ). (10) At steady state, n g ζ m = ν a . In addition, anode current I and set point current I 0 have to be equal, so n g ζ m = ν a | I=I0 = ν a0 . Hence, a pressure decrease has to be compensated by an increase of ζ m or a decrease of ν a0 .

Figure 10 .

 10 Figure 10. Electron energy distribution function EEDF is plotted for different pressure along with the reduced electron ionization impact frequency ζ e -= σ i e -(v e -)v e -obtained from cross section.

Figure 11 .

 11 Figure 11. Electron mean energy versus discharge voltage obtained from PIC simulations. Data corresponding to the constricted mode are well fitted by a linear curve (red dashed curve).

Table 2 .

 2 Simulations conditions used for comparison with experimental discharge modes

		A	Case B Case C
	Mode	Constricted Diffuse Diffuse
	Pressure (10 -2 mbar)	1.36	6.07	6.07
	Current (mA)	1	1	2

Table 3 .

 3 Comparative of experimental, simulation and Child Langmuir model results for mode transition prediction

Table 4 .

 4 Simulated discharge voltage for various anode radii (Ra) and end plates holes radii (R end ) for p = 3.88.10 -3 mbar.


Adrian Balan

Rakesh Kumar

Mohamed Boukhicha

Olivier Beyssac

Jean-Claude Bouillard

Dario Taverna

William Sacks

Massimiliano Marangolo

Emmanuelle Lacaze

Walter Escoffier

Emanuelle Lacaze

Jean-Marie Poumirol

Abhay Shukla
email: abhay.shukla@upmc.fr

Anodic bonded graphene

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

Introduction

The discovery of graphene [START_REF] Berger | Ultrathin epitaxial graphite: 2d electron gas properties and a route toward graphene-based nanoelectronics[END_REF][START_REF] Novoselov | Two-dimensional atomic crystals[END_REF] has led to a frenetic development of the field [START_REF] Castro Neto | The electronic properties of graphene[END_REF], motivated by the spectacular results obtained by various groups and also by the potential that graphene may have with respect to future applications in microelectronics. Several methods [START_REF] Geim | Graphene: Status and Prospects[END_REF] have now been developed for producing graphene, including the original methods of epitaxial growth on SiC substrates [START_REF] Berger | Ultrathin epitaxial graphite: 2d electron gas properties and a route toward graphene-based nanoelectronics[END_REF] and micro-mechanical cleaving [START_REF] Novoselov | Two-dimensional atomic crystals[END_REF]. More recent methods include exfoliation [START_REF] Hernandez | High-yield production of graphene by liquid-phase exfoliation of graphite[END_REF][START_REF] Dmitriy | Preparation and characterization of graphene oxide paper[END_REF] and CVD growth on metallic substrates [START_REF] Reina | Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition[END_REF][START_REF] Soo | Large-scale pattern growth of graphene films for stretchable transparent electrodes[END_REF].

Each of these methods is relevant for specific applications and here we present a method of fabrication of graphene that we have developed which may be classified as a variant of micro-mechanical cleaving. As we have shown in an earlier publication [START_REF] Shukla | Graphene made easy: High quality, large-area samples[END_REF] we obtain high quality samples of few layer graphene on Pyrex glass substrates. The quality is seen both from the Raman spectra and from transport measurements. The dimensions that we obtain are typically from a few tens to a few hundred microns for monolayer and bilayer samples while we have obtained trilayer samples of mm size [START_REF] Shukla | Graphene made easy: High quality, large-area samples[END_REF]. A further advantage of this method is that it can be used, as in the case of micromechanical cleaving, for fabricating ultra-thin samples of virtually any layered material in a straightforward manner.

Our aim in this paper is to provide a critical analysis of this method including aspects of fabrication such that the reader may have the necessary elements for replicating it, followed by results of characterization of samples thus obtained. In section 2 we give a brief introduction to the anodic bonding method and its application to the fabrication of ultra-thin samples of layered materials. In section 3 we give results from the Raman characterization of these samples including examples of Raman 'mapping'. In section 4 we show results from local probe microscopy on these samples and discuss some Scanning Tunneling Microscopy (STM) results. Finally in section 5 we discuss some transport measurements on simple Field Effect Transistor (FET) devices.

