F Medjdoub
email: farid.medjdoub@uni-ulm.de

M Alomari

J.-F Carlin

M Gonschorek

E Feltin

M A Py

N Grandjean

E Kohn

Barrier layer scaling of InAlN/GaN HEMTs

Keywords: InAlN barrier, HEMT, GaN-based, scaling, Microwave devices

We discuss the characteristics of In0.17Al0.83N/GaN High Electron Mobility Transistors (HEMTs) with barrier thicknesses between 33 nm and 3 nm, grown on sapphire substrates by MOCVD. The maximum drain current (at VG = +2.0 V) decreased with decreasing barrier thickness due to the gate forward drive limitation and residual surface depletion effect. Full pinch-off and low leakage is observed. Even with 3nm ultra thin barrier the heterostructure and contacts are thermally highly stable (up to 1000°C).

I. INTRODUCTION

Many high speed applications require devices with high power, high efficiency and high linearity at mmwave frequencies and beyond. In this field, GaAs and InP-based HEMTs with gatelengths below 100 nm have been used employing gate recess [START_REF] Streit | High performance Wband InAlAs-InGaAs-InP HEMT[END_REF][START_REF] Medjdoub | InP HEMT downscaling for power applications at W band[END_REF], however limited in power by their breakdown strength. On the other hand GaN-based HEMTs with AlGaN barriers have enabled more than 1 order of magnitude higher power densities.

Nevertheless, it has been difficult to extend the frequency of operation into V-band (50-75 GHz) [START_REF] Quay | AlGaN/GaN HEMTs on SiC: towards power operation at V-band[END_REF] and W-band (75-111 GHz) [START_REF] Micovic | GaN HFET for W-band power applications[END_REF] due to their difficult technology. To obtain very high frequency performance, it is necessary to shrink the gate length while maintaining the structural aspect ratio of gate length to gate to channel separation [START_REF] Awano | Short-channel effects in subquarter-micrometergate HEMTs: simulation and experiment[END_REF][START_REF] Jessen | Short-Channel Effect Limitations on High-Frequency Operation of AlGaN/GaN HEMTs for T-Gate Devices[END_REF]. However in planar AlGaN/GaN devices, the channel sheet carrier density starts to decrease already at a barrier thickness of around 20 nm [START_REF] Tilak | Influence of Barrier Thickness on the High-Power Performance of AlGaN/GaN HEMTs[END_REF][START_REF] Higashiwaki | Barrier Thickness Dependence of Electrical Properties and DC Device Characteristics of AlGaN/GaN Heterostructure Field-Effect Transistors Grown by Plasma-Assisted Molecular-Beam Epitaxy[END_REF]. Consequently, surface depletion of the interfacial induced polarization occurs for thin barriers due to the location of the polarization dipole countercharge on the surface in a deep level (often referred as surface donor [START_REF] Ibbetson | Polarization effects, surface states, and the source of electrons in AlGaN/GaN heterostructure field effect transistors[END_REF]). This level is approx. 1.6 eV below the conduction band edge in the case of AlGaN/GaN with 30 % Al-content. The deep surface donor level can only be removed by surface passivation like with Si-Nitride [START_REF] Hyungtak | Effects of SiN passivation and high-electric field on AlGaN-GaN HFET degradation[END_REF] and indeed 10 nm barrier thickness AlGaN/GaN MISHEMTs have been demonstrated using a 2 nm SiN dielectric [START_REF] Higashiwaki | AlGaN/GaN MIS-HFETs with fT of 163 GHz using cat-CVD SiN gate-insulating and passivation Layers[END_REF]. Recently, 3.5 nm barrier highly strained AlN/GaN MISHEMTs have been demonstrated with an extremely high current level [START_REF] Cao | Ultrathin MBE-Grown AlN/GaN HEMTs with record high current densities[END_REF]. However, these approaches rely on the deposition of an amorphous defect free dielectric overlayer with high dielectric strength and are therefore difficult to realize. These problems have prevented the use of gate lengths smaller than 100 nm in planar devices without apparition of short channel effects. Another possibility to obtain a high HEMT aspect ratio is the use of a gate recess.

However, AlGaN possesses a ceramic-like stability and cannot easily be etched with wet solutions. However, dry etching of the strained barrier layer may generate defects. In fact defects in the strained barrier layer have been found one of the main limiting factors in reliability [START_REF] Joh | Mechanism for electrical degradation of GaN High Electron Mobility Transistor[END_REF]. Thus, although AlGaN/GaN recess techniques have been developed [START_REF] Chini | Power and linearity characteristics of field-plated recessed-gate AlGaN-GaN HEMTs[END_REF], planar approaches may still be more robust.

Recently, In0.17Al0.83N has been proposed as an alternative to AlGaN barrier layers [START_REF] Kuzmik | Power Electronics on InAlN/(In)GaN: Prospect for a Record Performance[END_REF]. In this case a high interfacial sheet charge density of 2.8 x 10 13 cm -2 has been predicted and has indeed been verified by CV and Hall measurements [START_REF] Gonschorek | High electron mobility lattice matched AlInN/GaN field-effect transistor heterostructures[END_REF] and by the analysis of HEMT devices with output current densities above 2 A/mm [START_REF] Medjdoub | Can InAlN/GaN be an alternative to high power / high temperature AlGaN/GaN devices?[END_REF].

Hall measurements on heterojunctions with various barrier thicknesses have indicated a constant sheet charge density down to approx. 10 nm barrier thickness and even at a barrier thickness of 5 nm a sheet charge density of 1.6 x 10 13 cm -2 is observed [START_REF] Medjdoub | Barrier layer downscaling of InAlN/GaN HEMTs[END_REF]. These results have encouraged this investigation of barrier thickness scaling of InAlN/GaN HEMT structures.

II. MATERIAL GROWTH AND DEVICE FABRICATION

The heterostructures were grown by Metal Organic Chemical Vapor Deposition on sapphire substrates.

The structures consisted of a 2 μm thick GaN buffer, a 1.0 nm thick AlN spacer and various In0.17Al0.83N barrier layer thicknesses. The total barrier thicknesses (InAlN barrier and AlN spacer) have been between 33 nm and 3.0 nm. The heterostructures have been characterized by HRXRD (High Resolution X-Ray Diffraction). The 3.0 nm barrier thickness is not detectable by HRXRD, which on the other hand indicates the growth of an extremely thin InAlN film. The sheet carrier densities (NS Hall) of the different layers resulting from Hall measurements are shown in Fig. 1. The HEMT structures were fabricated as follows: Mesa isolation was performed by dry etching using argon plasma. Ti/Al/Ni/Au metal stack annealed at 870°C for 30 s has been used for the ohmic contacts.

The drain-source distance was 3 μm. Ni/Au Schottky gates with 0.25 µm footprint were defined by e-beam lithography and placed asymmetric between source and drain, 1 μm apart from source contact.

III. SURFACE BARRIER PROPERTIES

The Hall measurements indicate a reduction in channel sheet charge density below 10 nm barrier thickness. To understand this behaviour, a model of the InAlN surface properties has been developed, especially the location and density of surface traps. The Atlas simulation tool was used to simulate the sheet charge density as a function of the barrier thickness, not distinguishing between the 1 nm AlN spacer layer and the InAlN barrier layer to simplify modelling (no influence of the AlN smoothing layer on the sheet charge density has been observed experimentally). We have considered three cases: (1) a high (infinite) density of surface traps comparable to the case of the GaAs surface, (2) a deep donor sheet charge density located near the surface equal to the interfacial polarization dipole charge corresponding to the surface donor model of AlGaN [START_REF] Ibbetson | Polarization effects, surface states, and the source of electrons in AlGaN/GaN heterostructure field effect transistors[END_REF] and (3) a fully ionized surface sheet charge density equal to the interfacial polarization dipole charge in combination with a discrete surface state level (donor-like traps) of identical density, located on the surface. Case (1) would result in an accessible channel sheet charge density decreasing with barrier thickness. This is not observed. Case [START_REF] Medjdoub | InP HEMT downscaling for power applications at W band[END_REF] only results in the full undepleted sheet charge density for thin barriers, if the surface donor level is located within the conduction band. In this case the polarization countercharge would fully obey the semiconductor carrier distribution statistics, which is not obvious for example for adsorbed molecules. In case (3) the surface potential is only dominated by the surface state properties, where this state could be also considered as a chemical redox couple on the free surface. This third case is illustrated with Fig. 1. Fitting the surface potential (φs) dependence on barrier thickness with the experimental data, reveal a surface state level between 0.4 and 0.6 eV below the conduction band edge (in this case 0.6 eV). This value is considerably lower than the one discussed in [START_REF] Ibbetson | Polarization effects, surface states, and the source of electrons in AlGaN/GaN heterostructure field effect transistors[END_REF]. In consequence planar InAlN/GaN HEMTs can be fabricated with barrier thicknesses down to the tunnelling thickness in a well controlled way and can be evaluated even before passivation.

IV. DEVICE CHARACTERISTICS

The dependence of maximum HEMT output current (IDSS) on barrier thickness shown in fig. 2 V as can be extracted from the extrapolation in fig. 3. In the insert (Fig. 2) the FET output characteristics are shown for the 0.25 µm gatelength FETs (w = 50 µm) with 15 nm and 3.0 nm total barrier thicknesses. For thick barriers IDSS is approx. 1.8 A/mm and decreases to 0.6 A/mm for the 3.0 nm barrier. It can be seen that current compression is not seen in the thin barrier device prior to forward gate breakdown. Thus, for thin barriers IDSS is gate barrier limited and not NS limited. After passivation by PECVD Si3N4 at 300 °C the saturation current IDSS at VG = +2 V is increased, especially for the thin barrier devices. This may again be a consequence of a reduction or removal of the surface potential φs in the access regions together with a modification of the field redistribution at the point where the gate metal, the barrier layer surface and the passivation layer meet.

Scaling of the barrier thickness could not easily be controlled down to 3 nm by HRXRD and verification would be needed by TEM. However, scaling the FET pinch-off voltage as shown in Fig. 3 results in a linear relationship for all data points, confirming the growth data. After passivation, a maximum output current of 0.9 A/mm has been obtained in conjunction with pinch-off voltage of VP = -1.0 V for the 3.0 nm barrier device. Enhancement mode of operation would be reached for a total barrier thickness of 2.0 nm.

