M M Timm

E Oliviero

W Sun

Séverine Gomès

Georges Hamaoui

P F P Fichtner

N Frety

S Gomes

resistivity and thermal conductivity of amorphous CrSi2 thin films

Introduction

Transition metal silicides (TMSi 2 ) have favorable physical properties that make them suitable for applications in microelectronics [START_REF] Murarka | Silicide thin films and their applications in microelectronics[END_REF], ceramics [START_REF] Chen | Transition metal silicides: fundamentals, preparation and catalytic applications[END_REF], spintronic [START_REF] Jin | Synthesis, Characterization and Physical Properties of Transition Metal Silicide Nanowires[END_REF] and thermoelectricity [START_REF] Fedorov | Thermoelectrics of transition metal silicides[END_REF][START_REF] Nozariasbmarz | Thermoelectric silicides A review[END_REF]. Among them, CrSi 2 is a narrow bandgap p-type semiconductor with a high electrical conductivity considered for a wide range of applications, from electro-optic interconnects [START_REF] Bost | Semiconducting silicides as potential materials for electro-optic very large scale integrated circuit interconnects[END_REF], Schottky junctions [START_REF] Barus | Analysis of I-V measurements on CrSi2-Si Schottky structures in a wide temperature range[END_REF][START_REF] Anilturk | Electrical transport at a nonideal CrSi2-Si junction[END_REF], to IR detection devices [START_REF] Bost | An investigation of the optical constants and band gap of chromium disilicide[END_REF][START_REF] Goroshko | Enhancement of near IR ensitivity of silicon-based photodetectors[END_REF]. Furthermore, CrSi 2 is chemically and thermally stable at high temperatures. Combined with its high electrical conductivity and Seebeck coefficient, CrSi 2 becomes an engaging material for potential use in thermoelectric applications [START_REF] El Qader | Structural, electrical, and thermoelectric properties of CrSi2 thin films[END_REF][START_REF] Dasgupta | Structural, thermal, and electrical properties of CrSi2[END_REF][START_REF] Goroshko | Formation and Thermoelectric Properties of Si/CrSi2/Si (001) Heterostructures with Stressed Chromium Disilicide Nanocrystallites[END_REF][START_REF] Nishida | The crystal growth and thermoelectric properties of chromium disilicide[END_REF]. However, CrSi 2 has a high thermal conductivity, which negatively affects the figure of merit ZT and hinders its use as practical thermoelectric devices [START_REF] El Qader | Structural, electrical, and thermoelectric properties of CrSi2 thin films[END_REF][START_REF] Pandey | Origin of enhanced thermoelectric properties of doped CrSi2[END_REF]. Nevertheless, due to its extensive range of potential applications, it is of great interest to understand the behavior of CrSi 2 under different conditions.