Anodic bonding and sample fabrication

Anodic bonding is a much employed technique in the silicon industry [START_REF] Wallis | Field assisted glass-metal sealing[END_REF] and is still widely used [START_REF] Collart | Spherically bent analyzers for resonant inelastic x-ray scattering with intrinsic resolution below 200 mev[END_REF] for bonding glass to conductive material owing to the good bond quality (silicon-on-insulator bonding or SOI bonding). It allows the joining of two solids without intervening layers like glue and is typically performed between a sodium-bearing glass substrate (for example Pyrex 7740 from Corning) and a silicon wafer, the two materials being fairly well matched in terms of thermal expansion. At high temperatures, (¿ 200 • C) Na 2 O in Pyrex glass decomposes into one O - 2 and two Na + ions. The mobility of the smaller sodium ions is high and the presence of an electric field (the applied potential difference being typically of the order of 1.7 kV) across the glass substrate [START_REF] Albaugh | Electrode phenomena during anodic bonding of silicon to sodium borosilicate glass[END_REF] makes them migrate towards the cathode at its back. They leave behind a negative space charge in the region of the interface arising from the static oxygen ions. These create an electrostatic field with the positive charges in the silicon wafer. Since the space charge layer is typically a few hundred nanometers to a few microns thick, the field created across this layer is very high and pulls the wafer into intimate contact with the substrate, ultimately leading to the formation of stable Si-O-Si bonds and resulting in a wafer bonded to the glass substrate.

Conducting materials which readily oxidize can thus be bonded to Pyrex substrates through a similar mechanism.

In the case of graphite the initial bonding is predominantly electrostatic in nature as we will show with the help of our Raman results in section 3. This is also borne out by the fact that the Raman spectra do not show any sign of formation of graphene oxide. The nature of the two surfaces to be bonded is of importance and a thorough cleaning of the glass substrate with standard laboratory procedure is necessary. We use graphite flake precursors of mineral origin. HOPG graphite for example is not an ideal precursor as graphene obtained from it rarely exceeds lateral dimensions of a few micrometers. We have had good results both from flakes obtained from Madagascar graphite as well as small (typically mm size) graphite inclusions in calcite. The flake must be carefully cleaved before being put into contact with the clean glass surface so as to insure a clean interface. The substrate must contain alkali ions which are mobile under certain conditions and we have used various borosilicate glasses and obtained good results. The surface to be bonded of course needs to be optically polished. Other than these precautions, the bonding can be carried out in normal laboratory surroundings.

We have consistently obtained good results for an interface temperature of 200-250 • C and an applied potential difference between the cathode on the glass substrate and the anode in contact with graphite in the range of 1700-1800 kV. The typical time of application of the potential difference is ten minutes at the end of which the flake is effectively bonded to the substrate. The last step in the procedure is to cleave the flake off the glass substrate using either a scalpel blade or adhesive tape or both. Visual identification with the help of a microscope is then sufficient to isolate graphene remaining on the substrate often in the company of thicker graphite flakes. This identification is somewhat more difficult than in the case of a SiO 2 /Si substrate because it is based only on optical contrast. This is the reason that our optical micrographs do not resembles the 'standard' ones for graphene on SiO 2 /Si substrates. With experience one can identify monolayer samples, and few layer samples are easily visible.

Raman spectroscopy

Raman spectroscopy has fast become a method of choice both for characterizing graphene samples as shown by several early papers [START_REF] Ferrari | Raman spectrum of graphene and graphene layers[END_REF][START_REF] Graf | Spatially resolved raman spectroscopy of single-and few-layer graphene[END_REF], as well as studying electron phonon coupling in graphene [START_REF] Bonini | Phonon anharmonicities in graphite and graphene[END_REF] and its variation with doping. Indeed the 2D lineshape is very sensitive to the number of layers in the graphene sample. The lineshape also depends on the nature of the substrate [START_REF] Casiraghi | Raman fingerprint of charged impurities in graphene[END_REF][START_REF] Shukla | Graphene made easy: High quality, large-area samples[END_REF], typically in a bilayer the multiple peak nature is more obvious in graphene bilayers on SiO 2 /Si substrates. Here we show two examples of the application of Raman mapping of our graphene samples. Raman maps and spectra were acquired using a Renishaw InVia Reflex microspectrometer.

A 514 nm Laser Physics argon laser was focused on the sample by a DMLM Leica microscope with a 50x (NA=0.75) objective. The Rayleigh diffusion was eliminated by edge filters and the signal was finally dispersed using a 1800 gr/mm grating. In the first example we show how optical identification of graphene is corroborated by Raman measurements. In the second we show measurements on a sample which is doped insitu due to strong electrostatic interaction with the substrate. It is now well-known that doping of graphene provokes a shift in the G and 2D lines [START_REF] Das | Monitoring dopants by raman scattering in an electrochemically top-gated graphene transistor[END_REF][START_REF] Gava | Probing the electrostatic environment of bilayer graphene using raman spectra[END_REF]. The G line shift is a reliable method for extracting a corresponding doping level. Notably, in bilayer graphene doping can induce a splitting of the G line [START_REF] Gava | Probing the electrostatic environment of bilayer graphene using raman spectra[END_REF].

In Figure 1(a) we show an optical micrograph of a graphene sample which is a monolayer island (darker part of the image) inside a bilayer. The Raman map of a part of the sample is shown in Figure 1(b). The color code corresponds to the width of the 2D line and clearly demarcates the bilayer part from the monolayer. The representative linehapes for this sample are shown in Figure 1(c), both for the G line and the 2D line.