V. THIN BARRIER THERMAL STABILITY

Even at the smallest barrier thickness the stability of the barrier layer and contact/barrier layer interfaces can tolerate high processing temperatures up to 1000 °C, promising potentially high robustness of these ultra thin barrier devices. In the stress experiment (in vacuum), the samples have been heated in intervals of 100°C and kept at each temperature for 30 min. The temperature has been determined as described earlier [START_REF] Kuzmik | Power Electronics on InAlN/(In)GaN: Prospect for a Record Performance[END_REF]. At each temperature step the sample has been cooled down to Room Temperature (RT) and data were taken to evaluate permanent degradation. Temperatures above 1000°C were not possible because of Au-melting. The room temperature DC characteristics after cycling and final heating for 30 min at 1000°C are shown in Fig. 4. In spite of the extensive thermal stressing, the gate forward drive capability, the open channel current and pinch voltage are preserved. No gate sinking has occurred. Thus, the metallurgical interface properties and the properties of the InAlN/AlN barrier layer stack have not been degraded and ohmic as well as Schottky contacts have also not suffered. Thus, stable interface compounds have been formed. Their analysis will have to be subject of an additional materials oriented investigation. As shown in fig. 4, the reverse Schottky barrier leakage after temperature stress is almost identical to the virgin one. In contrast, Ni/Au Schottky barrier contact on Al0.3Ga0.7N/GaN devices with 25 nm barrier thickness show ohmic behaviour after such a temperature stress experiment.

VI. CONCLUSION

Barrier layer scaling of InAlN/GaN HEMTs has been studied for the first time down to 3 nm. The results indicate that InAlN/GaN HEMT device structures can be reliably designed and fabricated with barrier layer thicknesses approaching the tunnelling thickness and approaching enhancement mode characteristics. The thermal stability of 3 nm barrier thickness InAlN/GaN devices has been also evaluated up to 1000°C. No degradation of the InAlN/GaN interface towards the GaN buffer and the surface metals occurred at such high temperatures, which is promising for highly reliable ultra short, ultra high frequency GaN-based devices. Semienhancement mode of operation (VP = -1.0 V) with 0.9 A/mm has been obtained. Improving the gate breakdown characteristics further, would enable enhancement mode of operation with channel current densities well above those, which can be obtained by the conventional AlGaN/GaN counterparts.

ACKNOWLEDGMENT

This work was carried out in the frame of the European project UltraGaN (contract 6903). Thus, sincere thanks go to all members of this consortium for many stimulating, fruitful and helpful discussions. The growth material of this work is supported by the Swiss National Science Foundation. The authors would like to acknowledge Mr. Y. Men for e-beam processing.

Figure captions

  follows the trend observed for the sheet charge density obtained from Hall-measurements NS Hall shown in fig. 1. However, the correlation between IDSS and NS Hall for the free surface is only indirect. Underneath the gate no reduction in sheet charge density with thinner barrier is expected, if no stress is added by the gate metal. The surface potential of the free surface can only act in the source/gate and gate/drain access regions mainly as parasitic current limiter on the FET open channel current. In this first analysis the open channel current has been approximated by the saturated output current IDSS obtained at VG = + 2.0 V, and the correlation with Ns Hall cannot be quantitative. At this gate bias gate leakage becomes noticeable, which is consistent with a Schottky barrier height of approx. 1.7

Figure 1 :

 1 Figure 1: Simulated NS Hall data as a function of the In0.17Al0.83N/GaN HEMT barrier thickness. The experimental results are indicated by the squares. Corresponding electron mobility are approx. 1000 cm 2 /Vs down to 9 nm barrier thickness and 600 cm 2 /Vs for the 6 nm and 3 nm barrier thicknesses.

Figure 2 :

 2 Figure 2: DC maximum output current density (IDSS) at VG = + 2V as function of barrier thicknesses of In0.17Al0.83N/GaN HEMTs at room temperature. The inserts show the DC output characteristics of a 15 nm (VGS swept from +2 V to -10 V by step of 2 V) and 3.0 nm barrier thickness (VGS swept from +2 V to -1 V by step of 1 V) 0.2550 μm 2 In0.17Al0.83N/GaN HEMT.

Figure 3 :

 3 Figure 3: Pinch-off voltage dependence on barrier thicknesses of In0.17Al0.83N/GaN HEMTs.

Figure 4 :Fig. 1 : 10 Fig. 2 :Fig. 3 :

 411023 Figure 4: DC output characteristics of a 3.0 nm In0.17Al0.83N/GaN HEMT after 30 min 1000°C thermal stress. The corresponding R.T. characteristics are shown in the insert of fig. 2. VGS swept from +2 V to -1 V by step of 1 V. The insert shows the Schottky diode characteristics at R.T. and after 30 min 1000°C thermal stress.


C Pietzka

A Denisenko
email: andrej.denisenko@uni-ulm.de

M Alomari

F Medjdoub

J.-F Carlin

E Feltin

N Grandjean

E Kohn

Effect of Anodic Oxidation on the Characteristics of Lattice-Matched AlInN/GaN Heterostructures

Keywords: Lattice-matched, AlInN, anodic oxidation, surface states

The effect of anodic oxidation on the electronic characteristics of lattice-matched AlInN/GaN heterostructures was investigated using field-effect transistor (FET) structures with the gate areas in direct contact with the electrolytes. The gate surface of the FETs was subjected to anodic oxidation in 0.1 M KOH. The oxidized heterostructures were analyzed by electrochemical impedance spectroscopy and by modeling the characteristics of the electrolyte-gate FETs and the energy-band diagram of the heterostructures. This analysis suggested that the anodic treatment induced a bulk oxidation of the AlInN barrier. The Fermi level at the oxidized AlInN surface was shifted deep into the bandgap. The oxidation led to a reduction of the carrier mobility and to partial depletion of the channel.

INTRODUCTION

Recently the AlInN/GaN material system has become of interest for electronic applications due to its promising electronic properties, polarization effects, and high thermal stability. Significant progress in the growth of lattice-matched AlInN/GaN heterostructures has resulted in a polarizationinduced sheet charge density n s > 2 • 10 13 cm -2 in the two-dimensional (2D) channel in combination with a mobility l > 1000 cm 2 /V s at room temperature. 1,2 Lattice matching at the AlInN/GaN interface can be achieved at 17% indium content in the AlInN barrier, which leads to a stress-free structure. 3 The stress-free structure and the high current densities make AlInN/GaN heterostructures an ideal candidate for high-power and high-temperature applications. For example, the AlInN/GaN heterostructure has shown an extraordinary temperature stability, 4,5 which has been demonstrated by transistor operation at 1000°C in vacuum. 4 Despite this ceramic-like stability at high temperatures, the surface of the AlInN barrier layer could be subjected to oxidation and etching in strong acidic and alkaline electrolytes. 6,7 However, no effect of the chemical treatment on the electronic properties of the heterostructures has been investigated yet. The information on the morphological and electronic characteristics of the oxidized barrier layers is needed to assess surface trapping and charging effects in high-electron-mobility transistors (HEMTs). Besides, oxidation of III-nitride materials may improve the characteristics of Schottky gate diodes. For instance, the leakage current of Schottky diodes on bulk-doped GaN layers could be strongly suppressed by oxidation due to blocking of dislocation-related defect paths. 8 The oxidized AlInN/GaN heterostructures could also be used for chemical sensing devices, similar to the other III-nitrides and their compounds. [9][10][11][12][13][14] An interesting aspect related to this application field might be a low density of stress-induced defects expected for the lattice-matched heterostructures. This particular characteristic could possibly improve the stability of AlInN surface against corrosion in electrolytes, as it is well known that the chemical limits of III-nitrides depend very much on their crystalline quality. [15][16][17] A prerequisite of ion-sensing FETs on III-nitride materials is the presence of surface oxides bonds, 10,18 which can be due to native oxidation in the atmosphere 19 or induced by processing. However, chemical modification of the heterostructure surface may also induce charged electronic states which could pin the surface Fermi level and also affect the carrier transport in the 2D channel. In this paper we present our results on the effect of anodic oxidation in 0.1 M KOH electrolyte on the electronic properties of lattice-matched AlInN/GaN heterostructures. The anodic polarization of the heterostructure surface could also be related to the operation of ion-sensitive FETs under various bias conditions. Therefore, the evolution of the barrier and the channel characteristics under oxidation could be also useful to assess the performance of AlInN/GaN-based chemical sensors. In our study we investigated the effect of oxidation using the FET structures with gate areas in direct contact with electrolytes. The advantage of these structures is that they enable one to analyze the electrical equivalent circuit of the AlInN-electrolyte interface and to evaluate the transport properties in the 2D channel at the same time. The obtained results and the described methodology of the heterostructure evaluation could be used to optimize the oxidation processes of AlInN/GaN material for microwave HEMTs and chemical sensor applications.

EXPERIMENTAL

The lattice-matched In 0.17 Al 0.83 N/GaN heterostructures used in these experiments were grown by metalorganic chemical vapor deposition (MOCVD) on sapphire substrates with a thickness of the Al-InN barrier layer of approximately 10 nm. Details of the growth technique can be found elsewhere. 1,4,20 The sheet charge density n s of the electron gas in the 2D channel was 2.6 • 10 13 cm -2 , obtained from Hall measurements. The Hall mobility l H at room temperature was 950 cm 2 /V s. A set of FET macrostructures with gate areas open for contact with electrolytes was fabricated on these heterostructures, as shown by the schematic cross section of a single device structure in Fig. 1. The mesa was defined by dry etching in argon plasma. A Ti/Al/Ni/Au sequence was used for ohmic contact metallization, following by rapid thermal annealing of the deposited metallization at 890°C in N 2 atmosphere for 60 s. The gate areas of the FET were formed using a PTFE (Teflon) adhesive tape with a perforated opening, resulting in a gate length L G of 0.5 mm and a gate width of 1.6 mm.