Many studies have been performed in single and polycrystalline CrSi 2 , including bulk and thin films. It has been shown, for example, that the electrical properties of CrSi 2 are pretty sensitive to variations in the composite stoichiometry [START_REF] Burkov | Energy filtering enhancement of thermoelectric performance of nanocrystalline Cr1-x Six composites[END_REF], deposition methods [START_REF] Nava | Temperature dependence of semiconducting and structural properties of Cr Si thin films[END_REF][START_REF] Caricato | Morphological and structural characterizations of CrSi2 nanometric films deposited by laser ablation[END_REF], thermal treatments [START_REF] Novikov | Crystallization and Transport Properties of Amorphous Cr-Si Thin Film Thermoelectrics[END_REF] and the nature of the substrate [START_REF] Kim | Electrical properties of CrSix, Cr/CrSix/Cr/ CrSix, and CrSix/Si/CrSix/Si sputtered on alumina plates[END_REF][START_REF] Novikov | Enhancement of thermoelectric properties in nanocrystalline M -Si thin film composites ( M = Cr, Mn )[END_REF]. Most studies have focused on crystalline CrSi 2 , but in its amorphous form, CrSi 2 can also present interesting behaviors. Amorphous materials have a wide range of applications in microelectronics devices, such as thin-film transistors (TFTs) [START_REF] Park | Review of recent developments in amorphous oxide semiconductor thin-fi lm transistor devices[END_REF], memristors [START_REF] Nili | Nanoscale Resistive Switching in Amorphous Perovskite Oxide (a-SrTiO3) Memristors[END_REF], solid-state nonvolatile phase-change memory (PCM) [START_REF] Ha | Use of ordered mesoporous SiO2 as protection against thermal disturbance in phase-change memory[END_REF] and photovoltaic solar cells [START_REF] Schropp | New Developments in Amorphous Thin-Film Silicon Solar Cells[END_REF], and in thermoelectric (TE) devices [START_REF] Nolas | Amorphous Thermoelectrics[END_REF]. It has been suggested that some amorphous semiconductors can attain good thermoelectric properties due to their modest to good electrical conductivity (r) and poor thermal conductivity (j), improving the r/j ratio as required by the energy conversion figure of merit [START_REF] Nolas | Amorphous Thermoelectrics[END_REF][START_REF] Wu | Comparative Study of the Thermoelectric Properties of Amorphous Zn41Sb59 and Crystalline Zn4Sb3[END_REF][START_REF] Kim | Transparent Amorphous Oxide Semiconductor as Excellent Thermoelectric Materials[END_REF]. The performance of these devices depends on the operating temperature, also concerning their physical properties and thermal stability. This situation is challenging for amorphous structures and of great importance to optimize their use concerning device functionality and reliability.

In this sense, we explore the use of ion implantation to modify the electrical and thermal properties of amorphous CrSi 2 . The combination of ion implantation and in situ or ex situ thermal treatments allows tailoring several properties of semiconductors. It includes not only the electrical conductivity [START_REF] Sanders | The effect of argon implantation on the conductivity of boron implanted silicon[END_REF][START_REF] Uzan-Saguy | Electrical isolation of GaN by ion implantation damage: Experiment and model[END_REF][START_REF] Choudhary | Nitrogen ion implantation effects on the structural, optical and electrical properties of CdSe thin film[END_REF] but also the microstructure configuration to provide, for example, impurity gettering [START_REF] Fichtner | Copper gettering at half the projected ion range induced by lowenergy channeling He implantation into silicon[END_REF][START_REF] Brett | Trapping of Pd, Au, and Cu by implantation-induced nanocavities and dislocations in Si[END_REF], end-of-range defect engineering [START_REF] Isoda | Observation of silicon self-diffusion enhanced by the strain originated from end-ofrange defects using isotope multilayers[END_REF] or even strain relaxation effects [START_REF] Kato | Fabrication of high-quality strain relaxed SiGe(110) films by controlling defects via ion implantation[END_REF][START_REF] Luysberg | Effect of helium ion implantation and annealing on the relaxation behavior of pseudomorphic Si1-xGex buffer layers on Si (100) substrates[END_REF]. Ion beam modification techniques are explored to improve crystalline materials' electric and thermal properties via doping or by forming point and extended defects to act as phonon scattering centers, thus reducing the phonon contribution to the thermal conductivity. For amorphous materials, the situation is different. Ion implantation can still be used for doping, but the microstructure modification requires a different approach. Amorphous materials provide an extra degree of freedom concerning ion implantation experiments. Since the ion implantation/irradiation processes can occur at nonthermal equilibrium conditions, it allows modifying the compaction degree of the amorphous lattices or enables the formation of crystalline clusters and even the promotion of complete recrystallization resulting in a highly defective crystal lattice.