In Figure 2(a) we show an optical monograph of a large monolayer sample with small parts of bilayer and thicker graphene towards the top edge. In Figure 2(b) we show the corresponding Raman map of the sample where the colour code is based on fits of the measured spectra in the terms of reference spectra for monolayer (dark blue), bilayer (light blue) and few layer (white). Here we find that both the G and 2D lines are consistently shifted with respect to the values expected for undoped graphene.

In Figure 2(c) we show representative spectra for the G line and the 2D line in the monolayer, bilayer and few layer parts of the sample. The few layer graphene has an average thickness of 2 nm as measured by Atomic Force Microscopy (AFM) and should correspond to four layers. The shift in the Raman lines is considerable (G peak at 1611 cm -1 , 2D peak at 2706 cm -1 as seen, for example in comparison to the values shown for the first sample in Figure 1 and by comparison with theoretical estimations and earlier experimental data of doping as a function of gate voltage [START_REF] Gava | Probing the electrostatic environment of bilayer graphene using raman spectra[END_REF][START_REF] Gava | Ab initio study of gap opening and screening effects in gated bilayer graphene[END_REF][START_REF] Das | Monitoring dopants by raman scattering in an electrochemically top-gated graphene transistor[END_REF] we estimate the number of doped holes to be 4.5x10 13 cm -2 . Most interestingly, we find that the bilayer G line is consistently split(1585 cm -1 and 1600 cm -1 ) as predicted by calculations and the intrinsic doping estimation from the shifts of the G and 2D lines for the monolayer as well as the bilayer is coherent. Graphene samples are often intrinsically doped due to interaction with the susbstrate [START_REF] Casiraghi | Raman fingerprint of charged impurities in graphene[END_REF]. In our samples in general we do not find large shifts of the Raman lines. However, since our samples are bonded due to an electrostatic interaction we can expect enhanced interaction with the charged substrate, though one would expect this charging and hence the interaction to decay over time. We have found large shifts in the Raman lines in a few samples, as the one presented here. The enhanced intrinsic doping is a strong indication of the electrostatic nature of the bonding process in our samples as the space charge layer formed at the interface during the bonding acts as a polarized bottom gate.

Local probe microscopy

We performed room temperature UHV STM imaging on some of our samples and observed different phenomena related to defects and layer decoupling. The samples were initially earthed with photolithographically deposited gold contacts. However photolithography was found to considerably degrade the sample surface requiring additional annealing and cleaning and we now prefer to use mechanical masks to evaporate the gold contacts onto the sample and substrate.

In Figure 3(a) we show a large sample of graphene (about 100 µm in lateral dimensions). The darkest portions of the image correspond to the glass substrate and the lighter parts correspond to graphene layers. Raman spectra measured on two spots on this thinnest layer are shown in Figure 3(c). The measured lineshape of the 2D line resembles that of a single peak and hence one could conclude that the sample is a monolayer. However the full width at half maximum (FWHM) is large (48 cm -1 ) and could indicate a bilayer since the bilayer 2D line is a composite of more than one peak. The issue is resolved by confrontation with the surface topography obtained by AFM and with the images of the local density of states obtained by STM. An AFM scan of a portion of the sample indicated by a square in Figure 3(a), is shown in Figure 3(b). The average step height measured between the graphene and the substrate is about 1.2 nm indicating that we have indeed a bilayer sample. In Figure 4 we present some room temperature UHV STM images of our graphene samples. Figure 4(a) represents a moiré pattern obtained on the sample in Figure 3, confirming that it cannot be single layer graphene since the moiré pattern originates from a rotational misalignment between two superimposed graphene layers. In the large area STM scan shown here (the spots are not those of the equivalent graphite lattice, the lateral dimension of the scan being 90 nm) we observe two adjacent moiré domains differing by the periodicity of the spots and thus by the interlayer rotational misalignment. The relation [START_REF] Campanera | Density functional calculations on the intricacies of moiré patterns on graphite[END_REF] between the period D of the supperlattice and the rotation angle θ of the two layers is D = In order to investigate the effect of local perturbations on the measured Local Density of States (LDOS) in graphene we deposited a small quantity of Fe (nominally 0.2 ML) on this graphene sample. On large scale STM images the appearance of scattered dots confirmed the presence of Fe atoms or clusters on the graphene surface, similar to those observed on iron doped graphite [START_REF] Bourelle | Stm study of surfaces of kish graphite doped by iron[END_REF]. On a thicker edge of the sample corresponding to few layer graphene, we were able to image periodic oscillations in the local density of states induced by an isolated defect corresponding very probably to a Fe cluster (Figure 4(b)). The defect induces oscillations with a global three fold symmetry corresponding to similar earlier observations [START_REF] Mizes | Long-Range Electronic Perturbations Caused by Defects Using Scanning Tunneling Microscopy[END_REF]. The supperlattices formed have a lattice parameter √ 3 times greater and are rotated with 30 • with respect to the graphite lattice ( √ 3x √ 3R30 LDOS reconstruction). These oscillations are screened out away from the defect over distances of 2-4 nm.

