Keywords: UAV, MAV, flapping wing, numerical model, Bond Graph

Abstract-Micro air vehicles (MAVs) are becoming more popular over larger unmanned aerial vehicles (UAVs) as they are easily portable, more discreet and less dangerous in case of a crash. Among common types of MAVs, the flapping wing configuration shows incredibly potential flying skills such as hovering, flying backward and recovering after shock. However, designer faced many difficulties due to the microsize of the MAVs. In this work, we propose a numerical model based on Bond Graph for our flapping MAV which allows us to analyze the wing kinematic and thus predict the total lift force. This Bond Graph model employs the quasisteady aerodynamic theory and the Lagrange dynamic equation as the main principles. A proper wing kinematic which allows the enhancement of the total lift force could be derived from this model based on the non-linear optimization of the system's sensitivity parameters such as spring stiffness, working frequency and input voltage... A prototype is fabricated and characterized. Comparing the experiment and the simulation, the model is able to predict the wing movement and the mean lift force, and therefore could be used as a design tool. A take-off demonstration is provided to confirm our results.

INTRODUCTION

The performance of the existing designs of the flapping wing MAVs are worse compared to the fixed and rotary wing groups. Low Reynolds number condition leads to highly unsteady aerodynamics of such vehicles). Nevertheless, the perspective of potentially achieving the exceptional flying performances has prompted a significant amount of research on the kinematics and aerodynamics of flapping flight in nature [START_REF] Song | Three-dimensional flow and lift characteristics of a hovering ruby-throated hummingbird[END_REF], [START_REF] Sane | The aerodynamics of insect flight[END_REF], [START_REF] Lehmann | The mechanisms of lift enhancement in insect flight[END_REF]. Subsequently, several studies have considered how vehicle designs could mimic the function or the form of flying organisms [START_REF] De Croon | The DelFly[END_REF], [START_REF] Ristroph | Stable hovering of a jellyfish-like flying machine[END_REF], [START_REF] Bontemps | Modeling and evaluation of power transmission of flapping wing nano air vehicle[END_REF]. Here we introduce three outstanding existing insect-like flapping MAVs including DelFly, AV Hummingbird and Robobee. The DelFly is a tail fully controllable MAV [START_REF] De Croon | Design, aerodynamics and autonomy of the DelFly[END_REF]. Its Micro version is currently accepted as the smallest free flying controllable flapping wing MAV equipped with a camera and a video transmitter. This 10 cm wing span vehicle weighs 3.07 grams and can fly around 3 minutes. The 16 cm span and 19 g AV Hummingbird is a tailless remote controlled NAV built to mimic a hummingbird [START_REF] Keennon | Development of the Nano Hummingbird: A Tailless Flapping Wing Micro Air Vehicle[END_REF]. The tailless design makes it closer to a real flying humming bird, but leads to a passively unstable attitude. As a result, it needs a more advanced control system to stabilize the vehicle. Equipped with a small video camera ______________________________________ Le Anh Doan is with the University of Valenciennes, IEMN, CNRS UMR-8520, Valenciennes, France (e-mail: Leanh.Doan@etu-valenciennes.fr).

for the purpose of surveillance and reconnaissance, this vehicle operates in the air for up to 11 minutes at 18km/h. Inspired by the biology of a bee, the Harvard RoboBee [START_REF] Ma | Controlled Flight of a Biologically Inspired, Insect-Scale Robot[END_REF] is the smallest and lightest MAV that can perform controlled hovering. This vehicle weighs less than one-tenth of a gram, and flies using -artificial muscles‖ composed of materials that contract when a voltage is applied.

As the first example of MAV is passively stable with tail and the last one is limited in payload, it is thus decided to develop a flapping MAV mimicking the hummingbird. The developed MAV, however, is lighter in weight compared to the AV Hummingbird. To preserve the wing kinetic energy and to achieve a capable resonant system, we directly drive our flapping wing MAV using conventional DC motors coupled with helical springs. Flexible part that is added to the wing contributes to the wing's passive rotational movement. With the presence of elastic components in the system, higher efficiency would be achieved. The concept of this flapping MAV is shown in the Figure 1. This configuration allows us to utilize some of the off-the-shelf parts as well as available technology framework.

Figure 1: MAV concept Since the studied system combines the mechanical, the electromagnetic and the aerodynamics fields, a unique model Bond Graph formalism has been set up. This kind of formalism is widely used to model multiphysics systems and energetics efficiency. Combining the Lagrange dynamic equations with a quasi-steady model of aerodynamic forces, this model serves as a tool for diagnostic system's performance. Thanks to the simulation, the MAV is able to operate at its resonant frequency with a specific torsion spring. Further optimization on the stiffness of flexible part contributes to a proper phase shift between flapping and rotational movements, which certainly results in an enhancement of total lift force. To prove the rightness of the model, a prototype is fabricated with components whose characteristics are retrieved from the simulation. The wing's motion tracked with a high-speed camera and the mean lift force measurement are used to validate the model. A take-off demonstration is also provided to confirm our results.

This article is inspired by the work of L. Hines et. al in [START_REF] Hines | Liftoff of a motor-driven, flapping-wing microaerial vehicle capable of resonance[END_REF] with three main different contributions. Firstly, the flexible part made by a piece of rubber is positioned along the longest chord of the wing. This helps reducing the length of the wing offset which brings the wing and also the center of Bond Graph based design tool for a passive rotation flapping wing

Le Anh Doan*, Christophe Delebarre, Sebastien Grondel, Eric Cattan the lift closer to the actuator. This configuration also helps to vary the stiffness of the flexible part easily by modifying the rubber pieces ‗dimensions. Secondly, simulation part shows the effort to approach the resonance to maximum lift forces by choosing a proper torsional spring. Lastly, to fulfill a design tool, the effect of wing offset d w and wing flexure stiffness K w on rotational amplitude, lift force and phase shift is also taken into account.

MAV MODEL

2.1

The Word Bond Graph of the MAV The Word Bond Graph in Figure 2 reveals the main components of MAV dynamic model. The model is similar to the well-known block diagram, with the major difference that the -bonds‖ which link the elements together represent bidirectional exchange of physical power. Each bond depicts instantaneous flow of energy or power denoted by a pair of power variables called flow and effort. Flapping motion of wing is induced with a sinusoidal input voltage to each motor where A is the peak-to-peak voltage, f is the operating frequency. The bond next to the wave generator block would present the flow of electrical power and the power variables would be the current (i) and the voltage ( ), whose product is power (P in ). Likewise, the motor output angular velocity ( ) and the torque ( ) are flow and effort of the corresponding power (P mechanic ). A quasi-steady model is employed to model the aerodynamic forces on our passively rotating flapping wing with the aim to predict wing motion (flapping angle and rotational angle ) and lift.

2.2

Motor driver and geared motor model While motor driver can be easily presented by a modulated effort source MSe, a model for a DC motor connected through a gear reduction is a little more complicated as shown in Figure 3. The mature armature wingdings are introduced by the R 0 element. A gyrator GY with the armature constant, , converts input (v) voltage into angular velocity ( ). A transformer element TF stands for the gearbox with the gears ratio of . J m and b m are the rotor inertia and motor rotational damping values respectively. The R element right next to the TF element demonstrates the gearbox efficiency. Some power sensors are integrated in the Bond Graph model to inspect input power (P in ), dissipated power (P R0 , P bm and P eff ) and the power supplied to the wing (P mechanic ).

2.3

Aerodynamic model In the absence of skin friction, to model the aerodynamic forces on thin flapping wings, we employ a quasi-steady model [START_REF] Dickson | An integrative model of insect flight control[END_REF], where the instantaneous aerodynamic forces on the wing are approximated using Blade Element Method. In this method, each wing is divided into a set of cross-section strips, each of width dr, and at a mean radius r from the axis of flapping. The instantaneous forces may be represented as the sum of three components, each acting normal to the wing surface:

(1

)
where F trans is the translational force, F rot is the rotational force. The added air mass force F air is the inertia of the airflow generated by unsteady wing motion.

Wings model

The dynamic model of the wing is formulated in Lagrangian form with two coordinates of flapping ( ) and rotational angle ( ).

⃗ ⃗ ⃗ ⃗⃗ ⃗ ⃗⃗ (2) where T is the total kinetic energy of the system, equalizing the sum of the kinetic energies of the wing translation and rotation, and V is the potential energy of the system caused by the torsion spring and the flexible part with corresponding stiffness of K s and K w . m w is the wing mass and J w is the wing inertia. The motion equations of the wing could be drawn from the derivation of the Lagrangian equations.

( ̇) ⃗⃗⃗ ̇ (3) ( ̇) ⃗⃗⃗ ⃗⃗⃗ (4)
where the ⃗⃗⃗ is the driving flapping torque and also the torque provided by the motor. b w is the damping of flexible part. ⃗⃗⃗ and ⃗⃗⃗ are the moments due to the aerodynamic forces in the rotational and flapping direction. This approach for passive wing is inspired by the work of [START_REF] Hines | Liftoff of a motor-driven, flapping-wing microaerial vehicle capable of resonance[END_REF]. Reader should refer to this work for full derivation of dynamic equations.

It is noted that Equations 3 and 4 have the second order form of a spring-mass-damper system with nonlinear coefficients. They can be presented in Bond Graph by two different set of C, I, R elements for each dynamic coordinate as in Figure 4. Apart from the aerodynamic moment, the two above MSE elements also include moments caused by the gyroscopic effect and centrifugal force acted on flapping and rotational axes. Wing's Bond Graph elements can be found in Appendix To form the MAV's model, all of the sub-systems are linked together through the common bonds, presenting the power transfer as shown in Figure 5.