In this paper, we report on the effects of Ne and Al ion implantation on the microstructure, electrical and thermal properties of amorphous CrSi 2 thin films. The implantations are done at different fluences in substrates kept at room temperature or heated at 250 °C. The microstructure of the samples is characterized by transmission electron microscopy (TEM). We also use electrical resistivity and scanning thermal microscopy (SThM) measurements to assess electrical and thermal conductivities. The discussions consider Ne as an inert element and how the matrix responds to the implantation-induced atomic displacements, causing, for example, the formation of nanosized Ne bubbles and CrSi 2 crystallites. Al is a common dopant in crystalline CrSi 2 . Here we investigate its doping and irradiation effects throughout the amorphous CrSi 2 films. The results obtained demonstrate that the ion implantation/irradiation process causes significant modifications to the electrical and thermal properties of amorphous CrSi 2 thin films.

Experimental procedure

CrSi 2 amorphous films with an average thickness of 260 nm were deposited in Si (100) substrates by radio-frequency magnetron sputtering technique in a high-vacuum chamber (base pressure of 2 9 10 -6 mbar) with a high-purity CrSi 2 target (3'', 99.95%, Neyco) using the experimental protocol developed previously [START_REF] Moll | Elaboration et e ´tude des proprie ´te ´s thermoe ´lectriques du disiliciure de chrome sous forme de monocristal, de couche mince et de nanofil[END_REF]. The sputtering targets were operated at 120 W under a pressure of 0.51 Pa with a highpurity Ar flow (99.99%). A SiO 2 sublayer with a thickness of 25 nm was deposited by sputtering before the CrSi 2 layer. The depositions were done at room temperature with the multi-target Plassys MP4505 equipment (CTM, Universite ´de Montpellier, France). One series of samples contains Al implantations at distinct energies, calculated to produce a uniform concentration-depth distribution along the film. A second sample series has Ne implantations in a uniform concentration-depth distribution. The Ne and Al implantations were done at room temperature and at a temperature of 250 8C. The fluences of the Ne and Al implantations at 250 8C were adjusted to produce similar damage levels, computed in displacement per atom (dpa) units [START_REF] Stoller | On the use of SRIM for computing radiation damage exposure[END_REF]. The implantation profiles and fluences, simulated using the SRIM code (E d = 25 eV (Cr), E d = 15 eV (Si)) [START_REF] Ziegler | Stopping and range of ions in matter: SRIM[END_REF], lead to atomic concentrations of & 0.008 at.% for Al (room temperature implantation), of & 0.64 at.% for Al (250 8C implantation) and & 1 at.% for Ne. All the implantation experiments were done using a 500 kV HVEE ion implanter (Laborato ´rio de Implantac ¸a ˜o Io ˆnica, Instituto de Fı ´sica, UFRGS, Brazil) maintaining the beam current below 500 nA/cm 2 to avoid the heating of the substrate that could affect the sample microstructure. The sample's temperature during implantations is kept constant, feedback-controlled using a type K thermocouple and a Tantalum resistance attached to the sample holder. Table 1 summarizes the implantation parameters.

The composition and stoichiometry of the as-deposited CrSi 2 film were accessed via Rutherford backscattering spectrometry (RBS). The RBS measurements were performed using a 1.5 MeV He ? beam from a 3 MeV HVEE Tandem accelerator (Laborato ´rio de Implantac ¸a ˜oI o ˆnica, Instituto de Fı ´sica, UFRGS, Brazil). The microstructure evolution of the samples was analyzed via transmission electron microscopy (TEM) using cross-sectional samples prepared by focused ion beam (FIB) with the FEI Strata DB 235 equipment (IEMN, Lille, France). TEM observations were made using a JEOL JEM 2200FS transmission electron microscope operating at 200 kV (MEA Platform, Universite ´de Montpellier, France). Ne and Al distributions along the CrSi 2 film were assessed via scanning transmission electron microscopy-energy-dispersive X-ray spectrometry (STEM-EDX). The STEM-EDX measurements were performed using a 100 mm 2 Oxford Xmax TLE windowless detector. The crystallinity of the CrSi 2 samples considered in this study was assessed via the selected area electron diffraction (SAED) technique. The electrical resistivity (q) measurements on implanted and non-implanted CrSi 2 thin layers were carried out between 80 and 330 K with a magnetic field of 0.5 T by the van der Pauw method. The electric contacts were made using In droplets, and the measurements were performed with an Ecopia HMS 5000 four-point probe equipment (CTM, Universite de Montpellier, France). Scanning thermal microscopy (SThM) in active mode using the Wollaston wire probe [START_REF] Guen | Scanning thermal microscopy on samples of varying effective thermal conductivities and identical flat surfaces[END_REF] in ambient air conditions was used to characterize the thermal conductivity (j) dependence of the CrSi 2 films.