The electrochemical experiments were performed in a three-electrode glass cell with a platinum counterelectrode and a saturated calomel (SCE, V SCE = +242 mV versus normal hydrogen electrode) reference electrode. The fabricated structures were analyzed in 0.1 M H 2 SO 4 and 0.1 M KOH electrolytes. In the electrode mode (cyclic voltammetry and impedance spectroscopy measurements), the source and drain contacts of the FETs were short-circuited to achieve a constant potential along the channel and connected with the working electrode input of the potentiostat. Cyclic voltammetry and electrochemical impedance spectroscopy measurements were performed using a PARSTAT 2273 potentiostat (Princeton Applied Research). In the FET mode, the drain and the gate biases were controlled by Keithley 2420 and Keithley 238 source meters. The source-to-gate potential was adjusted to the potential of the SCE reference electrode using a personal computer (PC)-controlled Keithley 2000 electrometer. All measurements were performed in a grounded Faraday cage with the solutions deairated by nitrogen gas flow.

RESULTS AND DISCUSSION

As-Grown Surface

An atomic force microscopy (AFM) image of the as-grown surface is shown in Fig. 2a. The surface roughness by AFM was approximately 0.8 nm (RMS). The hole-like structures on the surface could be related to the presence of dislocation defects. The cyclic voltammetry scan of the as-grown structure in 0.1 M KOH is shown in Fig. 3, curve 1. The potential window of water electrolysis on this AlInN surface was %3 V. A small peak was observed only in the negative potential range prior to the onset of hydrogen evolution. According to Ref. 21 this peak can be associated with the chemisorption of hydronium ions, which substitute the indium-or aluminium-hydrogen surface bonds. No other activity on the surface was observed within the potential window. The background current was in the nA/cm 2 range, which points towards a high quality of the as-grown surface. Here it should be noted that strong anodic polarization of the as-grown surface led to irreversible changes of its electrochemical and electronic characteristics (discussed below). Therefore all measurements of the as-grown structures discussed here were performed within a limited potential range prior to application of the anodic treatment.

The electrochemical impedance spectroscopy of the as-grown structure indicated the presence of two RC elements in series, as shown in Fig. 4a insert. The corresponding Bode plot measured at 0.0 V versus SCE in 0.1 M KOH is shown in Fig. 4a. The fitting of the impedance spectra was performed using ZSimpWin Ò data analysis software. Prior to the fitting procedure, the value of the cell series resistance was evaluated subtracted from the impedance data. The fitting parameters of the equivalent circuit are listed in Table I. The capacitance C 1 obtained from the fit was approximately 0.72 lF/cm 2 , which corresponds to the expected AlInN barrier layer capacitance, taking into account its thickness t (10 nm to 12 nm) and dielectric constant e ($10). The corresponding parallel resistance R 1 was in the range of 10 MX cm 2 , which confirmed the absence of charge transport across this barrier layer in the electrolyte and correlates with the low background current in the voltammogram.

The second RC element was fitted using a constant-phase element (CPE) Q 2 with the complex admittance Q 0 (jx) n , where Q 0 and n are parameters and x = 2pf with frequency f, as shown in the inset to Fig. 4b. The value of the nominal capacitance at x = 1 related to the CPE Q 2 was %20 lF/cm 2 , which is typical for an electrical double-layer capacitance on metal electrodes. On the other hand, the as-grown AlInN surface might be characterized by the presence of a native oxide with a thickness of a few monolayers, as demonstrated for the case of AlGaN/GaN heterostructures. 19 The presence of a chemically active oxide layer on the as-grown surface was supported by measuring the opencircuit potential (OCP) in 0.1 M H 2 SO 4 and 0.1 M KOH electrolytes. For the as-grown surface the OCP values were +0.40 V and -0.19 V versus SCE, respectively. This yielded a pH sensitivity of %50 mV/pH, which approaches the theoretical NernstÕs limit of 59 mV/pH and which is typical for oxidized III-nitride surfaces. 10 Thus, the R 2 Q 2 element of the equivalent circuit can be attributed either to the electrical double layer on the AlInN surface or to a native surface oxide. The dispersion factor n of the CPE was about 0.80, which can be interpreted in terms of inhomogeneous surface characteristics. 22 The channel resistance between the source and drain contacts in air was %253 X/square, which is consistent with the sheet carrier density n s of and the mobility l from the Hall measurements. No noticeable changes of the channel resistance were observed when the as-grown structure was into the electrolyte solutions with no external gate electrode connected.

In the FET mode (with the gate electrode biased), the as-grown structure showed a high drain current but a very low transconductance. The corresponding transfer characteristics in 0.1 M H 2 SO 4 and 0.1 M KOH are shown in Fig. 5. Both curves showed a pH-induced shift, which is typical for ion-sensitive FETs. [9][10][11] In our experiments, pH sensing of the as-grown AlInN surface could be expected as well, regarding the pH-induced shift of the OCP values mentioned above. However, the projected pinch-off voltage estimated from the slope of the transfer characteristics would be about -6 V or more. This means that the gate bias range for channel modulation is located at high negative potentials corresponding to the range of oxygen evolution on the AlInN surface (note, that a negative gate potential means a positive potential of the surface in the electrode mode, see Fig. 3). At the low gate potentials corresponding to the window of water electrolysis with low gate leakage current, the as-grown FET operated in the current saturation range. This makes accurate modeling of the FET characteristics impossible.

Effect of Anodic Treatment in KOH Electrolyte

The effect of anodic treatments was studied by exposing the gate area of the FET in 0.1 M KOH at a potential of +2.5 V versus SCE, as marked in Fig. 3 by the black dot. The amount of charge transferred during the oxidation was about 1.6 mC/cm 2 ($1 • 10 16 electrons/cm 2 ). The cyclic voltammogram I.

Table I. Fitting Parameters of the Electrochemical Impedance Spectra in Fig. 4 Equiv. Circuit

C 1 F/cm 2 R 1 X cm 2 Q 2 :Q 0 S s 1/2 /cm 2 Q 2 : n R 1 X cm 2 (a)
7. of the oxidized surface revealed a pronounced suppression of the anodic current (curve 2 in Fig. 3).

A similar effect has been reported for n-doped GaN electrodes after anodic treatment in NaOH electrolyte 23,24 and has been explained by the formation of a surface oxide. The formation of an oxide layer on the AlInN surface could therefore also be expected in our experiments. However, no changes of the surface morphology were revealed after the oxidation. The AFM image of the treated surface is shown in Fig. 2b. No traces of corrosion, such as an increased surface roughness or a change of the dislocation areas, were revealed in comparison to the as-grown surface. However, the impedance characteristics of the AlInN-electrolyte interface were noticeably changed after the oxidation. The corresponding Bode plot is shown in Fig. 4b. The impedance spectrum was fitted now by a circuit consisting of an R 1 C 1 element and a single CPE Q 2 in series (Fig. 4b,inset). The dispersion factor n of the CPE was reduced after the oxidation treatment to about 0.4, while its numerical value at x = 1 was increased by a factor of 30 in comparison to the as-grown condition (see Table I). According to Ref. 25 a CPE with such a low value of n could be explained by a distributed RC network, which describes an inhomogeneous system. The parameters of the R 1 C 1 element representing the AlInN surface barrier layer were slightly changed after the oxidation. The capacitance C 1 obtained from the best fit was increased by 9% from 0.72 lF/cm 2 to %0.79 lF/cm 2 at 0.0 V potential in 0.1 M KOH. The corresponding parallel resistance R 1 was reduced by 40%, but was still remaining in the MX cm 2 range. Similar changes of the equivalent circuit parameters were observed also at other potentials both in 0.1 M KOH and 0.1 M H 2 SO 4 electrolytes.

The anodic treatment also led to a partial depletion of the 2D channel. The sheet resistance within the oxidized area was increased to 6.9 kX/square. The pinch-off voltage of the FET structures exposed to the electrolytes was shifted into the potential window with low gate leakage current. This enabled the full analysis of the FET characteristics. The direct-current (DC) output characteristics of the oxidized FET in 0.1 M H 2 SO 4 and the corresponding transfer characteristics in the saturation range in 0.1 M H 2 SO 4 and 0.1 M KOH are shown in Fig. 6. The FET characteristics were analyzed using a charge-control model of a metal-insulator-semiconductor FET (MISFET) here applied to a heterostructure FET. [START_REF]HEMTs and HBTs: Devices, Fabrication and Circuits[END_REF] This model was adapted to our structures by taking into account the results of the impedance analysis of the electrolyte-heterostructure interface. Since the CPE Q 2 showed a high nominal value, its effect on the gate capacitance C GS in the DC mode could be ignored. Thus, the C GS value in the MISFET model was taken equal to the C 1 element, which is 0.79 lF/cm 2 . This value was also close to the expected capacitance of the AlInN barrier layer. In addition, the series resistance of the source and drain areas (formed by the part of the channel covered with the adhesive tape) R S and R D of 150 X were also taken into account. To fit the data close to the pinch-off, a parallel source-to-drain shunt resistance R DS of 20 kX was also included into the model (see Fig. 6b The fitted value of the carrier mobility in the FET channel was about 200 cm 2 /V s, compared to 950 cm 2 /V s of the as-grown structure. Taking the value of the sheet resistance of 6.9 kX/square in air, the sheet charge density within the exposed area of the heterostructure was estimated to be %4.5 • 10 12 cm -2 . To understand the origin of the depletion effect by the anodic treatment, the energy band diagram of the AlInN/GaN heterostructure was analyzed using 2D numerical simulation (Silvaco Ò ). A schematic band diagram of the heterostructure is shown in Fig. 7a, including the polarization effects of the GaN-based material system. The polarization-induced charge at the AlInN/ GaN interface was calculated using the theoretical model in Ref. 27. The parameters of this model were adjusted to the material properties of the AlInN barrier, the GaN buffer, and the conduction-band offset DE C of 0.91 eV at the AlInN/GaN interface. 28 In the 2D energy-band model the polarizationinduced field was simulated by a dipole charge of 2.6 • 10 13 cm -2 at the opposite surfaces of the AlInN barrier layer. The two-dimensional electron gas (2DEG) can be considered to be generated by a surface donor state. 29 The positive charge accumulated at this surface donor state will balance the 2DEG density in the channel, whereas the energy level of this surface state will determine the filling of the 2D channel at the heterostructure interface. Following this concept, we introduced a donor-like surface state in the 2D simulation model to represent the surface donor. The energy level and the density of the surface state on the as-grown surface were evaluated using the experimental dependence of the 2DEG density on the thickness of the AlInN barrier layer as reported in Ref. 20, AlInN/GaN heterostructures have been grown with different barrier thicknesses using the same MOCVD process as in our study. Figure 7b shows the calculated dependencies of the 2DEG density n S as a function of the barrier layer thickness t and the donor level energy E T (dashed curves). The experimental data points from Ref. 20 are shown in Fig. 6b by the square symbols. By fitting these data, an energy level of the surface donor E T at about 0.4 eV below the conduction band was found. The value of n s for the as-grown structure in our experiment, as marked by the symbol A, is also in close agreement with this trap energy level.