2.5

Parameters selection To limit the number of free parameters, our simulated motor is based on an off-the-shelf motor (GM15A). The motor characteristics are provided by the manufacturer and also in the work of [START_REF] Hines | Liftoff of a motor-driven, flapping-wing microaerial vehicle capable of resonance[END_REF]. The wing geometry characteristics is retrieved from a 3D model. The added helical springs could be chosen based on the desired flapping frequency. The maximum stiffness value is also limited by the toughness of the plastic gearbox of the motor.

2.6

Vertical take-off model In order for a fly vehicle to rise into air, the lift force (F Lift ) created must be at least greater than or equal to the force of gravity (F Gravity ) as shown in (5):

̈ ( 5
)
where z is the prototype's altitude and m MAV is the mass of prototype. Suppose that the prototype moves at relatively low speeds through the air, the viscous resistance f viscous is approximately proportional to its velocity:

̇ (6)
where b viscous is the viscous coefficient. A 1-junction is employed to present Equation 5 as in the vertical take-off block.

Figure 6: Vertical take-off block

SIMULATION

3.1

Sensitivity to spring stiffness and driving frequency Three stiffness values of springs are tested on a range of input frequency from 1 to 20 Hz. The system input voltage isheld at 2V to limit the high flapping angle of the least stiff elastic element. Different colors are used to specify different elastic element stiffness systems as shown in following figures. While the system with the highest spring stiffness resonates at 8 Hz, resonance of the lowest one could not be determined. Maximum flapping amplitude of the remaining system occurs at 3 Hz.

Peak lift force, however, happens beyond the flapping resonance. At low flapping frequencies, the wing speed is too low to produce remarkable lift or wing rotation. In ascending order of stiffness, maximum mean lift occurs at 4 Hz, 7 Hz and 10 Hz respectively. It is noted that, the highest spring's stiffness brought the peak of lift close to the frequency at which the resonance happens. Performing this system at its resonance frequency allows us to optimize the system efficiency without much decreasing in lift. A compromise between wing speed and flapping angle amplitude is always the key-point to enhance the system performance. In conclusion, a system with maximum stiffness (2.956e3 mN.mm/rad) activating at 4.27 V and 10 Hz will be our ultimate solution.

3.3

Sensitivity to wing flexure stiffness The system with spring stiffness of 2.956e3 mN.mm/rad isstimulated by an input voltage of 4.27sin (2 10t). Some interesting discoveries could be seen in Figure 9. It is better to remind that phase shift is the difference between rotational angle and flapping angle, a negative phase shift means the former angle is lag behind the latter and vice versa. It is obvious that larger value of wing flexure stiffness results in smaller rotational amplitude. A lower value than the minimum bound of inspection range (1e-4 N.m/rad) causes over rotation and the wing exhibits erratic and unsteady behavior. Lower driving frequency could generate an acceptable wing motion, however it will not be the goal of this scope. Raising the stiffness does slowly increase the total lift force, on the other hand it also augments the angle of attack. If this angle exceeds , drag coefficient will play a major role while lift coefficient gradually degrades. This means thatthe total force is less directed vertically but more horizontally.

We always expect to achieve a phase shift close to because it leads to the highest rotational angle which occurs at midpoint of flapping trajectory where the maximum flapping velocity generates an ultimate total lift force. Fortunately, all of our desires are achieved with this configuration. This system has of rotation angle at mid-stroke and of phase shift at a flexure stiffness of 2.2e-4 N.m/rad.

3.4

Sensitivity to wing offset In this part, we use the same setting as the previous part except that the variable is now the wing offset. It is apparent that the wing translational velocity at the mid-chord of each wing strip is proportional to wing offset, so increasing the wing offset improves the translational force. This is well demonstrated in the range of 0 to 35 mm. However, increasing the wing offset also augments the moment inertia about the flapping axis and therefore reduces the flapping movement. While the wing size is kept constant, greater wing offset also effectively increases the damping force which can be limited given maximum motor torque. As can be seen in the Figure 10, when the wing offset is beyond 35 mm, the lift force starts to reduce. Another disadvantage of raising this parameter is that it moves the center of lift far from center gravity and makes the prototype more difficult to stabilize. Bigger size of the prototype is also another drawback. A table summarizes the optimized parameters can be found in Appendix.

3.5

Dynamic simulation results After running the simulation with all the parameters found from previous discussions, a satisfactory wing's motion is portrayed in Figure 11. The rotation trajectory is and lags behind the flapping by nearly . Notably, the force shown in Figure 11 The results from the added vertical take-off block are illustrated in the following picture. The MAV reaches the altitude of 6 cm after 0.5 s with a 100mA peak sinusoidal input current.

Figure 12: Simulation of vertical take-off

EXPERIMENT

After building the MAV model, we can now validate whether it reproduces the system behavior within acceptable bounds with the recommended parameters drawn from the optimization step. This section starts by the preparation of main components and materials for the MAV fabrication. Then a set of validation experiments including wings movement observation and total mean lift force estimation are conducted to confirm the rightness of our model.

4.1

Component fabrication A tiny DC pager-motor named GM15A with a planetary gearhead 25:1 reduction has been chosen as the main actuator. It is designed for a 3V nominal operation, giving 920RPM drawing 100mA. The motor is controlled by a motor driver named Pololu DRV8835. Our flapping system is induced with a Pulse Width Modulation (PWM) approximated sinusoidal voltage generated by our motor driver. The wing should be as light as possible but not flexible because it is supposed to be rigid except at the flexible part and near the rotational axis.

Figure 13: Fabricated Wing The wing has the length of 8.5 cm and the maximum chord length is 3.5cm. The flexible part is made by a piece of rubber and its rotational stiffness depends on its length. d w is the offset distance from the motor output shaft to the free end of leading edge. The membrane is made by very thin film with the thickness of 25 . The mass of the wing is 0.12 g measured by a microbalance Mettler Toledo.

4.2

Wing kinematic observation To observe the wing movement, a high speed camera has been positioned in front of the wing. The half prototype is activated by an input voltage of 4.5sin( 2) (V). A of rotational angle at the mid-stroke is the validation for the mathematical model Mean lift force measurement The wings with the appropriate kinematic were employed in the lift force measurement experiment. The whole set-up could be seen in the Figure 15. The moment created by the weight of the half prototype and the -Mass‖ have the same value but in the reverse direction counting at the center rotation of the level so that the level is in its balance position at the beginning. As soon as we activate the motor, the lift force is generated and measured by the load cell. A mean lift force of 4.5g could be seen in the next figure.

Figure 16: Force measurement From the simple equation of equivalent moments, the lift force must be 0.4 time as much as the force measured by the load cell and therefore we get the mean value of 1.8 g. The mean lift measured from the experiment is a little bit bigger than one from the simulation, however it is still good to predict the total lift. Remind that the total mass of full prototype is 2.8 g, so it should fly if we employed two motors.

4.4

Take-off demonstration A complete prototype was assembled with recommended configuration. It was let to move up and down unrestrictedly along two fixed parallel carbon tubes which, by the way, limit all other MAV's degree of freedom. At an input voltage of 4V peak-to-peak, the system starts to fly up. After a few seconds, a snapshot of the MAV motion (Figure 17) was taken, showing that it was above the initial position nearly 2 cm. It is not correlated to the take-off simulation due to the friction of the MAV with two carbon tubes.

CONCLUSION

In this study, a design tool based on a dynamic numerical model has been proposed to analyze the wing kinematic of a new MAV and thus predict the total lift force. An elastic element placed in parallel to the gearbox output shaft and wing allows operation at resonance. The effect of varying wing offset, elastic element stiffness, and elastic wing stiffness have been simulated in order to determine the best configuration.

After experiments, we can conclude that an adequate amount of lift force is produced to bring our prototype to the air, which validates the results of our design tool. As the lift weight ratio is equal 1.28, it is not possible to handle electrical circuits including microcontroller, motor driver, Bluetooth device and also battery at this time. Future work has to be developed on increasing the working frequency but maintaining the same wing kinematic. If we succeed to double the working frequency the lift force will increase by the factor of four and the MAV has more spare weights.

Figure 2 :

 2 Figure 2: The Word Bond Graph of the MAV

Figure 3 :

 3 Figure 3: Motor driver and motor with gearbox blocks The mature armature wingdings are introduced by the R 0 element. A gyrator GY with the armature constant, , converts input (v) voltage into angular velocity ( ). A transformer element TF stands for the gearbox with the gears ratio of . J m and b m are the rotor inertia and motor rotational damping values respectively. The R element right next to the TF element demonstrates the gearbox efficiency. Some power

Figure 4 :

 4 Figure 4: MAV's wing block

Figure 5 :

 5 Figure 5: Bond Graph presentation of MAV

Figure 7 :Figure 8 :

 78 Figure 8(a) illustrates well a nearly linear relationship between flapping amplitude and the input voltage in the range of examination. Flapping angle is inversely proportional to the value of spring's stiffness. In reality, the flapping amplitude should not exceed pi/2 to avoid wings ‗collision. With this

Figure 9 :

 9 Effect of wing flexure stiffness K w on rotational amplitude (a) lift force (b) and phase shift (c)

Figure 10 :

 10 Effect of wing offset d w on Flapping amplitude (a) and lift force (b).

Figure 11 :

 11 (a) is the cycle-averaged lift which is in the vertical body direction. The peak lift with the force of 17 mN occurs at each mid-stroke. Dynamic simulation of a proper wing trajectory (a) and resulting lift force (b)

Figure 14 :

 14 Figure 14: Diagram of wing observation experiment set-up

Figure 15 :

 15 Figure 15: Lift measurement set-upThe moment created by the weight of the half prototype and the -Mass‖ have the same value but in the reverse direction counting at the center rotation of the level so that the level is in its balance position at the beginning. As soon as we activate the motor, the lift force is generated and measured by the load cell. A mean lift force of 4.5g could be seen in the next figure.