The rich screening pattern surrounding such defects, has attracted theoretical attention in the past in the case of graphite and requires the calculation of local density of states. This is obtained by solving for the electronic structure of the graphite/graphene lattice with the relevant local perturbations (a non trivial problem). We observe here that the internal structure of the defect (a closer look will reveal that there are two maxima in the STM image of the defect) provokes different patterns in the oscillations as explained by Mizes et al. [START_REF] Mizes | Long-Range Electronic Perturbations Caused by Defects Using Scanning Tunneling Microscopy[END_REF]. In Figure 4(c) we show three different examples of these patterns.

Finally in Figure 4(d) we show an STM image of the expected honeycomb structure from a monolayer graphene sample. Though this honeycomb structure is a signature of an uncoupled graphene layer we have also observed it in few layer graphene, and even in graphite. This is possible [START_REF] Li | Scanning tunneling spectroscopy of graphene on graphite[END_REF] if the imaged portion under the STM tip is indeed decoupled from the underlying layers.

Transport measurements

In this last section we give typical results for transport measurements conducted on simple field effect transistor devices made from anodic bonded graphene. We show device characteristics of a top-gated monolayer (inset of Figure 5(b))and bilayer (inset of Figure 5(a)) graphene samples. These were patterned using photolithography and Au/Cr (30/10 nm) bilayer deposition. By measuring the resistance in two and fourpoint configurations the average contact resistance is estimated to be of the order of 50 ohms. For the gating of our devices we use electrochemical top-gating with a solid (Polyethylene oxide/LiClO 4 ) electrolyte [START_REF] Das | Monitoring dopants by raman scattering in an electrochemically top-gated graphene transistor[END_REF] which allows high doping levels with low gate voltages. On the addition of the electrolyte the sample resistance increases, possibly due to the addition of impurities.

In Figure 5 we show the square resistance of our devices calculated using the corresponding aspect ratio. The charge neutrality peak, characteristic of few-layer graphene, is found at gate voltages of 0.51V for the monolayer and 0.3 V for the bilayer. I-V characteristics (shown for the bilayer) between source and drain show linear behavior, excluding Schottky barriers at the contacts, and expectedly reflect the drop in resistance due to the field effect doping.

The solid electrolyte is based on the diffusion of Li ions and on the application of a reasonable (generally 2 volts or below) voltage between the gate and the source or drain electrode. The charge appearing on the gate electrode causes ionic diffusion in the solid electrolyte and the formation of a Debye layer over a characteristic distance corresponding to charge separation for the electrolyte. This distance is typically 1 or 2 nm for our case and since the Debye layer screens the applied field, it is also the effective distance over which the field is applied to the graphene sample. It can be seen that this field can be almost one order of magnitude higher with respect to standard dielectrics used since the latter are often much thicker. This is a considerable advantage. However there are also some drawbacks including the fact that the diffusion of Li ions is suppressed at low temperatures which means that to change the doping in the sample, one has to apply the corresponding gate voltage at room temperature.

Conclusion and Perspectives

The straightforward question that one can ask is the relevance of another method for making graphene. We hope to have shown that the method that we present is a simple and practical way of producing good quality and large area graphene on an insulating substrate for laboratory purposes. The graphene can only be as 'good' as the precursor used and the maximum size that we can hope to obtain for our samples is also limited by the quality of the precursor which means roughly mm size. This is however largely sufficient for most laboratory and research applications. We are now working on ways of transposing these graphene samples on other substrates, with a preference for mechanical transposition since chemical treatment often leaves unwanted residue on the sample which alters its electronic and mechanical properties.

Finally, as we have already stated, the method can also be used for producing ultra-thin films of other layered substances. We have prepared few atomic layer thick films of NbSe 2 and InSe for example and this aspect is particularly interesting. In only a few years graphene has become a huge and well trodden research field. However the

Figure 1 .

 1 Figure 1. (a) Optical micrograph of a graphene monolayer surrounded by a bilayer. (b) Raman map of a part of the sample shown in a) The width of the 2D line was used to define the colour code giving black for the monlayer area and red for the bilayer area. (c) representative Raman spectra for the G line and the 2D line from the monolayer and the bilayer.

Figure 2 .

 2 Figure 2. (a) Optical micrograph of a monlayer sample with thicker graphene layers towards the top edge of the photo. (b) Raman map of the same sample where the colour code is determined by fitting the spectrum measured at each point with respect to spectra measured on the monolayer (dark blue), bilayer (light blue) and few layer (white) parts of the sample. (c) Representative Raman spectra from the three area in the sample.

Figure 3 .