However, the n S value of 4.5 • 10 12 cm -2 after the oxidation cannot be explained by any reasonable trap level within the bandgap, provided the barrier thickness t remains 10 nm of the as-grown surface. This scenario is shown in Fig. 7b, where the dotted arrow connecting the points A and B¢ illustrates the shift of n S induced by a surface state level only. As one can see, this energy level would be located below the valence-band edge, which is not realistic. A more reasonable explanation of the observed effect would be a reduction of the AlInN barrier layer thickness by oxidation. Such a scenario is shown in Fig. 7b by the broken dotted arrow connecting the points A and B. The exact thickness of the reduced barrier according to this model could be estimated by knowing the energy level E ox T of the oxidationinduced states, as shown on the graph. But since no reliable data on the E ox T value are available up to now, no further details of the barrier composition can be determined yet. From the tunnelling limit of the AlInN barrier thickness ($1 nm to 2 nm), the oxidation-induced traps might be located at any energy level below 1 eV to 1.5 eV from the conduction-band edge, as shown by the dashed horizontal and vertical arrows in Fig. 7b.

On the other hand, the capacitance C 1 attributed to the barrier layer did not change essentially during the oxidation treatment, as revealed by the impedance spectroscopy measurements in Fig. 4b (also see Table I). This may be explained by a surface oxide layer, which substituted electrically a part of the barrier layer lost during the anodic oxidation treatment. Such explanation means bulk oxidation of the AlInN material rather than etching. In this case the insulating layer between the 2D channel and the surface would be a two-layer structure consisting of the thinner AlInN barrier layer and the native oxide layer on top. Here it is important to note that the value of n s is determined by the thickness of the polar AlInN layer, but not by the thickness of the nonpolar oxide. Therefore, the bulk oxidation would lead to a reduction of n s , but not to a significant change of the capacitance. However, no evidence of a separate RC element related to the surface oxide layer could be resolved by the electrochemical impedance spectroscopy, which was limited to the frequency range below 100 kHz. This might be explained by a compact structure of the oxide with negligible diffusion of ions from the electrolytes.

The proposed model may also explain the reduction of the mobility in the 2D channel after the oxidation. According to the model of carrier scattering in 2DEG systems, 30 the reduction of the insulating barrier separating the 2DEG and the charged stated at the oxidized interface would increase scattering at the remote ionized states. But no data on the preferential location of the oxidationinduced traps could be given yet. More study is needed to identify this oxide layer and its chemical composition.

CONCLUSIONS

In this paper we have described the effect of anodic treatments on the electronic characteristics of AlInN/GaN heterostructures. The grown heterostructure material was lattice-matched with about 17% indium content in the barrier layer. The carrier mobility in the 2D channel was 950 cm 2 /V s and the 2DEG density was 2.6 • 10 13 cm -2 . For the analysis, we prepared a set of field-effect transistor structures with open gate areas in direct contact with electrolyte solutions. Such structures enabled us to analyze the electrical equivalent circuit of the AlInN-electrolyte interface by electrochemical impedance spectroscopy in the electrode mode, the electronic characteristics of the heterostructure in the ion-sensitive FET mode, and to trace the evolution of these characteristics after the surface oxidation by in situ anodic treatment in 0.1 M KOH electrolyte. The obtained results were used to analyze the energy band structure of the AlInN/GaN heterostructures by 2D numerical simulation.

It was found that a strong anodic treatment of the heterostructure with the total charge transfer of 10 16 electrons/cm 2 most likely induced bulk oxidation of the AlInN barrier layer rather than etching. The oxidized barrier layer may represent a twolayer structure consisting of a thinned AlInN barrier and the oxide layer on top. The electrical resistance across this surface oxide layer in electrolytes was as high as the resistance across the AlInN barrier layer. This may suggest a compact structure of the oxide with negligible diffusion of ions from the electrolytes. No traces of the surface corrosion or selective etching by oxidation was revealed by AFM.

The Fermi level at the oxidized barrier surface was pinned deep (up to several eV energy) in the bandgap of AlInN. In comparison, the location of the surface Fermi level on the as-grown heterostructure was estimated to be about 0.4 eV below the conduction-band edge of AlInN. The oxidation-induced states reduced the 2DEG density in the channel to approximately 4.5 • 10 12 cm -2 . The carrier mobility in the 2D channel was also reduced by approximately a factor of 5, most likely due to enhanced scattering on the charged surface/interface states. Also, the oxidation-induced depletion shifted the characteristics of the electrolyte-gate FET into the gate potential range corresponding to the low gate leakage across the heterostructure-electrolyte interface.

However, the data on the chemical composition of the oxidized surface are still missing. A better understanding of the surface characteristics and the oxidation mechanisms might lead to improvement of the surface oxidation technology. The stable surface oxides of the AlInN/GaN heterostructures could be used in chemical sensor devices and for surface passivation of microwave power HEMT structures.

Fig. 1 .

 1 Fig. 1. Schematic cross section of an ion-sensitive FET on an AlInN/ GaN heterostructure.

Fig. 2 .

 2 Fig. 2. AFM images: (a) as-grown AlInN surface; (b) after anodic oxidation. The surface roughness (RMS) is approximately 0.8 nm in both cases.

Fig. 3 .

 3 Fig. 3. Linear cyclic voltammetry of the lattice-matched AlInN/GaN heterostructure. Curve 1: as-grown surface (the first scan); curve 2: after strong anodic oxidation with a total charge transfer of about 1 • 10 16 cm -2 . The potential of the oxidation process is marked by the arrow.

Fig. 4 .

 4 Fig. 4. Bode plots of the AlInN/GaN heterostructure at an electrode potential of 0.0 V versus SCE in 0.1 M KOH: (a) as-grown surface; (b) after anodic oxidation. The solid curves show the best fit using the equivalent circuits in the insert. The fitting parameters are listed in TableI.

Fig. 5 .

 5 Fig. 5. Transfer characteristics of the as-grown FET structure (before oxidation) in 0.1 M H 2 SO 4 and 0.1 M KOH electrolytes at a drain bias V DS of 1.5 V.

  inset). The fitting of the transfer characteristics by the MISFET model is shown in Fig. 6b by the solid curves. The pinch-off voltages derived from this fit were -0.80 V versus SCE in 0.1 M H 2 SO 4 and -0.25 V versus SCE in 0.1 M KOH. The pH-induced shift of the transfer characteristics correlated with the pH-dependent open-circuit potential in these electrolytes equal to +0.40 V and -0.17 V versus SCE, respectively.

Fig. 6 .

 6 Fig. 6. (a) The output characteristics of the ion-sensitive FET (ISFET) with the gate length L G of 0.5 mm in 0.1 M H 2 SO 4 and (b) the corresponding transfer characteristics in 0.1 M H 2 SO 4 and 0.1 M KOH at the drain bias of 1.5 V. The solid curves in (b) are the fits from a MISFET model. The equivalent circuit model including the parasitic resistance of the channel R DS is shown in the inset (see text).

Fig. 7 .

 7 Fig. 7. (a) Schematic band diagram of the AlInN/GaN heterostructure and (b) the 2DEG density calculated for various thicknesses t of the barrier layer and energy levels E T of the surface traps by 2D numerical simulation (dashed lines). The square symbols show the data points from Ref. 20 on the lattice-matched AlInN/GaN heterostructures (as-grown layers) with various barrier thicknesses t. The symbols A, B, and B¢ are explained in the text.

Pietzka, Denisenko, Alomari, Medjdoub, Carlin, Feltin, Grandjean, and Kohn

Effect of Anodic Oxidation on the Characteristics of Lattice-Matched AlInN/GaN Heterostructures

ACKNOWLEDGEMENTS

This work was supported by the European Project UltraGaN (co. #6903). The authors would like to thank D.M. Kolb (Inst. of Electrochemistry, Ulm University) for helpful discussions.


E Kohn

M Dipalo

M Alomari

F Medjdoub

J.-F Carlin

N Grandjean

S Delage

A Concept for Diamond Overlayers on Nitride Heterostructures

The development of diamond overlayers with high crystalline quality for high power devices in Si, GaAs or GaN, aimed at heat extraction from the top, has been a quest for many years. Recently CMOS circuits have been coated by untra-nano-crystalline-diamond (UNCD) grown at 350 °C, however with low thermal conductivity due to a substantial graphitic grain boundary content [1]. Usual growth conditions for nanocrystalline diamond (NCD) films of high thermal conductivity are a temperature above 600 °C in hydrogen atmosphere with high H* radical concentration and CH 4 growth chemistry. In fact, no such overlayer with high thermal conductivity has been developed to our knowledge up to now for GaAs or GaN. However, GaN based power devices are already presently seriously limited by their thermal losses and even the employment of diamond substrate heat spreaders may not be sufficient. Such substrate heat spreader configurations are usually realized by joining prefabricated semiconductor and diamond materials stacks by alloying or wafer bonding.

On the other hand, diamond overlayers of high crystalline quality could serve also other purposes like as inert electrochemical electrode for operation in highly corrosive and hazardous environments to detect (and oxidize) almost all hazardous organics like phenols or biomolecules like dopamine [2]. In addition, with a diamond electrode GaN based ChemFETs (in HEMT configuration) could operate in the amperometric as well as potentiometric mode, whereas Si-MOS-ISFETs can only operate in the potentiometric mode.

In this investigation NCD has been grown at conditions, which are typical for electrode applications [3], at a temperature of approx. 700 °C and CH 4 chemistry on an InAlN/GaN heterostructure with 10 nm InAlN thickness. The InAlN/GaN heterostructure can be grown lattice matched on GaN, thus avoiding stress induced defect formation at high temperature; and indeed (in contrast to the AlGaN/GaN system), it has shown to possess a high thermal stability up to at least 1000°C, when used in a HEMT configuration [4]. It was therefore chosen for this investigation.