Figure 17 :ACKNOWLEDGMENT

 17 Figure 17: Take-off demonstrationAPPENDIXAs Bond Graph model of the wing shown in this work is adapted from dynamic equations in[START_REF] Hines | Liftoff of a motor-driven, flapping-wing microaerial vehicle capable of resonance[END_REF], readers should refer to this reference for the full derivations. The following equation introduces only the complex Bond Graph elements related to the wing.


S Benzeghda

F Hobar

Didier Decoster

Jean-François Lampin

A Benzeghda
email: benzeghdasabah@gmail.com

Microwave Optical Switches Metal-Semiconductor-Metal Schottky based on GaAs

Keywords: Impulse response, MSM photodetector, space charge

Interdigitated metal-semiconductor-metal (MSM) photodetectors have received considerable attention for applications in microwave optical phoswitches. The impulse response of interdigitated metal-semiconductor-metal photoswich fabricated on GaAs non-intentional doped (NID) absorbing layer is investigated. The photodetector MSM is introduced in the microwave lines have active surfaces of 3x3 µm2 and electrode spacing of 0.2, 0.3, 0.5 and 1 µm.. The photocurrent response was measured after excitation and we found that the screening of the dark electric field and charge accumulation exceedingly modify the drift conditions of the photogenerated electrons and holes in active region of the MSM photoswich.

Introduction

High speed photodetectors are a key component of optoelectronic switching in coplanar transmission lines growth semiconductor material [START_REF] Wendorff | Optoelectronic switching in travelling-wave MSM photodetectors using photon energies below bandgap[END_REF]. Planar metalsemiconductor-metal (MSM) photosensing structures with Schottky barriers have attracted much attention, inherent advantages include: lower capacitance, a large photosensitive surface, easy to integrate, low dark current and faster responses compared to PIN photodetectors [START_REF] Crouse | Pavan Keshavareddy Numerical modeling of electromagnetic resonance enhanced silicon metal-semiconductor-metal photodetectors[END_REF], [START_REF] Glinz | Modeling of wide-area thin film metalsemiconductor-metal photodetectors for LIDAR applications[END_REF]. While mainly III-V semiconductor materials have been used for optoelectronic applications. This device is an interdigital comb (Ti(250 A˚)/Pt(250 A˚)/Au(4000 A˚)) leaving a free semiconductor surface between the two contacts which forms the active region in which light will be absorbed [START_REF] Decoster | Optoelectronic Sensors[END_REF]. It located in the central strip of coplanar line, deposited over a non-intentional doped GaAs layer (1014cm-3), with thickness equal to 1µm, epitaxially grown on a Semi-Insulating (S.I) substrate of 400 mm thick and crystal orientation <1 0 0>. The number of fingers (N) and inter electrode spacing (s) that is equal to the widths of fingers (l) and varies between 0.2, 0.3, 0.5 and 1 µm [START_REF] Zebentout | Effect of Dimensional Parameters on the Current of MSM Photodetector[END_REF].

Our work has been concentrated on the response behavior of the MSM photoswitch, and how optical illumination and polarization will give appear to variety effect [START_REF] Averine | Sachot Transit-time considerations in metal-semiconductor-metal photodiode under high illumination conditions[END_REF].

Results and discussion

The time resolved photocurrent response was studied experimentally by exciting the photoswich with femtosecond laser with low-jitter pulse (Maï Taï HP of Spectra-physics), pulse width 100 fs, wavelength 780nm, repetition rate 80MHz, Two microwave signal generators were used, the first for mounting up to 3 GHz (Rhode and Schwartz SMA 100A) and the second to go up 20 GHz (Anritsu MG3692A). The output signal was collected by tektronix DSA8200 oscilloscope. Depending on the characteristics of the laser pulse, including the wavelength and spectral width length, we arrive at a dispersion of 2 ps (fiber length of 1 meter, spectral width of the pulse at 0.8µm to 16 nm), which is much higher to the original duration of the laser pulses (200 fs). Taking the value of the injection efficiency and the data of the chromatic dispersion giving pulse duration of about 2 ps, if we take 1,1 mW through lensed fiber output, a power peak is obtained ~7 Watts.

The test bench microwave coupled to a fiber illuminating the device from above. When a semiconductor material is illuminated by photons, this allows electron-hole pairs generated by the absorption of light and they start to distribute with field dependent drift velocities and gives rise to a photocurrent [START_REF] Averine | Sachot Transit-time considerations in metal-semiconductor-metal photodiode under high illumination conditions[END_REF]. It may be noted that the pulse response of photoswitch at zero bias voltage is not negligible, the internal electric field is sufficient to collect the photogenerated carriers. The pulse shape is similar at different bias voltages; it has a faster rise time response, and faster initial sweep-out of the carriers, this fast leading part of the impulse response is chiefly due to the drift of electrons, followed by a slower long tail. The oscillations in the tail are most likely due to the impedance mismatch in the bond wires [START_REF] Averine | Sachot Transit-time considerations in metal-semiconductor-metal photodiode under high illumination conditions[END_REF][START_REF] Rosenzweig | Characterization of picosecond GaAs metal-semiconductor-metal photodetectors[END_REF].

To determine the lifetime of the carriers, we normalized the signal and we have shown in a logarithmic scale. Which of the following can we determine the lifetime of electrons by calculating the slope of the falling portion of the curve [START_REF] Siham | Interfaces optoélectroniques ultrarapides pour l'électronique supraconductrice à quantum de flux magnétique[END_REF].

Table I. -Pulse response of GaAs MSM3035 photoswitch

The rise and fall time (l0%-90% of the maximum), and FWHM (Full width at half maximum) for the GaAs MSM3035 are indicated in Table 1.

Figure 4 shows the measured photocurrent of MSM with 0,3µm finger distance with different bias voltage (0, 0.05, 0.25, 1 and 2.8V) and 1mW optical power level as function of time delay. After excitation, the carriers start to move with field dependent drift velocities, for GaAs, vd increases linearly with electric field for fields up to about 3.2 kV/cm. The finger distance is in submicron range it conducts to a full depletion region between electrodes, and this is result of the rapid extraction of electrons [START_REF] Siham | Interfaces optoélectroniques ultrarapides pour l'électronique supraconductrice à quantum de flux magnétique[END_REF][START_REF] Stephen | Nanoscale Tera-Hertz Metal[END_REF], this leads to faster rise time of 10ps, it is almost constant independent of voltage. The photogenerated carriers distribute in the semiconductor, they have a new velocity vectors. This fast rise time is flowed by fairly fast decay corresponding to the rapid collection of electrons and holes at contacts; this conducts to decrease of FWHM (Table I), subsequently to slower long tail which reflects the difference in drift velocities of electrons and holes [START_REF] Stephen | Nanoscale Tera-Hertz Metal[END_REF][START_REF] Lambsdorff | Subpicosecond characterization of carrier transport in GaAsmetal semiconductormetal photodiodes[END_REF].

With small interelectrode distances, although with increasing bias voltage, this leads to high electrical field, and ensure velocity saturation, due to the initial opposite movement of electrons and holes cloud toward contacts [START_REF] Klingenstein | Time resolved photocurrent response of metal semiconductor metal photodetectors to doublepulse excitation[END_REF], so the electric field is screened by the space charge. Near the anode, in the low field region electron-hole plasma is formed, this region contains more holes than electrons, where they move at much lower velocities [START_REF] Averine | Sachot Transit-time considerations in metal-semiconductor-metal photodiode under high illumination conditions[END_REF][START_REF] Lambsdorff | Subpicosecond characterization of carrier transport in GaAsmetal semiconductormetal photodiodes[END_REF][START_REF] Klingenstein | Time resolved photocurrent response of metal semiconductor metal photodetectors to doublepulse excitation[END_REF][START_REF] Kuhl | Influence of Space Charges on the Impulse Response of InGaAs Metal-Semiconductor-Metal Photodetectors[END_REF]14]. When substrate is deeper, these carriers are collected at much lower speeds [START_REF] Honkanen | Fabrication and modelling of SOI and GaAs MSM photodetectors and a GaAs-based integrated photoreceiver[END_REF], it explained as follows: inside the space-charge region, a strong field accelerates electrons towards the bulk just beyond this region, a retarding filed sweeps the electrons toward the surface [15], this leads to increase fall time.

In addition, increasing of carriers' velocity with increasing field usually called 'Negative Differential Resistivity' (NDR) often is mentioned 'Gun effect' [START_REF] Zhao | Time Response of Two-Dimensional Gas-Based Vertical Field Metal-Semiconductor-Metal Photodetectors[END_REF]. Gunn has discovered a new kind of current oscillations at microwave frequencies, in n-type GaAs, as the electric field increases, it leads to transfer of electrons from the high-mobility conduction band valley centered (low value of the wave vector k) (Γ valley), to higher energy, low-mobility satellite valleys ( L and X ), therefore, an increase in the electron effective mass and an increase in the density of electron states from the low-energy valley to the highenergy valley leads decrease in velocity [START_REF] Zhao | Time Response of Two-Dimensional Gas-Based Vertical Field Metal-Semiconductor-Metal Photodetectors[END_REF][START_REF] Eric | A study of picosecond, high-voltage photoconductive switching using electrooptic diagnostics and computer-aided analysis[END_REF][START_REF] Gunn | Instabilities of Current in III-V Semiconductors[END_REF], figure 5. The increase of FWHM and fall time of pulse response with increasing optical power is explained as follows.