 3 Figure 3. (a) Optical micrograph of a graphene sample used for STM imaging. (b) AFM imaging of the part of the sample shown in the red rectangle in (a). The step size with respect to the substrate indicates a height (1.2 nm, see profile in the inset) corresponding to a bilayer. (c) Respresentative Raman spectra from the sample. The 2D lineshape is that of a single Lorentzian and would indicate a monlayer, but the width suggests a thicker sample. The AFM measurement in (b) clearly points to a bilayer sample and the lineshape is explained by partial decoupling between the layers due to rotational misalignment (see text).

d 2 Figure 4 .

 24 Figure 4. (a) moiré domains imaged on the bilayer, indicating rotational misalignment between the layers for which we can calculate the angle (see text). (b) Fe atoms were deposited on the surface of the samples and on a thicker edge corresponding to few layer graphene we could observe perturbations in the local density of states introduced by an isolated defect. These perturbations are seen to be screened over a distance of 2-4 nm yielding the equivalent graphite lattice away form the defect. (c) A zoom into three areas from the image in (b) showing different surface patterns for the local density of states. Area 1 corresponds to the √ 3x √ 3R30 LDOS surface pattern while areas 2 and 3 are more subtle variations of this electronic reconstruction. (d) A monolayer graphene sample should have a signature honeycomb STM image and this is what we observe in this figure. The inset show a profile along the white line in the figure showing neighboring carbon atoms.

Figure 5 .

 5 Figure 5. Resistance as a function of top gate voltage for devices pictured in the inset. In the inset for figure 5 (b)we show as measured I-V characteristics as a function of the gate voltage. (a) bilayer, (b) monolayer.

discovery of graphene has opened up the general field of two dimensional materials and their intriguing properties and this field is still largely unexplored.

We acknowledge assistance from F. Gelebart and M. Morand, discussions with V. Etgens and F. Mauri and funding from DRITT(UPMC) and by E.U. EuroMagNet II program under contract 228043.


H A Barkad

A Soltani
email: ali.soltani@iemn.univ-lille1.fr

M Mattalah

J-C Gerbedoen

M Rousseau

J-C De Jaeger

A Benmoussa

V Mortet

K Haenen

B Benbakhti

M Moreau

R Dupuis

A Ougazzaden

Design, fabrication and physical analysis of TiN/AlN deep UV photodiodes

Keywords: Photodiodes, AlN, deep UV, Simulation

Deep-ultraviolet solar-blind photodiodes based on high-quality AlN films grown on sapphire substrates with a metal-semiconductor-metal configuration were simulated and fabricated. The Schottky contact is based on TiN metallisation. The material is characterised by micro-Raman spectroscopy and X-ray diffraction technique. The detector presents extremely low dark current of 100fA at -100V DC bias for a device area of 3.1 mm 2 . It also exhibits a rejection ratio between 180nm and 300nm of three orders of magnitude with a very sharp cut-off wavelength at 203nm (∼6.1eV).

The simulation, based on a 2D energy-balance model using COMSOL® software, permits to help the designer for the optimum topology determination by means of physical studies. The measurement results are in good agreement with the model predictions.

Introduction

Solar-blind deep-ultraviolet (DUV) photodetectors are able to operate at high temperature and in harsh environment thanks to their wide band gap structure. These devices present a variety of potential applications in the fields of automotive, aerospace, and military industries, as well as in environmental and biological research. These applications require high-performance solar-blind photodetectors with low dark current and high UV responsivity. DUV photodetectors fabricated from various wide band gap semiconductors such as wurtzite aluminium nitride (wAlN) [1], cubic boron nitride (cBN) [4] and diamond [5], were reported recently.

AlN is the only semiconductor with a large direct band gap (∼6.1eV) and it is available due to recent advances in epitaxial growth techniques [6] with very high quality epilayers obtained by metalorganic vapour physical epitaxy (MOVPE). In this paper, the design of the photodiodes is optimized by means of a 2D energy balance simulation based on COMSOL ® software.

The fabricated devices have a Metal/Semiconductor/Metal (MSM) configuration with circular interdigitated electrodes. The device's spectral response reveals a very sharp cut-off wavelength at 203nm (∼6.1eV).

Material characterization

AlN films are deposited on sapphire (Al 2 O 3 ) substrate by MOVPE. The growth temperature is 1100°C under 100Torr growth pressure. Trimethylaluminum and ammonia are used as the source materials and the carrier gas is hydrogen [6]. The AlN wurtzite structure belongs to the space group C 4 6υ P6 3 /mmc, associated to the irreducible representation Γ=A 1 +2B 1 +E 1 +2E 2 [7]. Crystalline phases and growth directions of AlN films were determined by θ/2θ X-Ray Diffraction (XRD) with CuKα (k=1.540598Å) radiation operated at 40kV and 30mA. The rocking curve scan was performed with a detector at 2θ=36.1° corresponding to the 2H-AlN (0002) reflection (JCPDS c25-1133) when the sample orientation θ was varied from 5° to 30°.