Fig. 1 shows the surface morphology of a 200 nm thick NCD film deposited onto this InAlN/GaN heterostructure in a 2 hr deposition process at 700 °C in a Hot Filament reactor and 0.3 % CH 4 in H 2 . Fig. 2 shows the corresponding Raman spectrum. Both, the grain structure and the sp 3 diamond peak are clearly identified. Fig. 3 shows the surface of the heterostructure mesa surrounded by the GaN buffer after diamond deposition. For diamond nucleation, a SiO 2 /amorphous Si PECVD (20nm/120nm) interfacial layer had been deposited. As can be seen, the InAlN surface morphology is not degraded, whereas blistering is seen on the GaN surface, again indicating the high ceramic-like stability of the InAlN surface. Ti/Al/Ni/Au ohmic contacts and Ni/Au Schottky contacts have been deposited after overgrowth and removal of the overlayer. The corresponding output characteristics are shown in fig. 4. Open channel current level, pinch-off voltage and transconductance are not essentially degraded. This means that the crystallinity of the heterostructure interface and the polarization discontinuity have not been degraded. On the other hand, the high leakage current illustrates the degradation of the GaN buffer. Thus, in respect to diamond overlayers for heat extraction, the InAlN/GaN heterosystem is viable candidate, however at the same time GaN buffer layer passivation is essential.

For the ChemFET application the NCD film has been deposited onto the Ni-gate contact pad. The diamond film had been quasi-metallically boron doped to allow amperometric operation. To avoid interfacial reactions between Ni and the overlayer, a Ta diffusion barrier had been inserted. Fig. 5 shows a voltamogram (IV characteristics) of the electrode (structure fig. 6), also in comparison to a typical a Pt electrode. The water dissociation window of 3.0 V and the low background current indicate the absence of any graphitic phases and pinholes, acting as metallic shorts.

To our knowledge this is the first successful attempt to overgrow a GaN-based heterostructure by high quality nanocrystalline diamond of high structural quality, opening a pathway to a number of novel concepts to III-nitride device structures.

  A. V. Sumant, et al. "Demonstration of low temperature ultrananocrystalline diamond (UNCD) film integration with CMOS devices for monolithically integrated diamond MEMS/NEMS-CMOS systems" NDNC 2007, abstr. 5-2, Osaka (Japan) [2] G.M. Swain; "Electroanalytical applications of diamond electrodes"; Thin Film Diamond II (Semiconductor and Semimetals vol. 77), Chapter 4, Elsevier 2004 [3] F. Medjdoub, et al. "Can InAlN/GaN be an alternative to high power / high temperature AlGaN/GaN devices?", Int. Electron Devices Meeting (IEDM), Tech. Digest, 927, 2006 [4] W. Haenni, P. Rychen, M. Fryda, C. Comninellis; "Industrial Applications of Diamond Electrodes"; Thin Film Diamond II (Semiconductor and Semimetals vol. 77), Chapter 5

Fig. 1 :Fig. 5 :

 15 Fig. 1: SEM morphology of 200 nm nanodiamond on InAlN/GaN Fig. 2: Raman spectrum of 200 nm nanodiamond on InAlN/GaN

Acknowledgement:

This work was supported in part by the European Union Marie Curie Network DRIVE (512224) and the European Union Research Project ULTRAGAN.


F Medjdoub
email: farid.medjdoub@uni-ulm.de

J.-F Carlin

M Gonschorek

E Feltin

M A Py

N Grandjean

E Kohn

ABOVE 2 A/mm DRAIN CURRENT DENSITY OF GaN HEMTS GROWN ON SAPPHIRE

Keywords: GaN HEMT, power, AlInN

We report on the investigation of an InAlN/GaN HEMT structure, delivering higher sheet carrier density than the commonly used AlGaN/GaN system. We achieved in a reproducible way more than 2 A/mm maximum drain current density for a gate length of 0.25 µm with unpassivated undoped devices realized on sapphire substrates. Small signal measurements yield a F T = 31 GHz and F MAX = 52 GHz, which illustrates the capability of these structures to operate at high frequencies. Moreover, the pulsed analysis indicates a more stable surface in the case of AlInN than that of AlGaN, attributed to the lattice matched growth of this barrier with 17 % In content on GaN, avoiding strain piezo polarization in the material.

Introduction

GaN is a universal material for a variety of electronic and optoelectronic applications such as light emitters, transistors, and sensors. Due to its wide band gap it is particularly interesting for high-frequency and high-power Field-Effect Transistors devices. However, many difficulties arise from the lattice mismatch and polar surface of the commonly used AlGaN/GaN heterojunction. Even if several improvements have been made in the growth and design of this material system in recent years the commercialization remains still difficult. Recently, the AlInN/GaN heterojunctions were proposed as alternative for power FETs [START_REF] Kuzmik | Power Electronics on InAlN/(In)GaN: Prospect for a Record Performance[END_REF].

In this paper we discuss results obtained from a GaN-based High Electron Mobility Transistor (HEMT) with an AlInN barrier. Applying Vegard's law for alloys of InN, AlN and GaN as indicated in Fig. 1, InAlN layers containing 17% Indium can be grown lattice matched on GaN and therefore have no piezoelectric stress component. This rather novel configuration promises high current densities up to 3 A/mm, the 2DEG being only induced by the difference in spontaneous polarization. Moreover, it offers a higher band offset between the GaN channel and the InAlN barrier as compared to AlGaN. However, the materials system is difficult to grow due to the distinctly different growth conditions required for GaN and InAlN and the difficulty to incorporate In and Al in the same lattice without clustering. However, these problems have been widely overcome in first promising investigations [START_REF] Neuburger | Unstrained InAlN/GaN FET[END_REF][START_REF] Medjdoub | Small signal characteristics of AlInN/GaN HEMTs[END_REF]. Medjdoub

F.

Growth and processing

Major issues for the growth of AlInN are the different requirements for the growth parameters to achieve good crystalline quality for binaries contained in the ternary alloy, namely InN and AlN. AlN tends to grow polycrystalline for growth temperatures below 1000 °C [START_REF] Watanabe | The growth of single crystalline GaN on a Si substrate using AIN as an intermediate layer[END_REF][START_REF] Krost | GaN-based epitaxy on silicon: stress measurements[END_REF]. On the other hand, InN can only be grown at low temperatures and high ammonia partial pressures, whereas high ammonia partial pressures have a strong effect on the AlN growth rate. Here, the studied structures (see Fig. 2) were grown by an AIXTRON MOCVD system on 2 inch diameter (0001) sapphire substrates [START_REF] Carlin | High-quality AlInN for high index contrast Bragg mirrors lattice matched to GaN[END_REF]. They consist of 2 µm thick GaN buffer, 1 nm thick AlN spacer layer and 13 nm thick AlInN barrier layer with 81% Al composition. The thin optimized AlN interlayer is used to reduce alloy disorder scattering and thus to improve the transport characteristics. Room temperature Hall effect measurements yielded a 2DEG sheet charge of 2.6×10 13 cm -2 with a record electron mobility of 1170 cm 2 /Vs and a sheet resistivity of 211 Ω/ [START_REF] Gonschorek | High electron mobility lattice-matched AlInN/GaN Field-Effect Transistor heterostructures[END_REF]. FETs have been realized as follows: The devices are mesa isolated by Ar plasma etching. Then, Ti/Al/Ni/Au ohmic contact stacks are alloyed at 890°C in nitrogen atmosphere to provide contact resistances of 0.7 Ω.mm, as measured by linear TLM. The submicron gates (L g = 0.25 µm) are defined with electron beam lithography. Ni/Au is deposited to produce the gate Schottky contacts. The HEMT devices are not yet passivated.

Experimental results

The output characteristics of a 0.25 µm gate length device with 25 µm gate width are shown in Fig. 4. The maximum drain current density at +2 V V GS exceeds 2 A/mm in a reproducible way (2.3 A/mm for the presented device), which is to our knowledge beyond the highest drain current density of any AlGaN/GaN HEMT structure, especially when fabricated on sapphire. It reflects the high sheet carrier density in this novel heterostructure. The transconductance peaks at 265 mS/mm (see Fig. 5) with 10 V applied to the drain and -5 V to the gate. The breakdown voltage at pinch-off is estimated to about 40 V for a device with a drain source spacing of 2.5 µm and is thought to be linked to residual buffer layer leakage. Pulse experiments were performed in a routine as used to assess the stability of AlGaN/GaN devices (described with more details in [START_REF] Gaquière | Pulsed bias/pulsed RF characterization measurement system of FET at constant intrinsic voltages[END_REF]) with 500 ns pulses (see Fig. 6).

All quiescent bias points (V DS0 , V GS0 ) are chosen in order to simultaneously eliminate the thermal effect (cold polarization) and to reveal the gate and drain lag effects: (V DS0 = 0 V, V GS0 = 0 V), (V DS0 = 0 V, V GS0 = pinch-off voltage) and (V DS0 = 10 V, V GS0 = pinch-off voltage). The first quiescent bias point is used as the reference to compare with the other bias conditions. A drop of 13 % (gate lag) and 38 % (drain lag) regarding the maximum drain current density is observed at V DS = 10 V. Compared to typical unpassivated AlGaN/GaN structures on sapphire, those changes are much lower, which indicate a more stable surface in the case of AlInN. The S-parameters of 0.25×50 µm 2 HEMTs were measured between 0.5 and 40 GHz. We achieved cut-off and maximum oscillation frequencies F t = 31 GHz and F max = 52 GHz (shown in figure 7) extrapolated from the current gain H 21 and the maximum available gain (MAG) at V GS = -6.5V and V DS = 10V respectively. These results show the capability of this structure to operate at high frequency. It reflects the high material quality already obtained and the maturity of the processing technology. We observed a good F MAX / F T ratio, which is may be directly correlated to the device power performances.

Conclusion

We have investigated the potential of AlInN/GaN structures for high power applications. We obtained more than 2 A/mm open channel current which corresponds to the highest maximum drain current density delivered by any GaN FET. We have also revealed the capabilities of these structures to operate at high frequencies. Pulsed measurements reveal a more stable surface in the case of AlInN than that of AlGaN. However, the breakdown voltage of these devices is still limited mainly because of a relatively low GaN buffer resistivity. The improvement of the GaN buffer resistivity associated with the utilization of a field plate technology should strongly improve the three-terminal breakdown voltage of these structures. Power measurements will be carried out on these devices.