The pulse energy is high enough to cause accumulation in the active region, screening of internal electric field, and formed an electron hole plasma near the anode, this region behave like virtual anode because a lot of holes are still staying in this region [START_REF] Averine | Sachot Transit-time considerations in metal-semiconductor-metal photodiode under high illumination conditions[END_REF][START_REF] Kuhl | Influence of Space Charges on the Impulse Response of InGaAs Metal-Semiconductor-Metal Photodetectors[END_REF].

The transient current response is limited by another factor, the trapping at the surface states of semiconductor states carriers create surface charge layer, [15,[START_REF] Shockley | Negative Resistance Arising from Transit Time in Semiconductor Diodes[END_REF], also, at active layer metal-semiconductor interface and/or at deep layer defects in band gab of the material can trap photgenerated carriers [START_REF] Lambsdorff | Subpicosecond characterization of carrier transport in GaAsmetal semiconductormetal photodiodes[END_REF][START_REF] Klingenstein | Time resolved photocurrent response of metal semiconductor metal photodetectors to doublepulse excitation[END_REF][START_REF] Kuhl | Influence of Space Charges on the Impulse Response of InGaAs Metal-Semiconductor-Metal Photodetectors[END_REF][START_REF] Hobar | Etude et realization de structure photoresistances interdigitées pour la detection optique[END_REF][START_REF] Salvestrini | Tuning of internal gain, dark current and cutoff wavelength of UV photodetectors using quasi-alloy of BGaN-GaN and BGaN-AlN superlattices[END_REF][START_REF] Holzman | Recombination-independent photogeneration of ultrashort electrical pulses[END_REF].

We can neglect the screening of the internal electric field to have no effect on the MSM photoswitch response if the space charges are much less than the total charge accumulated at the contacts [START_REF] Kuhl | Influence of Space Charges on the Impulse Response of InGaAs Metal-Semiconductor-Metal Photodetectors[END_REF][START_REF] Holzman | Recombination-independent photogeneration of ultrashort electrical pulses[END_REF][START_REF] Averine | Effect of optical excitation level on the msm-detector high-speed response, Ultrawideband and Ultrashort Impulse Signals[END_REF]. If each photon creates one electron-hole pair, the total number of the photogenerated carriers with total area L2 [START_REF] Averine | Sachot Transit-time considerations in metal-semiconductor-metal photodiode under high illumination conditions[END_REF][START_REF] Kuhl | Influence of Space Charges on the Impulse Response of InGaAs Metal-Semiconductor-Metal Photodetectors[END_REF]:

N = ୱ ୦ୡሺୱା୪ሻ ሺ1 -rሻሾ1 -expሺ-αdሻሿ (1)
Where d is the thickness of the active layer, α is the light absorption coefficient, r is the reflection coefficient, λ is the wavelength, c is velocity of light in a vacuum, h is the Planck constant, E is the optical pulse of energy. The condition can be expressed by [START_REF] Averine | Effect of optical excitation level on the msm-detector high-speed response, Ultrawideband and Ultrashort Impulse Signals[END_REF]:

E < க బ ሺக ౩ ାଵሻ మ ሺ୪ାୱሻ୦ୡ ସ୯ୱ మ ሺଵି୰ሻሾଵିୣ୶୮ሺିୢሻሿ ( 2
)
Where q is the elementary charge, V is the external bias voltage.

We finally obtain [START_REF] Kuhl | Influence of Space Charges on the Impulse Response of InGaAs Metal-Semiconductor-Metal Photodetectors[END_REF]:

N ୮୦ ≪ େ ୯ (3)
Where, C is the dark capacitance:

C = க బ ሺக ౩ ାଵሻ మ ସୱ (4)
The interdigited MSM photoswitch geometry can be optimized, in order to achieve a fast pulse response, the width of interelectrode gap [START_REF] Averine | Effect of optical excitation level on the msm-detector high-speed response, Ultrawideband and Ultrashort Impulse Signals[END_REF]: Where:

s = ୱା୪ ሾ6.5η ଶ + 1.08η + 2.37ሿ (
η = ୪ ୪ାୱ (6) M = 4.4RL ଶ ሺε ୱ + 1ሻ10 ିଵଶ (7)
χ is carrier driftdistance corrective coefficient V is velocity of ambipolar carrier drift By solving this equation, we found the optimum interelectrode gap:

s ୭୮୲ = ଶ ඥ2.2RV ୱୟ୲ ε ሺε ୱ + 1ሻ (8)
Vsat is the saturation velocity, for GaAs MSM (~105m/s), and R=50Ὡ [START_REF] Averine | Sachot Transit-time considerations in metal-semiconductor-metal photodiode under high illumination conditions[END_REF][START_REF] Averine | Effect of optical excitation level on the msm-detector high-speed response, Ultrawideband and Ultrashort Impulse Signals[END_REF]:

s ୭୮୲ ≈ 0.02L (9)

Conclusion

The impulse response of GaAs MSM photoswitch for different bias voltage and optical power and gap between fingers, for λ=780nm was studied experimentally. The pulse shape of the impulse response is similar, it achieves maximum after fast rise time due to fast electron drift velocity, flowed by fairly fast decay corresponding to the fast escape of electrons and holes to the interdigitated contacts, small Schottky contact spacing permitting rapid carrier extraction after photoexcitation, however, it followed by long tail. High electric field reaches a threshold level, the mobility of electrons decrease as the electric field is increased, due to the screening of internal field; thereby producing negative resistance (NDR). The trapping effect is another feasible way to reduce the carrier transit time. It is advantageous to use thin active layers to reduce the transient response.

As perspective, and for improvement this work, we'll modeled a GaN-based MSM to use it like photovoltaic ultraviolet detector.

Fig. 1 .

 1 Fig. 1. Metal-Semiconducteor-Metal

Fig. 2 .

 2 Fig. 2. The test bench microwave

Fig. 3 .Fig. 4 .

 34 Fig. 3. Pulse response of GaAs MSM312 photoswitch at λ=780µm with different bias voltages (Popt=1mW, A=3x3 µm2, s=l=1µm)

Fig. 5 .

 5 Fig. 5. Normalized amplitude of the temporal response of the MSM3035 at 2,8V to determine the lifetime of electrons and holes.

V

  m a l iz e d a m p l it u d e

Fig. 6 .Fig. 7 .

 67 Fig. 6. Gunn Effect

Table II

 II

		. -Pulse response of GaAs MSM3035
		photoswitch.
	P opt (mW)	RISE TIME	FALL	FWHM[ps]
		[ps]	TIME [ps]
	3	10	340	198
	1	14,8	280	140
	0,5	15	240	90

  [START_REF] Zebentout | Effect of Dimensional Parameters on the Current of MSM Photodetector[END_REF]

		0.08
							MSM 3028 250mV 3mW 100ps
							MSM 3028 250mV 1mW 100ps
		0.07					MSM 3028 250mV 500µW 100ps
		0.06
		0.05
	P h o t o c u rr e n t (m A )	0.03 0.04
		0.02
		0.01
		0
		4.1 -0.01	4.12	4.14	4.16	4.18	4.2	4.22
					Time(s)		x 10 -8


Nizar Emmanuel Defay

Ben Hassine

Patrick Emery

Guy Parat

Julie Abergel

Arnaud Devos

Tunability of aluminum nitride acoustic resonators: a phenomenological approach

Tunability of Aluminum Nitride Acoustic Resonators: A Phenomenological Approach

Emmanuel Defaÿ, nizar ben Hassine, Patrick Emery, Guy Parat, Julie abergel, and arnaud Devos Abstract-A phenomenological approach is developed to identify the physical parameters causing the dc-voltage-induced tunability of aluminum nitride (AlN) acoustic resonators, widely used for RF filters. The typical resonance frequency of these resonators varies from 2.038 GHz at -200 V to 2.062 GHz at +200 V. This indicates, based on these RF measurements versus dc bias and the model used, that the AlN stiffness variation versus dc bias is the prominent effect because both resonance and antiresonance experience a similar variation, respectively, 24 MHz and 19 MHz at 400 V. Picosecond ultrasonics were also used to prove independently that the acoustic velocity (and therefore AlN stiffness) is sensitive to dc bias and that the variation induced is comparable to that extracted from the resonance measurements. It turned out that the stiffness relative variation for an electric field of 1 V/µm extracted from picosecond ultrasonics is 54 ppm•µm/V. This is in good agreement with the value extracted from the RF measurements, namely 57.2 ppm•µm/V. The overall tunability of these AlN resonators reaches 1.1%, which is an interesting figure, although probably not high enough for genuine applications.