The FWHM of the (0002) peak measured in the θ-2θ XRD pattern of AlN film is 0.77° and its associated rocking curve 0.3°. The thickness of the thin film is 820nm.

The material quality is crucial since the optical performance in the extreme UV (EUV) is strongly influenced by the surface characteristics. High quality active layer is required and the thickness for total absorption should be optimal to obtain less defects, crack-free layers, low roughness, and good structural homogeneity. The doping level of the active layer is also one of the most critical parameters and should be below 10 15 cm -3 in order to achieve a depletion reach-through region for a reasonable electrode gap spacing and very low dark current.

Micro-Raman spectra are obtained using a Jobin Yvon spectrometer with three excitation sources at λ=266nm, 514nm and 785nm (see Figure 1). Optical modes marked by a star and a square correspond to sapphire substrate and the AlN thin film respectively. All the TO and LO modes of AlN and sapphire can be seen on the spectra of Danidov et al. [7] and McNeil et al. [8] demonstrating the crystalline state of the deposited AlN films. Table 1 gives a comparison of our results with those of references [7] and [8]. The spectrum under UV (@ 266nm) excitation exhibits clearly the A 1 (TO) at 621cm -1 , the E 2 2 (T0) at 660cm -1 and the A 1 (LO) at 891cm -1 . The E 1 2 mode at 247cm -1 is only observable under visible (λ=514nm) or near infrared (λ=785nm). These values are close to those of AlN crystal [7,8].

Modelling

In order to design and fabricate AlN based photodetectors for solar-blind applications, a 2D energybalance model using COMSOL® software was developed. This model is based on the conservation equations deduced from Boltzmann's equation and coupled to Poisson's equation as such:

div (-ε gradV)=q(n-p+N A + -N D -)
where ε is the permittivity, n the electron density, q is the absolute electron charge, N A + et N D + correspond ionized acceptors and donors density respectively.

Charge conservation equations:

For electrons:

( )

                + - = - = + ∂ ∂ → → e B n n n T nk div n E n µ v with U G v n div t n 1 ) ( ,
For holes:

( )

                  - = - = + ∂ ∂ → → e B p p p T pk div p E p µ v with U G v n div t p 1 ) (
Where G corresponds to electron-hole generation which is defined as: G(x)=φ(1-)α e -αx with (λ) the reflection coefficient and α(λ), the absorption coefficient versus wavelength. µ is the carriers mobility.

U is the bulk recombination rate, defined as: U=U rad +U SRH +U Aug , where U SRH corresponds to Shockley-Read-Hall rate recombination, U rad is the radiative recombination rate and U Aug is the Auger recombination rate.

The equations governing the energy flux are for electrons:

( ) ( ) ( ) w w w n E nv v n T k w div t nw w n n e B τ 0 . ) ( - - -       + - = ∂ ∂ → → →
, where τ w is the energy relaxation time and w is the average energy related to the average velocity n v and electron temperature T e .

L B e B n T k w T k v m w 2 3 , 2 3 2 1 0 2 * = + =
, where k B is the Boltzmann's constant.

The transport parameters such as electron mobility, energy relaxation time and electron effective mass, are obtained from stationary Monte-Carlo simulations at room temperature. In the charge conservation equation, a generation term is included to take into account the light excitation. In this model, the number of electron-hole pairs created per absorbed photon is always unity in the wavelength region 170-425 nm (no secondary ionizations).

Furthermore, the band-to-band and Shockley-Read-Hall recombination effects are introduced. From this model, numerical simulations are carried out to study and optimize the device parameters such as detector geometry (gap spacing and electrodes width) and applied bias.

The studied structure is a MSM photodiode. The 2D simulation is carried out for a 300 nm thick AlN (unintentionally doped n-type at 10 15 cm -3 ). The Schottky barrier height is 1eV and the total area is (10×10) µm 2 corresponding to a screen pixel whatever the considered topology. The photocurrent is calculated at 10V bias (V AK ) and the excitation power is fixed to 12 μW/cm 2 from 170 nm to 425 nm according to the available data of the absorption coefficient and the reflectivity of AlN [START_REF] Adachi | Optical Constants of Crystalline and Amorphous Semiconductors: Numerical Data and Graphical Information[END_REF].The photodiode design was optimized considering the different main parameters. The first structure parameter to be optimized is the finger spacing for a given bias. Figure 2 shows the evolution of the device responsivity versus the electrode spacing for a symmetric design with two fingers. In the simulation, the electrode length is 0.2 µm and the electrode spacing varies from 0.5 µm (submicron structure) to 8 µm. The V AK bias between the two electrodes varies from 0.1 V up to 15 V. The responsivity increases with this bias as it was expected and it is important to notice that it becomes higher for submicron devices [START_REF] Palacios | [END_REF].