Acknowledgement

This study is supported by the European Union under contract No. 6903

Figure 2 :Figure 3 :

 23 Figure 2: Schematic cross section of the AlInN/GaN HEMT devices Capacitance-Voltage profiling indicates a 2 DEG carrier density 14 nm below the surface at the AlInN/GaN interface (see Fig. 3), resulting in a sheet charge density in agreement with Hall measurements.

Figure 4 :Figure 5 :

 45 Figure 4: DC output characteristics of an Al 0.81 In 0.19 N/GaN HEMT 0.25×25 µm 2 .V GS swept from -10 V to 2 V by step of 2V

Figure 6 :

 6 Figure 6: Pulsed I D -V DS characteristics of an Al 0.81 In 0.19 N/GaN HEMT 0.25×50 µm 2 with three different quiescent bias points. V GS swept from -9 to 1 V by step of -2 V.

Above 2 AFigure 7 :

 27 Figure 7: Current gain cut-off frequency (⏐H 21 ⏐) and maximum oscillation frequency (extrapolated from the Maximum Available Gain) for a 0.25×50 µm 2 Al 0.81 In 0.19 N/GaN HEMT

  , J.-F. Carlin, M. Gonschorek, E. Feltin, M.A. Py, N. Grandjean, and E. Kohn

		-0.02
		-0.03		GaN			InN
	)
	2 Spontaneous Polarization (As/m	-0.08 -0.07 -0.06 -0.05 -0.04	AlN	In 0.17 Al 0.83 N
		-0.09
			3.1	3.2	3.3	3.4	3.5	3.6
					Lattice Constant (A)
	Figure 1: Spontaneous polarization over the lattice constant of InN, AlN, GaN. Lattice matched AlInN is
			obtained for AlInN with 17% Indium.


Thermal stability of 5 nm barrier InAlN/GaN HEMTs F. Medjdoub a, , M. Alomari a , J.-F. Carlin b , M. Gonschorek b , J.-F. Carlin b , E. Feltin b , M. A. Py b , N. Grandjean b and E. Kohn a a University of Ulm, EBS, Germany, farid.medjdoub@uni-ulm.de , b EPFL Lausanne, Switzerland.

InAlN/GaN is a new alternative heterostructure to AlGaN/GaN promising even higher power densities and high temperature operation [START_REF] Kuzmik | Power electronics on InAlN/(In)GaN: Prospect for a record performance[END_REF][START_REF] Medjdoub | Can InAlN/GaN be an alternative to high power / high temperature AlGaN/GaN devices?[END_REF]. Due to the high spontaneous polarization discontinuity maximum drain current densities of more than 2 A/mm and barrier thicknesses down to 5 nm have been demonstrated with semi-enhancement performance [START_REF] Medjdoub | Barrier layer downscaling of InAlN/GaN HEMTs[END_REF]. First RF power results have become available with 7.5 nm barrier [START_REF] Gillespie | Realization of InAlN/GaN Unstrained HEMTs on SiC Substrates with a 75 Å Barrier Layer[END_REF]. However no hints of the stability of these sub-10 nm barrier devices has been given.

In this paper, the properties of 5 nm barrier HEMT devices has been investigated in a temperature ramping experiment as used for AlGaN/GaN devices [START_REF] Daumiller | Evaluation of the temperature stability of AlGaN/GaN heterostructure FET's[END_REF] and InAlN/GaN devices with 13 nm barrier [START_REF] Medjdoub | Can InAlN/GaN be an alternative to high power / high temperature AlGaN/GaN devices?[END_REF] previously.

The devices have been tested up to 1000 °C in vacuum in an experiment, where the temperature had been ramped in 100 °C steps, keeping the device 20 min at each temperature and cooled to R. T. between each step. As shown in fig. 1 after cycling the maximum FET output current density is not degraded. This means that the polarization discontinuity is fully preserved and no interfacial diffusion has occurred. The pinch-off voltage remains identical and the Schottky diode leakage only marginally increased. No gate sinking, even on the nm-scale of the Ni metallization and no reduction in Schottky barrier height are observed (Fig. 2). The ohmic contacts, which have been alloyed at 850 °C by RTA show also exceptional stability. The initial alloying cycle still kept the alloy front just out of reach for high tunnelling probability. It has been moved within this distance in a very controlled way, however without degrading the contact resistance by the 1000 °C post alloying cycle.

High current densities in combination with thin barriers are only possible for low surface potentials and not possible for unpassivated AlGaN/GaN devices, where the surface potential is pinned at approx 1.65 eV (and only unpinned by the passivation process) [START_REF] Ibbetson | Polarization effects, surface states, and the source of electrons in AlGaN/GaN heterostructure field effect transistors[END_REF]. In the case of the InAlN barrier the surface potential is located approx. 0.4 eV below the conduction band even in the unpassivated case. (see Fig. 3). Due to this low value a high current density of 1.1 A/mm can still be obtained in planar 5 nm barrier devices with a pinch-off voltage of -2 V and a maximum forward gate bias of +2 V resulting in semi-enhancement operation.

In conclusion, the InAlN/GaN heterojunction is exceptionally stable even for barrier thicknesses below 10 nm. Experiments for barrier thicknesses of approx. 2.5 nm (verified by a pinch-off voltage of -0.8 V) are presently performed and results will be reported. Room temperature characteristics are shown in Fig. 4.

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :

 1234 Fig. 1: 0.25×50 µm DC output characteristics at room temperature and after 1000°C stress of 5 nm barrier thickness InAlN/GaN HEMTs


F Medjdoub
email: farid.medjdoub@uni-ulm.de

J.-F Carlin

M Gonschorek

E Feltin

M A Py

N Grandjean

E Kohn

Above 2 A/mm drain current density of GaN HEMTs grown on Sapphire

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

In this paper we discuss a result obtained from a GaN-based High Electron Mobility Transistor (HEMT) with AlInN barrier. This rather novel configuration promises high current densities up to 3 A/mm in a lattice matched configuration., the 2DEG being induced by the differences in spontaneous polarization [START_REF] Kuzmik | Power Electronics on InAlN/(In)GaN: Prospect for a Record Performance[END_REF][START_REF] Medjdoub | Novel AlInN/GaN HEMT structures for high power performance[END_REF]. Moreover, it offers a higher band offset between the GaN channel and the InAlN barrier as compared to AlGaN. However, the materials system is difficult to grow due to the distinctly different growth conditions required for GaN and InAlN and the difficulty to incorporate In and Al in the same lattice without clustering. Indeed these properties have been seen in first promising investigations on Silicon substrate [START_REF] Neuburger | Unstrained InAlN/GaN FET[END_REF].

The studied structures were grown by Metal Organic Vapor Phase Epitaxy system on 2 inch diameter (0001) sapphire substrate [START_REF] Carlin | High-quality AlInN for high index contrast Bragg mirrors lattice matched to GaN[END_REF]. They consist of 2 µm thick GaN buffer, 1 nm thick AlN spacer layer and 13 nm thick AlInN barrier layer with 81% Al composition (described in Fig. 1). Room temperature Hall effect measurements yielded a polarizationinduced 2DEG sheet charge of 2.6×10 13 cm -2 with a record electron mobility of 1170 cm 2 /Vs and a sheet resistivity of 211 Ω/ [START_REF] Gonschorek | High electron mobility lattice-matched AlInN/GaN field-effect transistor heterostructures[END_REF]. The devices are MESA isolated with Ar plasma etching. Then, a Ti/Al/Ni/Au ohmic contact stack is alloyed at 890°C in nitrogen atmosphere to provide contact resistances of 0.7 Ω.mm, as measured by linear TLM. The submicron gates (L g = 0.25 µm) are defined with electron beam lithography. Ni/Au is deposited to produce the gate Schottky contacts. The HEMT devices are not yet passivated.

The output characteristics of a 0.25 µm gate length device with 25 µm gate width are shown in Fig. 2 and3. The maximum drain current density at +2V V GS exceeds 2 A/mm in a reproducible way (2.33 A/mm for the presented device), which is to our knowledge beyond the highest drain current density of any AlGaN/GaN HEMT structure, especially when fabricated on sapphire. The transconductance peaks at 265 mS/mm with 10V applied to the drain and -5V to the gate. The breakdown voltage is estimated to about 40V at pinch-off for a device with a drain source spacing of 2.5 µm and is thought to be linked to residual buffer layer leakage.

The improvement of the GaN buffer resistivity associated to the utilization of a Field Plate technology should strongly improve the three-terminal breakdown voltage of these structures. Small signal, pulsed and power measurements will be carried out on these devices. V GS swept from -8V to 2V by step of 2V

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Schematic cross section of the AlInN/GaN HEMT devices

Acknowledgement

This study was supported by the European Union under contract No. 6903


Behrouz Maham
email: behrouz@unik.no

Mérouane Debbah
email: merouane.debbah@supelec.fr

Are Hjørungnes

Energy-Efficient Cooperative Routing in BER Constrained Multihop Networks

Due to the limited energy supplies of nodes, in many applications like wireless sensor networks energy-efficiency is crucial for extending the lifetime of these networks. We study the routing problem for multihop wireless ad hoc networks based on cooperative transmission. The source node wants to transmit messages to a single destination. Other nodes in the network may operate as relay nodes. In this paper, we propose a cooperative multihop routing for the purpose of power savings, constrained on a required bit error rate (BER) at the destination. We derive analytical results for line network topology. It is shown that energy savings of 100% are achievable in line networks with a large number of nodes for BER = 10 -4 constraint at the destination.