a luminum nitride (aln) thin films are widely used for their piezoelectric properties, especially for acoustic in the literature [START_REF] Volatier | switchable and tunable strontium titanate electrostrictive bulk acoustic wave resonator integrated with a bragg mirror[END_REF]- [START_REF] Yamada | Tuning of direct current bias-induced resonances in micromachined ba 0.3 sr 0.7 Tio 3 thin-film capacitors[END_REF]. They all belong to the perovskite material family. Indeed, perovskite materials exhibit frequency agility that can reach 5% under 200 kV/cm dc electric field for Pb(Zr,Ti)o 3 (PZT) thin films [START_REF] Conde | characterization of sol-gel PZT in thin film bulk acoustic resonators[END_REF]. In [START_REF] Conde | characterization of sol-gel PZT in thin film bulk acoustic resonators[END_REF], the authors also reported a k t 2 as high as 9.5%. The main drawback of all perovskite-based resonators is their low Qfactor compared with aln. Indeed, for PZT, typical values are around 50 in the gigahertz range, whereas Q can reach 250 for (ba,sr)Tio 3 (bsT) films [START_REF] Vorobiev | Tunable thin film bulk acoustic wave resonators with improved q factor[END_REF]. Paraelectric bsT exhibits higher Q than ferroelectric PZT because there are no domain walls in the paraelectric phase. Indeed, several studies showed that the domain walls involved in purely ferroelectric materials as PZT can induce high acoustic losses and therefore low Q in resonators [START_REF] Muralt | Is there a better material for thin film baW applications than aln?[END_REF] and lossy coplanar waveguide (cPW) rF lines [START_REF] Defaÿ | Ferroelectric properties of Pb(Zr,Ti)o 3 thin films until 40 GHz[END_REF]. If one considers a trade-off among all of the requirements, the best published results for tunable perovskite based resonators in the gighertz range are Q = 120, k t 2 = 6%, and Δf r /f r = -2% at E dc = 600 kV/cm obtained with bsT [START_REF] Yamada | Tuning of direct current bias-induced resonances in micromachined ba 0.3 sr 0.7 Tio 3 thin-film capacitors[END_REF]. The perovskites tunability s is mainly due to the high sensitivity of their dielectric susceptibility versus the electric field. This induces large variations of the electromechanical coupling versus the dc field. However, note that the antiresonance frequency, f a , of all acoustic resonators (and more generally of all electromechanical resonators) is independent of the electromechanical coupling as described in the IEEE Standard on Piezoelectricty [9, Eq. (60)] and explained in more detail in [START_REF] Defay | Integration of Ferroelectric and Piezoelectric Thin Films[END_REF] for the special case of tunable acoustic resonators. actually, for a piezoelectric layer alone, f a only depends on its thickness, density, and stiffness at constant electric displacement. Therefore, the antiresonance variations are due to the stiffness sensitivity versus the applied dc field, or to couplings other than the piezoelectric coupling. Indeed, in the case of bsT, Vendik et al. proposed the non-linear electrostrictive effect to explain the antiresonance frequency variations of bsT resonators versus the dc field [START_REF] Vendika | modeling tunable bulk acoustic resonators based on induced piezoelectric effect in baTio 3 and ba 0.25 sr 0.75 Tio 3 films[END_REF]. concerning aln tunability, it has been shown by cheng et al. that the positions of the resonances of an aln resonator are shifted by applying a dc voltage together with the rF signal [START_REF] Cheng | Frequency-temperature compensation of piezoelectric resonators by electric Dc bias field[END_REF]. For the antiresonance frequency, the origin of this phenomenon was attributed by lanz et al. to a change in the elastic constant of the piezoelectric material when a bias is applied [START_REF] Lanz | Piezoelectric thin films for bulk acoustic wave resonator applications: From processing to microwave filters[END_REF]. although several authors used a thermodynamic model to explain the resonance frequency variations of perovskite materials (paraelectric or ferroelectric), this method has never been used to explain the tunability of purely piezoelectric material as aln. In this paper, we propose to use this phenomenological approach for aln resonators. We will show that aln tunability is complex, resonators in the gigahertz range. Their main application is rF filters for mobile phones [START_REF] Ella | advancement of mEms into rF-filter applications[END_REF]. The success of aln is related to two main features. First, the electromechanical coupling of aln thickness mode, k t 2 , is around 7%, which fits very well with the standard telecom bandwidth (bW) requirements for wideband code division multiple access (W-cDma). W-cDma typically exhibits 80-mHz bW and a central frequency, f 0 , of 2.14 GHz (bW/f 0 = 3.74%). second, aln exhibits a very low internal viscosity, which induces very high resonance quality factors (Q). The best reported results reach 2500 around 2 GHz [START_REF] Fattinger | Everything you always wanted to know about baW[END_REF]. It is also highly desired for the next generation of filters to exhibit tunable properties. Indeed, adding tunability would allow a dramatic simplification of the rF architectures in use. For instance, a narrow bW tunable filter (a so-called channel filter) could replace the large bW head filter, voltage-controlled oscillator (Vco), and mixer. Therefore, investigating the bias-induced frequency variation of acoustic resonators is a very interesting topic for the next generation of rF filters. Highly tunable resonators in the gigahertz range have already been reported E. Defaÿ, n. ben Hassine, G. Parat, and J. abergel are with cEa lETI minatec campus, Grenoble, France (e-mail: edefay@cea.fr).

P. Emery and a. Devos are with the Institut d' Electronique, de microelectronique et de nanotechnologie, Umr cnrs 8250, Villeneuve d'ascq, France.

because it exhibits many different contributions. Finally, we will prove, by using an independent experiment on the same sample (namely, picosecond ultrasonics), that the stiffness variation is indeed the main contribution to the tunability of aln resonators when a dc bias is applied.

First, the proper variables must be chosen to describe the model. In the case of aln piezoelectric thin-film resonators, the electric displacement D is a good choice for the electrical variable because the antiresonance frequency of the thickness mode is independent of the coupling. This dramatically simplifies the model [START_REF] Defay | Integration of Ferroelectric and Piezoelectric Thin Films[END_REF]. note that this antiresonance behavior is unique and does not apply to the resonance frequency. The strain, S, is chosen as the mechanical variable because thin films are clamped to the substrate. Therefore, in-plane strains are zero, and because we neglect the shear strain, the out-of-plane strain is the only strain remaining, which greatly simplifies the model. Therefore, Helmholtz free energy F, whose natural variables are D and S, appears to be the best choice for such a case. The free energy expansion in the case of perovskites is simplified because only the even terms are considered, for both the symmetrical and the non-symmetrical ferroelectric phases. Indeed, it is supposed that the free energy expression is very similar regardless of the phase of the perovskite considered (paraelectric or ferroelectric). Therefore, because the symmetrical paraelectric phase exhibits the simplest free energy, this is chosen in the non-symmetrical case. However, there is no reason a priori to cancel out the odd coefficients of aln free energy. aln is pyroelectric and, therefore, non-symmetrical. moreover, there is no such thing as a symmetrical prototype phase for the wurtzite phase. In this case, the expansion considered reaches the third degree as written in [START_REF] Ella | advancement of mEms into rF-filter applications[END_REF]. The only dimension considered is in the thickness direction. because the only non-zero terms of S and D are along the thickness direction, the free-energy expansion reduces to

F F c S D h DS GD S NS D OD RS - = + - + + + + 0 0 2 0 2 0 2 2 3 3 1 2 1 2 1 3 1 3 D S β , (1)
where c 0 D , β 0 s , and h 0 are, respectively, the stiffness coefficient at fixed D, impermittivity at fixed S, and the piezoelectric coefficient. G and N are both electrostrictive coefficients. O and R are, respectively, representative of non-linear variations of the impermittivity and of the stiffness. The partial derivatives of F with respect to S and D are, respectively:

∂ ∂      = = - + + + ∂ ∂      = = - + F S T c S h D GD NSD RS F D E D h S G D D S S 0 0 2 2 0 0 2 2 β D DS NS OD + + 2 2
.

(

) 2
let us add a dc field to aln by considering that this dc field is much larger than the ac field. If one only keeps the linear terms, (2) then is

T c ND RS S h GD NS D E h GD NS = + + + -+ + = -+ + ( ) ( ) ( ) 0 0 0 2 2 2 2 2 2 D dc dc dc dc dc dc S S G S OD D + + + ( ). β 0 2 2 S dc dc (3)
because the dc stress in the thickness direction is zero, the dc strain, S dc , is

S h D c dc dc D = 0 0 . (4)
by substituting ( 4) into (3), one finally obtains:

T c N Rh c D S h G Nh c = + +                     + -+ +     0 0 0 0 0 0 2 2 D D dc D                  = -+ +                    D D E h G Nh c D dc D dc 0 0 0 2   + + +                     S Gh c O D D S D dc β 0 0 0 2 . (5)
These equations are actually equivalent to the (S, D) piezoelectric formalism [9] by using the following substitutions (here, we considered that D dc = E dc /β 0 s , which is true for the first order):

h h G Nh c E c c N Rh c E = - +           = + +          
2 = + +           Gh c O E . ( 6
)
This formalism is now used to fit the frequency variations observed on the aln resonators. The standard mason model is used here. Each layer of the stack, simply elastic or piezoelectric, is represented by its acoustic behavior. In this paper, the aln-based resonator is made of a bragg mirror (sin/sioc/sin/sioc) deposited on a 200-mm-thick si substrate. molybdenum bottom and top electrodes are deposited by sputtering, and 1.25-μmthick aln is deposited by dc-pulsed sputtering. Fabrication details are given elsewhere [START_REF] N. Ben Hassine | linear variation of aluminum nitride capacitance versus voltage induced by a piezoelectric-electrostrictive coupling[END_REF]. Electrical tests are performed by adding a dc voltage to the rF signal. because the breakdown electric field of aln is around 5 mV/ cm, the resonators used in this study withstand several hundred volts. The dc bias applied between top and bottom electrodes is varied between -200 and +200 V. a network analyzer is used to measure the S parameters of the aln resonators. The resonance, f s , and antiresonance, f p , frequencies are extracted from these data by measuring the maximum of the real parts of admittance Y and impedance Z, respectively. one can note that f p and f s -identified as the maximum of each curve-experience an increase with the dc bias; respectively, +0.90% and +1.07% between -200 and +200 V. moreover, the order of magnitude of the shift is similar for both f s and f p . This is very different from what can be observed in perovskite materials, such as bsT, for which f s exhibits a shift around 5 times greater than does f p . The frequency difference between f s and f p only depends on the coupling factor k t 2 . Therefore, the coupling variation is much lower in aln than in perovskites. This is confirmed by the extracted data shown in Table I. Indeed, k t 2 experiences a shift of less than 5% versus 400 V bias, whereas it can change by more than one order of magnitude in bsT. Physically, the main difference with perovskites is the weak dielectric constant ε variation. considering the maximum relative variation of ε, it reaches 7% for aln compared with 60% for bsT and 90% for PZT. This is the key to explain the low k t 2 variation. because f p does not depend on k t 2 , its variation is only explained by stiffness variation. because the variation is similar for f p and f s , stiffness variation is the main parameter explaining the frequencies variations. an important point is that the quality factor of f p and f s (Q p and Q s ) is not affected by the bias voltage. This is an important point for tunable applications, for which it is important to have a high Q-factor.