The second structure parameter to be optimized is the finger width and device symmetry. For this study, different finger widths and device symmetries were simulated at 10V. Figure 3 shows the theoretical evolution of the responsivity versus the photon energy for different symmetric (sym) or asymmetric (asym) topologies as anode width A L , cathode width K L , number of fingers N, spacing between electrodes S and the fill factor K (active area/AlN area ratio). It can be noted that the responsivity increases when the finger width becomes submicron. The shadowing of the active area by the interdigitated contacts is the main drawback of the MSM-detector and results in loss of efficiency; furthermore the photoresponsivity is driven by the device fill factor becoming worst for the asym1 structure.

The dark current was also simulated for different spacing S showing a difference of several magnitudes for submicronic devices. Furthermore, an increase of this current versus V AK is observed for a lower value of V AK when the spacing drops. Consequently, a compromise must be found between the dark current level and the photoresponse.

To improve the device responsivity, semitransparent electrodes associated to a submicronic spaced and asymmetric topology can be considered on a large area device. All these solutions provide better detector characteristics in terms of dark current, UV/visible contrast, linearity and enhancement in responsivity.

Device fabrication

The devices were fabricated by depositing TiN/Au Schottky contacts [12] on the AlN surface using magnetron sputtering deposition. First, the AlN films are cleaned with an aceton/isopropanol standard process and then etched in Ar + plasma with a substrate bias of -170 V for 90 s just before metal deposition in order to remove the native oxide on the AlN surface. Then, Au/TiN (150/5 nm) metallization is deposited on the AlN surface. An optical lithography is used to define the patterns.

The electrodes are obtained after an Au etching with a potassium iodide solution (KI) and a TiN etching using H 2 SO 4 /H 2 O 2 (piranha) solution. After deposition, the samples are annealed at 500°C for 40 min in N 2 atmosphere to improve the stability of the electrode contacts and to form the Schottky contact. To achieve high homogeneity of the electric field between the contacts, the electrodes are patterned with a circular interdigitated configuration (see Figure 4). The diameter of the devices varies from 1 to 3 mm diameter. The finger width is 2 µm, and the interspacing between the contact pads is 5 µm. The electrode fill factor is more than 62%. The advantage of this structure is a better noise factor in dark condition compared to submicronic devices [13].

Results and discussion

Current-voltage (I-V) characteristics are obtained using an Agilent 4155C with femtoampere (fA) resolution. Figure 5 shows the typical I-V characteristics of AlN-based MSM detectors measured in dark at room temperature in N 2 atmosphere before and after annealing. The device exhibits a very low dark current around 100 fA at ±100 V bias after annealing. The differential resistance measured between two fingers at a distance of 8 µm, varies from 2.1×10 14 Ω to 1.5×10 15 Ω before and after annealing, respectively between 0V and ±100V bias. The fabricated device exhibits a breakdown voltage (V BR ) higher than 100V corresponding to an electric field of 125 kV/cm.

Figure 6a shows the responsivity spectral measurement carried out on 2 mm diameter devices at ±100V DC bias, in the wavelength range from 180 nm to 300 nm (6.85 eV to 4 eV). The AlN MSM photodiodes have a peak responsivity around 2×10 -3 A/W at 200 nm and a very sharp cut-off around 203 nm (6.1eV). This cut-off wavelength is close to AlN based MSM photodetectors obtained by Dahal et al. [14]. A rejection ratio of more than three orders of magnitude is observed between 200 nm and 300 nm. With our responsivity of 2 mA/W, the experimental specific detectivity @ 200 nm (V AK =± 100V) correspond to 1.054×10 14 cm 2 H 1/2 /W.

Note that different solar-blind UV photodetectors based on wide band gap semiconductors have been reported. MSM diamond photoconductor [5] presented a rejection ratio of 1.6×10 4 between 210 nm (maximum photoresponse of 48 mA/W) and 400 nm at 5 V bias (S=5 and W=2 μm). MSM photodiode fabricated on cBN [4] films grown on silicon substrate showed at -30V a rejection ratio of four order of magnitude between the maximal response (i.e. 180 nm, 21 mA/W) and 250 nm (S=5 and W=2 μm). Finally AlN MSM photodiode [3] presented a rejection ratio of 1.2×10 4 between 200 nm (maximum photoresponse of 4.5 mA/W) and 360 nm at 30 V bias (S=4 and W=2 μm).