I. INTRODUCTION

Energy consumption in multihop wireless networks is a crucial issue that needs to be addressed at all the layers of communication system, from the hardware up to the application. In this paper, we focus on energy savings in routing problem in which messages may be transmitted via multiple radio hops. After substantial research efforts in the last several years, routing for multihop wireless networks becomes a well-understood and broadly investigated problem [1], [START_REF] Li | Energy-efficient cooperative routing in multi-hop wireless ad hoc networks[END_REF]. Nevertheless, with the emergence of new multiple antennas technology, existing routing solutions in the traditional radio transmission model are not efficient anymore. For instance, it is feasible to coordinate the multiple transmissions from multiple transmitters to one receiver simultaneously. As a result, transmitting signals with the same channel from several different nodes to the same receiver simultaneously are not considered collision but instead could be combined at the receiver to obtain stronger signal strength. In [START_REF] Boyer | Multihop diversity in wireless relaying channels[END_REF], the concept of multihop diversity is introduced where the benefits of spatial diversity are achieved from the concurrent reception of signals that have been transmitted by multiple previous terminals along the single primary route. This scheme exploits the broadcast nature of wireless networks where the communications channel is shared among multiple terminals. On the other hand, the routing problem in the cooperative radio transmission model is studied in [START_REF] Khandani | Cooperative routing in static wireless networks[END_REF], where it is allowed that multiple nodes along a path coordinate together to transmit a message to the next hop as long as the combined signal at the receiver satisfies a given SNR threshold value.

This work was supported by the Research Council of Norway through the project 176773/S10 entitled "Optimized Heterogeneous Multiuser MIMO Networks -OptiMO" and the AURORA project entitled "Communications under uncertain topologies". In this paper, a cooperative multihop routing is proposed for Rayleigh fading channels. The investigated system can achieve considerable power savings compared to non-cooperative multihop transmission, when there is a bit error rate (BER) QoS requirement at the destination node. We derive a simple closedform solution for power allocation among the transmitting nodes at each phase. Simulation results show that, using the proposed power allocation strategies, considerable gains are obtained comparing to the non-cooperative multihop transmission.

II. SYSTEM MODEL AND PROTOCOL DESCRIPTION

We consider an arbitrary N -relay wireless network, where information is to be transmitted from a source to a destination. Due to the broadcast nature of the wireless channel, some relays can overhear the transmitted information, and thus, can cooperate with the source to send its data. The wireless link between any two nodes in the network is modeled as a Rayleigh fading narrowband channel. The channel fades for different links are assumed to be statistically independent. The additive noise at all receiving terminals is modeled as zero-mean complex Gaussian random variables with variance N 0 . For medium access, the relays are assumed to transmit over orthogonal channels, thus no interrelay interference is considered in the signal model.

Following [START_REF] Khandani | Cooperative routing in static wireless networks[END_REF], we also assume that each transmission is either a broadcast transmission where a single node is transmitting the information, and the information is received by multiple nodes, or a cooperative transmission where multiple nodes simultaneously send the information to a single receiver. Various scenarios for the cooperation among the relays can be implemented. A general cooperation scenario, m-cooperation,

(1 ≤ m ≤ N ), can be implemented in which each relay combines the signals received from the previous relays and along with that received from the source.

For a general scheme m-cooperation, (1 ≤ m ≤ N ), each receiving node decodes the information after combining the signals received from the previous m transmitting nodes. Fig. 1 shows a wireless multihop network consisting of a source node s, N relays, and a destination node d, which is operating under m-cooperation scenario. The cooperation protocol has N + 1 phases. In Phase 1, the source transmits the information, and the received signal at the destination and the ith relay can be modeled, respectively, as

y 0 = P 0 f 0 s + w 0 , (1)
y 0,i = P 0 f 0,i s + v i , (2)
where P 0 is the average total transmitted symbol energy of the source, since we assume the information bearing symbols s's have zero-mean and unit variance, w 0 and v i are complex zero-mean white Gaussian noise, and f i,j , i = 0, 1, . . . , N , j = 1, 2 . . . , N + 1, are complex Gaussian random variables with zero-mean and variances σ 2 i,j , respectively. In Phase 2, relay nodes are sorted based on their received received SNR, such that relay 1 has the highest received SNR. Generally, in Phase n, 2 ≤ n ≤ N + 1, the previous min{m, n} nodes are transmitting their signal toward the next node. Similar to [START_REF] Khandani | Cooperative routing in static wireless networks[END_REF], we assume that transmitters are able to adjust their phases in such a way that the received signal at the nth receiving node in Phase n is

y n = P 0 |f 0,n | u(m -n) s + n-1 i=max(1,n-m) P i |f i,n | ŝi + v n , (3)
where the function u(x) = 1, when x ≥ 0, and otherwise is zero and the symbol ŝi is re-encoded symbol at the ith relay.

III. BER-BASED LINK COST FORMULATION

In this section, our objective is to find the optimal power allocation required for successful transmission from a set of transmitting nodes to a set of receivers. In order to derive explicit expressions for the link costs, we consider three distinct cases described as follows.

A. Point-to-Point Link Cost

The simplest case is the case where only one node is transmitting within a time slot to a single target node. For decoding the message reliably, the BER must be less than the threshold value BER max .

Assuming a Rayleigh fading link with variance of σ 2 0 in the network, M -PSK or M -QAM modulations, and coherent detection, the average probability of error can be obtain as [5, Eq. ( 6)],

P b e = c π 1 - gP 0 σ 2 0 2N 0 + gP 0 σ 2 0 , (4)
where the parameters c and g are dependent on the modulation type. Using (4), the minimum required power, and hence, the point-to-point link cost is given by

C(tx 1 , rx 1 ) = P 0 = 2N 0 g σ 2 0 1 1 1 - 2 BER max c 2 -1 , (5)
where tx 1 and rx 1 denote the transmitter and receiver nodes, respectively. Since BER max ≪ 1, the link cost in ( 5) can be approximated as

C(tx 1 , rx 1 ) ≈ N 0 g σ 2 0 c 2 BER max . (6)

B. Point-to-Multipoint Link Cost

In this case, we assume a transmitter node tx 1 broadcast its information toward a set of receiving nodes Rx = {rx 1 , rx 2 , . . . , rx m }. Assuming that omnidirectional antennas are used, the signal transmitted by the node tx 1 is received by all nodes within a transmission radius proportional to the transmission power. Hence, a broadcast link can be treated as a set of point-to-point links, and the cost of reaching a set of nodes is the maximum of the costs for reaching each of the nodes in the target set. Thus, the minimum power required for the broadcast transmission, denoted by C(tx 1 , Rx), is given by [START_REF] Boyd | Convex Optimization[END_REF] where C(tx 1 , rx i ) is found from [START_REF] Simon | Digital Communication over Fading Channels: A Unified Approach to Performance Analysis[END_REF].

C(tx 1 , Rx)= max {C(tx 1 , rx 1 ), C(tx 1 , rx 2 ), . . . , C(tx 1 , rx m )},

C. Multipoint-to-Point Cooperative Link Cost

In this case, a set of multiple nodes Tx = {tx 1 , tx 2 , . . . , tx m } cooperate to transmit the same information to a single receiver node rx 1 . Assuming coherent detection at the receiving node, the signals simply add up at the receiver, and acceptable decoding is possible as long as the received BER becomes less than BER max . Now, we are going to derive a tractable BER formula at the receiving node rx 1 , which leads to a closed-form power allocation strategy among the cooperative nodes. Therefore, we use the approach proposed in [START_REF] Ribeiro | Symbol error probablity for general cooperative links[END_REF] to derive the BER expressions for the high SNR regime. That is [6, Eq. ( 10)]

P b e ≃ c t+1 i=1 (2i -1) 2(t + 1)g t+1 t! ∂ t p γ (0) ∂γ t , (8)
where

∂ t pγ (0) ∂γ t
is the tth order derivative of the pdf of the equivalent channel, and the derivatives of p γ (γ) up to order (t-1) are supposed to be zero. Using (3), the received SNR at the receiving node can be written as γ = m i=1 γ i , where

γ i = Pi|fi| 2 N0
with f i denotes the channel between the ith transmitter and the receiving node.

In [START_REF] Ribeiro | Symbol error probablity for general cooperative links[END_REF], the following proposition is proposed, which can be used to calculate the BER expression in [START_REF] Wang | Highperformance cooperative demodulation with decode-and-forward relays[END_REF].

Proposition 1: Consider a finite set of nonnegative random variables {γ 1 , γ 2 , . . . , γ m } whose pdfs p 1 , p 2 , . . . , p m have nonzero values at zero, and denote these values as p 1 (0), p 2 (0), . . . , p m (0). If γ = m i=1 γ i , then all the derivatives of p γ (γ) evaluated at zero up to order m -2 are zero, while the (m -1)th order derivative is given by

∂ m-1 p γ (0) ∂γ m-1 = m i=1 p i (0). (9)
Using Proposition 1 and (8), we get

P b e ≈ c m i=1 (2i -1) 2g m m! m i=1 p i (0). (10)
Hence, using (10) and the fact that the value of an exponential distribution with mean

Piσ 2 i N0 at zero is N0 Piσ 2 i
, the average BER expression can be approximated as

P b e ≈ c m i=1 (2i -1) 2g m m! m i=1 N 0 P i σ 2 i . (11)
The total transmitted power for the multipoint-to-point case is m i=1 P i . Therefore, the power allocation problem, which has a required BER constraint on the receiving node, can be formulated as min

m i=1 P i , s.t. c 2g m m! m i=1 (2i -1)N 0 P i σ 2 i ≤ BER max , P i ≥ 0, for i = 1, . . . , m. (12)
Before deriving the optimal solution for the problem given in (12), the following theorem is needed.

Theorem 1: The optimum power allocation P * 1 , . . . , P * m in the optimization problem stated in (12) is unique.

Proof: The objective function in (12) is a linear function of the power allocation parameters, and thus, it is a convex function. Hence, it is enough to prove that the first constraint in (12), i.e.,

f (P 1 , . . . , P m ) = c 2g m m! m i=1 (2i -1)N 0 P i σ 2 i -BER max , (13)
with [START_REF] Boyd | Convex Optimization[END_REF], it can be verified that f (P 1 , . . . , P m ) is a posynomial function, which is a convex function.

D f = {P i ∈ (0, ∞), i ∈ {1, . . . , m} | f (P 1 , . . . , P m ) ≤ 0}, f : D f -→ R, is a convex function. From
The optimal power allocation strategy for high SNRs is found in the following. However, since the approximate BER expression derived in (11) is an upper-bound on BER, this result can be used reliably.