Values extracted from the fitting process are G + (Nh 0 )/(c 0 D ) = 6.2 • 10 9 , N + (Rh 0 )/(c 0 D ) = 1.38 • 10 11 , and (Gh 0 )/(c 0 D ) + O = 1.14 • 10 10 . It is very important to note that every parameter involved in these expressions matters and none of them is negligible. That means that the whole set of third-order terms involved in the free energy described by (1) must be taken into account to perform a successful fit. This is a huge difference compared with perovskites, for which the large dielectric constant variation allows many parameters to be neglected [START_REF] Yamada | Tuning of direct current bias-induced resonances in micromachined ba 0.3 sr 0.7 Tio 3 thin-film capacitors[END_REF].

This model approach gives accurate results. It is also possible to confirm the variation of aln stiffness by performing an independent experiment, using picosecond ultrasonics. The set-up is based on a tunable Ti:sapphire laser. Experimental details can be found elsewhere [START_REF] Devos | strong oscillation detected by picosecond ultrasonics in silicon: Evidence for an electronic structure effect[END_REF]. The absorption of a first optical pulse (the pump pulse) generates a thermal strain pulse whose extension can be less than 10 nm. The resulting wave packet of longitudinal phonons propagates in the sample normally to the surface at the corresponding sound velocity. The reflection at the film/substrate interface yields returning echoes which are detected at the surface by another delayed optical pulse (the probe pulse). by adjusting the delay between pump and probe pulses, it is possible to get a very precise measurement of the round trip time, from which one can deduce the sound velocity or the film thickness [START_REF] Grahn | Picosecond ultrasonics[END_REF].

For these experiments, we prepared the following stack, from the top layer to the substrate: indium tin oxide (ITo; 200 nm)/aln (1.25 μm)/mo (150 nm)/si substrate. all of the layers were sputtered on a 200-mm (100) si substrate. The ITo layer was patterned by wet etching, providing the top electrode of the capacitors. because the aln and mo layers are not patterned, there is no direct access to the bottom mo layer. The dc voltage is therefore applied by connecting the probes to two ITo top electrodes exhibiting very different surfaces (S large = 30 × S small ). These electrodes define two in-series capacitors. because the surface ratio is large, the voltage is mainly applied to the smallest capacitor, which is actually the device under test. The actual dc voltage range applied to the device under test is ±150 V. The area of the smallest capacitor where the laser probe is focused is 0.25 mm 2 . Thus, we measure the round trip time of the optically generated acoustic wave by observing the reflectivity step as seen by the probe pulse. This provides the opportunity to observe how the dc voltage U affects the round-trip time. This reflectivity step corresponds to two different contri- butions as shown in Fig. 2: the reverse of the pulse generated in ITo and detected at the aln/mo interface (path 1 in Fig. 2) and a pulse generated in the mo and detected at the ITo/aln interface (path 2 in Fig. 2). both of these pulses have made one round trip in the ITo and one way through the aln layer. We performed three experiments, at -150, 0, and +150 V. Fig. 2 shows the evolution of the step's temporal position for the three different voltage conditions. one can see that the application of a positive dc voltage induces a decrease of the time of flight in aln, assuming that ITo parameters are not affected by the electrical conditions. The relative variation of the time of flight per volt ΔT/TV is -25 ppm/V. To go further, we have to take into account the thickness change resulting from the piezoelectric effect of aln. The acoustic velocity of the longitudinal wave, V l , is linked to density, ρ, and the stiffness coefficient c D (that is to say, c 33 D in the case of c-oriented aln) by [START_REF] Defay | Integration of Ferroelectric and Piezoelectric Thin Films[END_REF] V c

l D = ρ . (7)
Velocity is thickness t divided by the time of flight T, whereas density is mass m divided by volume Sd (where S is the top electrode surface area). Therefore, c D can be written as

c mt ST D = 2 . (8)
because the variations are very small, one can obtain the stiffness variation versus the dc bias by a basic differential calculation, which gives

d d d D D c c t t T T = -2 . (9)
Here, we consider that the mass does not change. moreover, the surface is also kept constant as the film is clamped on the si substrate. The term dt/t is the out-ofplane strain resulting from the piezoelectric effect, which is d 33 E. because E = -U/t, ( 9) can be written

d d D D c c d U t T T = - - 33 2 . ( 10
)
It is important to know the sign of d 33 , which depends on the direction of aln polarization P. In a previous study, P was observed to point in the downward direction, indicating a negative sign for d 33 [START_REF] N. Ben Hassine | linear variation of aluminum nitride capacitance versus voltage induced by a piezoelectric-electrostrictive coupling[END_REF]. This induces a thickness increase when a positive bias is applied, because the electric field is also in the downward position when U > 0. Finally, by using [START_REF] Defay | Integration of Ferroelectric and Piezoelectric Thin Films[END_REF], the relative stiffness variation per volt is calculated to be 54 ppm/V by using d 33 = -4.7 ppm/V obtained from the resonator extraction. note that the piezoelectric term is only 4 ppm/V. because the stiffness variation is an intrinsic behavior of the material, this value is usually reported at a constant field by multiplying the results obtained by the thickness. The stiffness relative variation for an electric field of 1 V/μm is therefore 54 ppm•μm/V. This result can be compared with the stiffness variation extracted from the antiresonance frequency, f a , shift. Indeed, f a is linked to thickness t, stiffness at fixed D, c D , and density, ρ, by

f t c c S mt a D D = = 1 2 1 2 ρ , (11)
neglecting the electrode and bragg mirror influence [9], [START_REF] Defay | Integration of Ferroelectric and Piezoelectric Thin Films[END_REF]. by adopting the same differential calculation (also valid without taking into account electrodes and bragg mirror because they are supposed to be unaffected by bias voltage) one obtains

d d D D a a c c d U t f f = - + 33 2 . ( 12
)
The measured relative variation of f a versus voltage is 21 ppm/V. by using d 33 = -4.7 pm/V and by reporting the variation for a 1-μm-thick film, [START_REF] Cheng | Frequency-temperature compensation of piezoelectric resonators by electric Dc bias field[END_REF] gives 57.2 ppm•μm/V for the relative variation of stiffness for an electric field of 1 V/μm. This value is in very good agreement with the value obtained by picosecond ultrasonics. lanz et al. reported f a variation reaching 72 ppm/V for 380-nm-thick aln film [START_REF] Lanz | Piezoelectric thin films for bulk acoustic wave resonator applications: From processing to microwave filters[END_REF]. by using this result in [START_REF] Cheng | Frequency-temperature compensation of piezoelectric resonators by electric Dc bias field[END_REF] and supposing that d 33 is identical to ours, the relative stiffness variation is calculated to be 59.4 ppm•μm/V. This result is also in very good agreement with the value obtained by picosecond ultrasonics and confirmed the hypothesis proposed by lanz that the antiresonance frequency variation versus bias is mainly due to the stiffness variation experienced by aln. as a final remark, note that we identified the stiffness coefficient c as observed by picosecond ultrasonics with c D as characterized from the baW antiresonance. although the electrical conditions exhibited by picosecond ultrasonics are not very clear yet, this identification is mainly justified by the similar resonance and antiresonance variations. The coupling coefficient k t 2 reaches 7% in state-of-the-art aln thin films, suggesting that the electrical conditions should not dramatically change the stiffness behavior of these films. This statement should obviously be reconsidered in the case of highly coupled perosvkite materials such as PZT or bsT.

We can discuss these results in terms of applications and, more specifically, rF tunable resonators and filters. Indeed, if we take the results in Fig. 1, the maximum variation of the resonance and antiresonance frequencies obtained by comparing the frequencies at -200 and +200 V is typically 1%. because the variation of the electromechanical coupling is also weak, the bandwidth of an rF filter made of these resonators would be almost constant. For rF filters, the ability to change the central frequency without changing the bandwidth is an interesting feature. However, the minimum desired frequency variation is around 3.8%, as discussed in the introduction. This order of magnitude is almost, but not completely, reached. another obvious drawback for applications is the huge voltage (400 V) required to promote this tuning. This is not affordable in a mobile phone. However, note that to generate the tuning, only the electric field matters. Therefore, decreasing the thickness of the active layer is probably something to consider, although it will increase the resonance frequency accordingly. a trade-off is perhaps achievable. It is also interesting to compare these results with the temperature variations experienced by the resonance frequencies, that is to say, the temperature coefficients of frequency (Tcf) of these resonators. We obtained Tcf p = -23 ppm/°c and Tcf s = -21 ppm/°c [START_REF] Hassine | Etude de la fiabilité des composants à ondes acoustiques de volume (baW) pour des applications radio fréquence[END_REF]. Therefore, if one considers that the whole temperature range is typically 100°c, the frequency variations reach approximately -0.2%, compared with roughly +1% for the maximum dc bias variation. It is worth noting that these resonators are not compensated with respect to temperature. Very efficient temperature compensation can be achieved by adding silicon dioxide in the resonator stack. such compensated aln resonators integrated in an oscillator exhibit a temperature drift of better than 40 ppm throughout the entire 120°c temperature range [START_REF] Petit | Temperature compensated baW resonator and its integrated thermistor for a 2.5GHz electrical thermally compensated oscillator[END_REF]. as a final remark, at this stage, the most interesting outcome of this study for applications is probably the following. There is a way to change the stiffness of an acoustic resonator that induces a frequency variation without notably affecting its bandwidth or its quality factor. That is to say, being able to vary stiffness is probably a very good way to achieve efficient tunable acoustic resonators.

acknowledgments

The authors thank G. caruyer, a. Volatier, and c. Zinck for fruitful discussions at the early stages of this study. references

  Fig. 1 shows (a) re(Y) and (b) re(Z) versus frequency in the vicinity of the thickness mode resonance versus the dc bias. Each curve is represented together with its fitting curve, as obtained from the mason model.