Spectra of the sub-band gap absorption are usually decomposed into "band to defect" and "band to tail" type transitions. In the first case, the photocurrent I ph proportionally varies to the capture cross section of traps by the relationship: σ(hν)≡∆I ph /(I dark .φ 0 .t) (1) with ∆I ph =I ph -I dark . The illumination time is 62 ms and the incident photon flux per cm 2 is φ 0 =P/(S.E) where P is the incident power and E the photon energy. The active area is equal to 2.2 mm 2 . The results of the fitting parameters obtained from Inkson's model [15] are listed in Table 2. Three trap levels are deduced at 3.5, 5.45 and 5.8 eV up to the valence band. The capture cross sections of ET 2 and ET 3 show that those transitions are indirect. The transition ET 3 is responsible for the exponential increase at the absorption edge commonly described by the Urbach's rule [16] and follows the relation:

I ph (E)=I ph0 exp[(E-E G )/E U ] (2),
where I ph0 is a material parameter, E G is the band gap and E U is the Urbach energy representing the width of the exponential absorption edge. In the particular case of III-nitrides, the Urbach's tale is mainly a consequence of the carrier interaction with lattice and lattice imperfections (defects, impurities…). The photocurrent curve is fitted with the Urbach expression (2) with the different parameters E G , I ph0 and E U corresponding to 6eV, 4,45×10 -4 and 0.13 eV, respectively. Figure 6b corresponds to the simulated responsitivity using the optical constants of single crystal of AlN with two light polarization states: E⊥c and E//c (c is the optical axis). Data for simulation are taken from

Adachi [START_REF] Adachi | Optical Constants of Crystalline and Amorphous Semiconductors: Numerical Data and Graphical Information[END_REF].The responsivity curve is fitted from ( 2 the shortest value reported up to now on this semiconductor. The rejection ratio (the ratio of responsivity between 200 nm and 300 nm) is found to be over three orders of magnitude.

Figure captions

Table 1: Peak positions of Raman modes (in cm -1 ) of different wurtzite AlN and Al 2 O 3 structures [7].

Our sample corresponds to the AlN/Al 2 O 3 structure.

Table 2: Results of the fitting parameters obtained from the fitting curve of Figure 7a. The different level of traps ET 1 , ET 2 , ET 3 and the parameters P 1 permit to determine the capture cross sections (σ). Optical modes (cm -1 )

AlN bulk single crystal [7] wAlN thin film [7] AlN (our sample)

  ) with different parameters E G , I ph0 and E U corresponding to 6.1 eV, 5.22 mA/W and 0.22 eV, respectively for E⊥c and 5.95 eV, 2.5 mA/W and 0.47 eV for E//c. The experimental values are close to the theoretical predictions considering bulk AlN material showing a good crystallographic quality of the deposited thin films. Conclusion In summary, solar-blind DUV MSM photodetectors were simulated and fabricated on high quality wurtzite AlN films by MOVPE with a circular inter-digitated electrode design. The device topology is optimized from a 2D energy-balance simulation where the generation and recombination terms are included. The Schottky contact is based on a TiN metallisation process. A very low dark current around 100 fA at ±100V DC bias is obtained. The photodetectors show a high sensitivity to DUV light and a sharp cut-off wavelength around 203nm, close to the band gap of bulk AlN. This corresponds to

Figure 1 :

 1 Figure 1: Micro-Raman spectra collected from the AlN film grown on a sapphire substrate. The squares are assigned to wurtzite AlN modes and the stars are assigned to sapphire substrate modes.

Figure 2 :

 2 Figure 2: Responsivity evolution versus the electrode spacing for a symmetric structure with two 0.2 µm length electrodes for different biases under radiation excitation (λ=6.1 eV).

Figure 3 :

 3 Figure3: Responsivity versus the photon energy for different topologies: A L is the anode length, K L is the cathode length, N is the electrode number and K is the corresponding fill factor. The distance S between 2 electrodes varies between 0.5 µm and 1 µm. V AK =10V.

Figure 4 :

 4 Figure 4: Optical microscopy image of a 2mm diameter AlN photodiode (a) and zoom view of the circular inter-digitated electrodes in the centre area (b).

Figure 5 :

 5 Figure 5: Current-Voltage characteristics of a 2 mm diameter AlN photodiode measured in dark conditions at room temperature before and after annealing at 500°C.

Figure 6 :

 6 Figure 6: Spectral responsivity of an AlN MSM photodiode measured at ±100V DC bias under irradiation in the 180-300 nm range (a). Simulated responsivity from data [10] for two light polarization states (E//c and E⊥c) (b).

  Figure 1

  Photons energy (eV)

Figure 6

 6 Figure 6

Table 1

 1

	capture cross sections	E T (eV)	P 1	σ σ σ σ(cm 2 )
	σ C =P 1 (hν-E T1 ) 3/2 /hν	ET 1	3.5	2.2×10 -5	-
	σ C =P 1 (hν-E T2 ) 3/2 /hν	ET 2	5.45	6×10 -4	1.25×10 -11
	σ C =P 1 (hν-E T3 ) 3/2 /hν	ET 3	5.8	0.018 1.426×10 -11

Table 2

 2

N K A L K L S 10 0,5 0,5 0,5 0,5 7 0,65 0,5 0,5 1 6 0,55 1 0,5 1 7 0,3 1 1 0,5 5 0,5 1 1 1 4 0,4 2 1 1