Proposition 2: For the set of m transmitters, which send a common signal toward the destination, the optimum transmit power coefficients in (12) satisfy the following equations

P i = Ψ (m) BER max N 0 σ 2 i m k=1 k =i N 0 P k σ 2 k , i = 1, . . . , m, (14)
where

Ψ (m) = c m i=1 (2i -1) 2g m m! , ( 15
)
Proof: The Lagrangian of the problem stated in (12) is

L(P 1 , . . . , P m ) = m i=1 P i + λf (P 1 , . . . , P m ). (16)
For nodes i = 1, . . . , m with nonzero transmitter powers, the Kuhn-Tucker conditions are

∂ ∂P i L(P 1 , . . . , P m ) = 1 + λ ∂ ∂P i f (P 1 , . . . , P m ) = 0, (17)
where

∂ ∂P i f (P 1 , . . . , P m ) = -Ψ (m) N 0 P 2 i σ 2 i m k=1 k =i N 0 P k σ 2 k . (18)
Using ( 17) and (18), we have

P 2 i = λ Ψ (m) N 0 P 2 i σ 2 i m k=1 k =i N 0 P k σ 2 k , (19)
for i = 1, . . . , m. Since the strong duality condition [7, Eq. (5.48)] holds for convex optimization problems, we have λf (P 1 , . . . , P m ) = 0 for the optimum point. If we assume Lagrange multiplier has a positive value, we have f (P 1 , . . . , P m ) = 0, which is equivalent to

BER max = Ψ (m) m k=1 N 0 P k σ 2 k . (20)
Dividing both sides of equalities ( 19) and (20), we can find the Lagrange multiplier as

λ = P i BER max . (21)
P * i = Ψ (m) BER max m k=1 N 0 σ 2 k 1 m . ( 22
)
Proof: In Theorem 1, we have shown, this problem has a unique solution. Now, using Proposition 2, by the fact that the problem in (12) should have a unique solution, we put initial values P * 1 , . . . , P * m in ( 14), and we observe that the closedform solution as ( 22) is achieved, which satisfies the set of equations in ( 14).

An interesting property of P * i derived in ( 22) is that it is just dependent on the product of all path-loss coefficients of links. Therefore, P * i s can be calculated in a decentralized manner by broadcasting the product term from the receiving node toward the transmitting nodes. Using Theorem 2, the resulting cooperative link cost C(Tx, rx 1 ), defined as the optimal total power, is given by

C(Tx, rx 1 ) = m i=1 P * i = m Ψ (m) BER max m k=1 N 0 σ 2 k 1 m . (23)

IV. ENERGY SAVINGS VIA COOPERATIVE ROUTING

The problem of finding the optimal cooperative route from the source node to the destination node can be mapped to a Dynamic Programming (DP) problem [START_REF] Khandani | Cooperative routing in static wireless networks[END_REF]. As the network nodes are allowed only to either fully cooperate or broadcast, finding the best cooperative path from the source node to the destination has a special layered structure. In [START_REF] Khandani | Cooperative routing in static wireless networks[END_REF], it is shown that in a network with N + 1 nodes, which has 2 N nodes in the cooperation graph, standard shortest path algorithms have a complexity of O(2 N ). Hence, finding the optimal cooperative route in an arbitrary network becomes computationally intractable for larger networks. For this reason, we restrict the cooperation to nodes along the optimal noncooperative route. That is, at each transmission slot, all nodes that have received the information cooperate to send the information to the next node along the minimum energy noncooperative route [START_REF] Khandani | Cooperative routing in static wireless networks[END_REF]. Therefore, with the help of the link cost expressed in Subsections III-A and III-B, the minimum-energy non-cooperative route is first selected, which has N intermediate relays. Then, nodes along the optimal non-cooperative route cooperate to transmit the source information toward the destination. That is, at each transmission slot, all nodes that have received the information cooperate to send the information to the next node along the minimum energy non-cooperative route. In the nth transmission slot, the reliable set is Tx n = {s, r 1 , . . . , r n-1 }, which is including the source node and the previous relays r i , i = 1, . . . , n -1. The link cost associated with the nodes in Tx n , which cooperate to send the information to the next node n, follows from (23), and is given by

C(Tx n , n) = n Ψ (n) BER max n-1 k=0 N 0 σ 2 k,n 1 n . ( 24
)
Note that the nth node denotes the nth relay when n ≤ N , and the destination node when n = N +1. Therefore, the total transmission power for the cooperative multihop system is

P T (coop)= N +1 n=1 C(Tx n , n) = N +1 n=1 n Ψ (n) BER max n-1 k=0 N 0 σ 2 k,n 1 n . (25)
For the case of m-cooperation scheme, in which just previous closest nodes cooperate to transmit along the noncooperative route, P T (cooperative) in (25) can be modified to

P T (m-coop) = N +1 n=1 C m (Tx n , n) = m n=1 n Ψ (n) BER max n-1 k=0 N 0 σ 2 k,n 1 n + N +1 n=m+1 m Ψ (m) BER max n-1 k=n-m N 0 σ 2 k,n 1 m . (26)
The energy savings for a cooperative routing strategy relative to the optimal noncooperative strategy is defined as

Energy Savings = P T (noncoop) -P T (coop) P T (noncoop) , (27)
where P T (coop) is computed in (25) and (26) for the case of full-cooperation and m-cooperation routings, respectively. P T (noncoop) denotes the total transmission power for the noncooperative multihop strategy. Using (6), P T (noncoop) can be calculated as

P T (noncoop) = c 2 BER max N n=0 N 0 g σ 2 n,n+1 . (28)
For each of these topologies, we derive the optimal noncooperative route and obtain a lower bound on the optimal energy savings achievable by cooperative routing. The bound is obtained by deriving analytical expressions for energy savings for a sub-optimal cooperative route, where cooperation is restricted to nodes along the optimal non-cooperative route. That is, at each transmission slot, all nodes that have received the information cooperate to send the information to the next node along the minimum energy non-cooperative route.

A. BER Upper-Bounds at the Destination Node

In this subsection, we will view the system from our end-toend equivalent BER perspective. That is, we represent BER max in each step in terms of the required BER at the destination.

In the case of non-cooperative multihop system in which N relays are in cascade, when BPSK is used, the BER P b n at nth node is affected by all previous n -1 hops and can be iteratively calculated according to the recursion [8]

P b n = (1 -P b n-1 )P b n-1,n + P b n-1 (1 -P b n-1,n ), (29)
with P b 0 = 0, where P b n-1,n is the BER from the (n -1)th node to the nth node. The end-to-end BER at the destination is given by using n = N + 1 in (29). Since the BER at the destination should be less than the required BER QoS, it is enough to consider the upper-bound for the BER. Thus, for any general constellation, the BER can be bounded as P b n ≤ (1 -P b n-1 )P b n-1,n +P b n-1 . Assuming the power allocation strategies derive in Section III, P b n bound can be written as

P b n ≤ 1 - (1 -BER max ) n .
For the case of cooperative routing, the following upperbound can be obtained in the nth node

P b n ≤ 1 -   (1 -P b n-1,n ) n-1 i=max{1,n-m} (1 -P b i )   . ( 30
)
If the power allocation strategy derived in ( 22) is used, (30) can be rewritten as

P b n ≤ 1 -   (1 -BER max ) n-1 i=max{1,n-m} (1 -P b i )   . (31)
To get an insight into the relationship between the endto-end BER P b N +1 and BER max , the upper-bound on P b N +1

when full cooperation is used can be represented as

P b N +1 ≤ 1 -(1 -BER max )
2 n-1 .

V. SIMULATION RESULTS

In this section, we present some simulation results to quantify the energy savings due to the proposed cooperative routing scheme. We consider a regular line topology where nodes are located at unit distance from each other on a straight line. The optimal non-cooperative routing in this network is to always send the information to the next nearest node in the direction of the destination. From ( 6), (28), and by assuming that σ 2 i,j is proportional to the inverse of the distance squared, the total power required for non-cooperative transmission can be calculated as

P T (noncoop) = (N + 1) cN 0 2 g BER max . (32)
Since we restrict the cooperation to nodes along the optimal non-cooperative route, the total transmitted power for full-cooperation and m-cooperation in line networks can be obtained from ( 25) and (26), respectively. Thus, by replacing Ψ (n) from (28) and σ 2 i,j = 1/|i -j| 2 , we have

P T (cooperative) = N +1 n=1 n c n i=1 (2i -1) BER max N n 0 n! 2g n 1 n , (33)
P T (m-coop) = m n=1 n c n i=1 (2i -1) BER max N n 0 n! 2g n 1 n + (N -m + 1)m c m i=1 (2i -1) BER max N m 0 m! 2g m 1 m , (34)
In Fig. 2, we compare the achieved energy savings of the proposed cooperative routing with respect to the non-cooperative multihop scenario, in which satisfying the BER max at each step is used as a performance criteria. For the BER max = 10 -4 , it can be observed that using the full cooperation scheme around 99% saving in energy is achieved when 4 relays are employed. Since the corresponding curve has an optimum performance when N = 4, we consider the m = 5 cooperation as an appropriate scheme. As it can be observed from Fig. 2, increasing the number of nodes in the network, 100% savings in energy is achievable. For the case of BER max = 10 -4 , the same characteristics can be seen.

Fig. 3 demonstrates a lower-bound on the obtainable energy saving in line networks, when the required BER at the destination, i.e., BER d , should be satisfied. We use (31) to get a reliable power allocation at transmitting nodes to fulfil the required BER QoS at the destination node. For two cases of BER d = 10 -3 and BER d = 10 -4 , vast amount of energy savings are obtainable. Since the maximum values of the curves corresponding to the full-cooperation routing occur when 5 and 8 relays are used, the 6-cooperation and 9-cooperation are used for BER d = 10 -3 and BER d = 10 -4 cases, respectively.

Fig. 1 .

 1 Fig. 1. Wireless multihop network under m-cooperation.

  coop.; BER max = 10 -4 m=5-coop.; BER max = 10 -4 Full coop.; BER max = 10 -5 m=5-coop.; BER max = 10 -5

Fig. 2 .

 2 Fig. 2. The average energy savings curves versus the number of transmitting nodes, (N + 1), employing full-cooperation and m-cooperation with two different BERmax constraints.

Fig. 3 .

 3 Fig. 3. The average energy savings curves versus the number of transmitting nodes (N + 1) relay networks employing full-cooperation and m-cooperation with two different BER d constraints.