Fig. 1 .

 1 Fig. 1. (a) real part of the admittance versus dc bias applied between -200 and +200 V, (b) real part of the impedance versus dc bias applied between -200 and +200 V. all curves are represented with their fit as obtained from the mason model.

Fig. 2 .

 2 Fig. 2. (left) cross-section of the sample and acoustic paths of the pulses that are responsible for the studied reflectivity step. (right) magnification of the studied reflectivity step detected at three different dc voltages (+150, 0, -150 V).

TablE I .

 I Extracted Parameters of the model obtained by Fitting the Experimental Data shown in Fig. 1.

	V bias (V)	-200	0	200
	c D (n/m 2 )	3.83•10 11	3.87•10 11	3.91•10 11
	k t 2	0.059	0.0577	0.0563
	ε s	10	9.7	9.3


A Telia
email: a_telia@hotmail.fr

A Bellakhdar

L Semra

A Soltani

Strain relaxation and thermal effects on the drain conductance in AlGaN/GaN HEMT

Keywords: GaN, HEMT, AlGaN/GaN, drain conductance, self heating. I

In this work the effects of technological and electrical parameters of Al m Ga 1-m N/GaN High Electron Mobility Transistors (HEMT) on the output conductance "g d " was studied in the first part. In the second part, the effect on g d of thermal and self heating in the device was analyzed. This study was made by taking into account the effects of spontaneous and piezoelectric polarizations in the heterojunction AlGaN/GaN and the relaxation of the crystal lattice induced by the aluminum mole fraction. An analytical model for the electrons concentration "n s " in the 2DEG and the current I ds in the channel for strong inversion regime was presented. By solving the Poisson equation and Schrödinger self-consistent calculations, including the spontaneous and piezoelectric polarizations, lattice relaxation, self heating and thermal effects the current I ds was calculated. The output conductance g d according to the voltage V ds are deduced thanks to the analytical description of the calculated current I ds .

INTRODUCTION

High-electron-mobility transistors (HEMTs) based on Al m Ga (1-m) N/GaN heterostructures are very promising for applications in high power and high temperature electronics operating at high frequencies owing to their outstanding electrical properties such as high breakdown voltage, high peak electron velocity and high sheet electron concentration in two-dimensional electron gas (2DEG) structures [START_REF] Wu | Very high breakdown voltage and large transconductance realized on GaN heterojunction field effect transistors[END_REF][START_REF] Miyoshi | High-electronmobility AlGaN/AlN/GaN heterostructures grown on 100mm-diam epitaxial AlN/sapphire templates by metalorganic vapor phase epitaxy[END_REF]. Performance of HEMTs can be estimated from some important electrical quantities that characterize them and which are good indicators for the performance of high frequency component. They must therefore have a high transconductance, low output conductance and a high cutoff frequency.

The output conductance characterizes a transistor in the saturated operation. It measures changes in the drain current relative to changes in drain voltage V ds , with constant gate voltage V gs . Calculating the power gain of the component requires knowledge of the output conductance, which allows, among other to account for the saturation drain current. From a physical standpoint, the drain conductance g d characterizes the confinement of electrons in the channel.

In this paper, we presented an analytical model for the electrons concentration "n s " in the 2DEG and the current I ds in the channel of HEMT for strong inversion regime by solving the Poisson equation and Schrödinger self-consistent calculations. Thanks to the analytical description of the current I ds we are able to demonstrate changes in the output conductance g d according to the voltage V ds , including the spontaneous and piezoelectric polarization, lattice relaxation, self-heating and thermal effects on the Al m Ga (1-m) N/GaN HEMT drain conductance g d .

II.

THEORETICAL ANALYSIS Fig. 1 shows the cross-sectional schematic of the AlGaN/GaN HEMT structure. The device structure consists of a thick undoped Gallium Nitride (GaN) layer grown on a substrate (SiC, Sapphire or Al 2 O 3 ) where an undoped layer Al m Ga 1-m N of thickness (d i ) called spacer was grown in order to reduce impurity scattering. Then, a doped layer Al m Ga 1-m N of thickness (d d ) and concentration (N d ) was grown. On the last layer, the gate was realized. When a gate voltage V gs is applied through a Schottky contact of this structure, the sheet carrier concentration "n s " of the 2DEG formed at the AlGaN/GaN heterojunction at a given temperature is given by [START_REF] Gangwania | Modeling and analysis of fully strained and partially relaxed lattice mismatched AlGaN/GaN HEMT for high temperature applications[END_REF].

( ) ( ) ( ) ( ) ( ) ( ) , , , s g s t h c d i m n x T m V V T m V x q d d d ε = - - + +Δ
Where q is the electron charge, m is the Al

mole fraction in Al m Ga 1-m N, ε(m) is the AlGaN dielectric constant, d d is the AlGaN barrier Undoped GaN n -AlmGa1-mN Undoped AlmGa1-mN Substrate (SiC,Al2O3,Si) dd di L 2DEG (1) 1 1 Figure 1. Schematic AlGaN/GaN HEMT structure used
thickness, d i is the spacer layer thickness, d Δ is the effective thickness of 2-DEG, V gs is the applied gate-source voltage and V c (x) is the channel potential at a position x. The threshold voltage V th (T,m) dependant of the temperature T and the mole fraction of the AlGaN/GaN HEMT is given by the following expression:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , , 2
d d th m C d i m qN d V T m m E T m d d m m σ φ ε ε = -Δ - - +
With N d is the doping density of AlGaN layer and φ m) the Schottky barrier height given by [START_REF] Ambacher | Two dimensional electron gases induced by spontaneous and piezoelectric polarization undoped and doped AlGaN/GaN heterostuctures[END_REF].

φ (m) = 0.84+1.3m [eV]
(3) and ΔE c (T,m) is the conduction band discontinuity at the Al m Ga 1-m N/ GaN interface given by [START_REF] Manju | Temperature and polarization dependent polynomial based non-linear analytical model for gate capacitance of Al mGa1_mN/GaN MODFET[END_REF]:

( ) , 0.70 ( , ) ( ) AlGaN GaN C g g E T m E T m E T ⎡ ⎤ Δ = - ⎣ ⎦ [eV]
The major factor which controls 2-DEG is the aluminium composition of AlGaN. The variation of Al mole fraction affects directly the polarization induced sheet charge density. The polarization induced charge density σ pz (m) is given by [START_REF] Ambacher | Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N-and Ga-face AlGaN/GaN heterostuctures[END_REF].

( ) ( )

( ) 1 (
3 3 1 3 3 1 3 3 3 3 1 3 (1 ) ( ) ( ) ( )
- ⎧ ⎛ ⎞ - ⎛ ⎞⎛ ⎞ × - → ≤ ≤ ⎪ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎛ ⎞ ⎛ ⎞ ⎪ ⎜ ⎟ ⎜ ⎟ - ⎛ ⎞ ⎪ ⎜ ⎟ ⎜ ⎟ = × - → ≤ ≤ ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎪ ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ ⎝ ⎠ → ≤ ≤ ⎩ ⎪ ⎪ ⎪ ⎪
Where a(0) and a(m) are the lattice constants, C 13 (m) and C 33 (m) are elastic constants and e 31 (m) and e 33 (m) are piezoelectric constants of Al m Ga l-m N layer. In the above expression, it has been assumed that GaN layer is fully relaxed. This is a reasonable assumption since the thickness of GaN is much larger than that of strained AlGaN layer and thus P pz (GaN)~0. The drain current I ds in the 2-DEG channel can be obtained from the current density equation by [START_REF] Sze | Physics of Semiconductor Devices[END_REF]:

( ) ( ) ( ) ( ) ( ) , , , , , , , C s ds s B dV x n T m x dx I T m x wq T x dn T m x K T q d x μ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ + ⎜ ⎟ ⎝ ⎠
Where w is the channel width, K B is the Boltzmann constant, V c (x) is a channel potential and ( ) , T x μ is the temperature dependent mobility given by [START_REF] Gangwania | Modeling and analysis of fully strained and partially relaxed lattice mismatched AlGaN/GaN HEMT for high temperature applications[END_REF]:

( ) ( ) 0 1 ( ) , 1 1 C T T x dV x E dx μ μ = + 1 0 ( ) c sat c s a t E v E T E v μ = -
Where μ 0 (T) is the low field mobility, E c is the critical electric field and sat v is the saturation velocity. Using equations ( 1) and ( 8) in equation ( 7), and integrating along the channel we can obtain:

( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 3 1 , , 4 , , , 2 ,

ds A T m A T m A T m A T m I T m

A T m

- + - = With ( ) ( ) ( ) ( ) ( ) ( ) 0 1 2 0 2 2 ( ) , 2
( ) 2 2 ( ) c s a t s d c sat d s d d i T E v A T m R R E v w T q D m R R R m q D d d μ μ ε ε ⎛ ⎞ - = + ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ - + ⎜ ⎟ + + ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 2 0 ' 2 2 0 2 ( ) , , 2 2 ( ) (
)

ds s d gs s d d i c s a t ds c sat w T q D m A T m V R R V T m R R m q Dd d T E v L V E v μ ε ε μ ⎛ ⎞ ⎜ ⎟ = +- + ⎜ ⎟ + + ⎝ ⎠ ⎛ ⎞ - --⎜ ⎟ ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) 2 2 0 ' 3 2 2 ( ) , , 2 2 ( )
ds gs ds d i w T q D m V A T m V T m V m q D d d μ ε ε ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = - ⎜ ⎟ ⎜ ⎟ + + ⎝ ⎠ ⎝ ⎠ ( ) ( ) ' , , B gs gs th K T V T m V V T m q = - -
Where R s and R d are the source and drain resistances respectively and V ds is drain to source voltage.

(10)

(11) (12) (13)
is Planck constant. The drain conductance is an important microwave parameter that determines the maximum voltage gain attainable from the device. The drain conductance of the Al m Ga 1-m N/GaN HEMT is evaluated as [START_REF] Gangwania | Modeling and analysis of fully strained and partially relaxed lattice mismatched AlGaN/GaN HEMT for high temperature applications[END_REF]:

( ) ( ) , , gs ds d V ds I T m g T m V ∂ = ∂ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 3 2 2 2 2 1 3 2 1 3 1 , 2 , , / , / 1 1 , , 4 , , , 4 , , 2
,

ds ds d A T m A T m A T m V A T m V g T m A T m A T m A T m A T m A T m A T m A T m ⎛ ⎞ ⎛ ⎞ ∂ ∂ ⎜ ⎟ ⎜ ⎟ ∂ ∂ -- ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ - - ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠ With: ( ) ( ) ( ) ( ) ( ) ( ) 1 2 0 3 ' , 0 , ( ) , ds C s a t s d ds C sat gs ds ds A T m V A T m T E v A R R V E v A T m A V m V V ∂ = ∂ ∂ ⎛ ⎞ μ - = + -⎜ ⎟ ∂ ⎝ ⎠ ∂ = - ∂ ( ) ( ) ( ) 2 0 2 2 ( ) 2
( )

d i w T q D m A m q D d d μ ε ε = + +
The electrons mobility depends strongly on temperature and electric field present which it decreases with increasing temperature.

III. RESULTS AND DISCUSSION

Fig. 2 shows the dependence of drain-source current (I ds ) versus drain-source voltage (V ds ) for Al mole fraction of AlGaN varying from 0.20 to 0.75 at room temperature (300 K). We note that the current I ds increase when the value of the mole fraction "m" increase. The current I ds depend strongly on the Al content of the top layer. This is essentially due to the increasing of the channel electron density induced by the enhanced polarization charge when an Al contents vary from 0.20 to 0.75 [START_REF] Sacconi | Spontarneous and piezoelectric polarization effects on the output characteristics of AlGaN/GaN heterostructure modulation doped FETs[END_REF]. The increase in aluminum mole fraction induces a strong discontinuity in the conduction band and a strong piezoelectric polarization, this is due to an increase in the concentration of 2DEG in the channel, and therefore it induces a large current value. . We observe that the drain conductance increases with an increase in the Al mole fraction and decreases with an increase in drain voltage. Increasing the aluminum mole fraction in the heterojunction AlGaN/GaN induces a strong discontinuity in the conduction band and a strong piezoelectric polarization. The discontinuity of the conduction band ΔE C resulting from this phenomenon is more important leading to electron confinement. The electric field breakdown becomes higher in the GaN based devices. Therefore, the carrier densities and mobility are higher, so the output conductance increases. On the other hand, we notice that the output conductance decreases when V ds increases and eventually becomes zero in the saturation region when the drain-source current saturates. We notice that for m ≥ 0.38, the relaxation effect of the crystal lattice decreases the output conductance. The difference between the curves with and without relaxation is very large when the crystal lattice is fully relaxed (m = 0.75) and less important when the AlGaN layer is partially relaxed (m = 0.45). Indeed, for m ≥ 0.38, the material is relaxed and the 2DEG density continues to increase, which affects the drain conductance g d . The influence of relaxation is greater for higher values of molar fraction (m ≥ 0.38). This is a consequence of the lack of influence piezoelectric polarization due to the disappearance of the mechanical stress on the AlGaN layer. We note that the curves of the drain conductance g d for m=0.25 (Fig 5-a) are superimposed in both cases with and without relaxation of the crystal lattice for each value of the concentration of the doping layer Al m Ga 1-m N. While for m = 0.45 (Fig 5-b), a significant difference is observed, this gap decreases the value of g d . We note that if the concentration of doping N d increase the drain conductance g d increases. This is due to the increased density of 2DEG in the quantum well, so there is an increase of spontaneous polarizations and piezoelectric which exists at the interface of the AlGaN/GaN heterojunction and improved containment of electrons into the well potential.

Fig. 6 shows the transfer characteristics of an AlGaN/GaN HEMT on the gate-source voltage when considering the thermal effect and the effect of self-heating. A degradation of drain-source current with increasing temperature of 300 K to 475 K was observed, this degradation becomes important when one considers the effect of self-heating. Fig. 7 shows the current-voltage characteristics of a HEMT based heterostructure AlGaN/GaN when considering the thermal effect and the self-heating (dashed lines). These characteristics show a negative resistance at higher drain voltage level. It is observed that, with increasing temperature from 300K to 425K, explicitly from an external source or by injecting more current in the transistor, a drain current decay. This is due to a decrease in the low field mobility of electrons where the electron transport in 2DEG channel is dominated by scattering with phonons. The self-heating is a local increase of crystal temperature due to dissipated Joule electric power. It is seen that the saturation drain current drops significantly when the temperature rises above room temperature from 300K to 425K. Also the saturation drain current does not vary much with the higher drain voltages.

Our results can be compared with those obtained by Gangwani et al. [START_REF] Gangwania | Modeling and analysis of fully strained and partially relaxed lattice mismatched AlGaN/GaN HEMT for high temperature applications[END_REF]. The striking constant behaviour of drain current suggests the usefulness of these devices for high power applications [START_REF] Parvesh | High Temperature Performance of AlGaN/GaN HEMT[END_REF][START_REF] Mosbahi | Electrical chacterization of AlGaN/GaN HEMTs on Si substrate[END_REF]. Drain conductance g d is deduced from the current I ds . When self-heating is not considered (solid lines) and when selfheating is considered (dashed lines).

Fig. 8 shows the drain conductance versus the drain-source voltage when considering the thermal effect and the self-heating (dashed lines). Then a strong increase in temperature significantly alters the performance of transistors. View the decrease in mobility by increasing the temperature from 300K to 425K, the saturation current decreases then increases the drain conductance.

IV. CONCLUSION

In this work, we have presented the simulation results obtained by studying the influence of different technological and electrical parameters, thermal and self-heating on the electrical characteristics determining the performance of the HEMT-based heterostructure AlGaN/GaN as the output conductance with and without the effect of relaxation of the crystal lattice in the HEMT AlGaN/GaN.

We note that, the increasing of the aluminum mole fraction "m" and concentration "N d " increase the carrier density "n s " in the 2DEG. Although, the increase of the doped layer thickness increases the output conductance. The influence of the crystal lattice relaxation on the electrical performance acts directly on the threshold voltage if the value of the mole fraction exceeds 0.38. We noted that when the temperature increases, the electron mobility decreases, this is due to a deterioration of transport properties which is dominated by the scattering with phonons. This decrease in mobility leads to a drop in drain current therefore a degradation of electrical characteristics that determine performance of the transistor.

=

  is conduction band density of state of 2DEG system, m* is effective mass of electron and h

Figure 2 .

 2 Figure 2. Variation of drain current Ids with voltage Vds at different Al mole fractions m for Vgs= -1.5V.

Fig. 3

 3 Fig. 3 shows, at room temperature, the variation of the Al m Ga 1-m N/GaN HEMTs output conductance versus drain voltage at gate voltage fixed at -1.5 V for different Al mole fractions, for partially relaxed (PR) 0.38 m ≥ and fully strained (FS) structures 0.38 m ≤. We observe that the drain conductance increases with an increase in the Al mole fraction and decreases with an increase in drain voltage. Increasing the aluminum mole fraction in the heterojunction AlGaN/GaN induces a strong discontinuity in the conduction band and a strong piezoelectric polarization. The discontinuity of the conduction band ΔE C resulting from this phenomenon is more important leading to electron confinement. The electric field breakdown becomes higher in the GaN based devices. Therefore, the carrier densities and mobility are higher, so the output conductance increases. On the other hand, we notice that the output conductance decreases when V ds increases and eventually becomes zero in the saturation region when the drain-source current saturates. We notice that for m ≥ 0.38, the relaxation effect of the crystal lattice decreases the output conductance. The difference between the curves with and without relaxation is very large when the crystal lattice is fully relaxed (m = 0.75) and less important when the AlGaN layer is partially relaxed (m = 0.45). Indeed, for m ≥ 0.38, the material is relaxed and the 2DEG density continues to increase, which affects the drain conductance g d . The influence of relaxation is greater for higher values of molar fraction (m ≥ 0.38). This is a consequence of the lack of influence piezoelectric polarization due to the disappearance of the mechanical stress on the AlGaN layer.

Figure 3 .

 3 Figure 3. Variation of drain conductance as a function of drain-

Figure 5

 5 Figure 5(a,b). variation of drain conductance as a function of drain-source voltage at different doping density « ND » with and without relaxation. relaxation for m=0.25 and m=0.45 respectively.

Figure 6 .Figure 7 .

 67 Figure 6. the transfer characteristics of HEMT with and without effect of self-heating and thermal effect for different temperature

Figure 8 .

 8 Figure 8. Dependence of drain conductance gd on drain voltage Vds for various values of temperature for Vgs= -2.5V and m= 0.25.When self-heating is not considered (solid lines) and when selfheating is considered (dashed lines).