Results and discussion

As-deposited sample characterization

Figure 1a shows cross-sectional TEM micrographs of the as-deposited, non-implanted CrSi 2 sample. The film was deposited over a Si wafer containing a 25 nm SiO 2 layer. The S arrow points to the surface of the film. For the FIB sample preparation, the film was covered by a thin carbon layer and then by a Pt film also appearing in the micrograph. The CrSi 2 film deposition process results in columnar structures due to the low mobility of the atoms adsorbed by the substrate surface during the sputtering deposition process. This type of growth is predicted for room temperature depositions by the Thornton model in zone 1 [START_REF] Thornton | Influence of apparatus geometry and deposition conditions on the structure and topography of thick sputtered coatings[END_REF][START_REF] Alfonso | Thin Film Growth Through Sputtering Technique and Its Applications[END_REF]. Figure 1b shows the SAED pattern from the CrSi 2 film depicted in Fig. 1a. In cross-sectional TEM specimens, SAED measurements are sensitive to the formation of small crystallites with characteristic dimensions above 1 nm. The observation of a diffuse and broad ring pattern indicates that the sample can be considered amorphous within the sensitivity level of the technique. Figure 1c depicts the RBS spectrum. It is well fitted by a SIMNRA code [START_REF] Mayer | SIMNRA User's Guide[END_REF] simulation (continuous line), demonstrating that the CrSi 2 film is stoichiometric. throughout the film (white disks surrounded by a dark ring produced by bright-field underfocus imaging conditions). Figure 2a shows that Ne bubble formation preferentially occurs along with the columnar structures of the film. The Al implantation to a concentration of & 0.008 at.% (Fig. 2b) does not affect the columnar structure arrangement. Also, it does not cause any other observable microstructure modification concerning the pristine structure shown in Fig. 1a. The scattered points in Fig. 2c,d Inert gases have an extremely low solubility limit and tend to agglomerate into more stable gas-vacancy complexes leading to bubble formation [START_REF] Da Silva | Formation of bubbles and extended defects in He implanted (1 0 0) Si at elevated temperatures[END_REF][START_REF] Oliviero | Damage accumulation in neon implanted silicon[END_REF][START_REF] Oliviero | Characterization of neon implantation damage in silicon[END_REF]. The same behavior also occurs in amorphous structures where the concept of vacancies correlates with small free volume pockets in the network structure [START_REF] Qiang-Li | Gas densities in helium bubbles in nickel measured by small angle neutron scattering[END_REF]. The resolution limit for underfocus phase-contrast imaging conditions in TEM observations is about 0.7 nm. Therefore, Fig. 2a reveals only the visible bubbles, coarsened by gettering additional Ne atoms or by interacting with other surrounding bubbles via, for example, irradiation-induced bubble migration and coalescence process [START_REF] Oliviero | Damage accumulation in neon implanted silicon[END_REF]. The same atomic mobility behavior discussed for the Ne atoms may also occur for the Al-implanted sample. In this case, the atomic mobility can be inferred by the difference in the Al concentration-depth profile measured by EDX and predicted by the SRIM code (Fig. 2d).

Implantations at room temperature

Ne Implantation at 250 8C

The effects of the Ne implantation at 250 8C in the CrSi 2 film structure can be inferred by the TEM micrographs presented in Fig. 3a which shows that the implantation effects are quite similar within the entire film thickness.

Figure 3b shows an enlarged micrograph from a region of Fig. 3a. This figure better illustrates the presence of crystal grains (darker domains obtained via bright-field diffraction contrast) and the homogeneously dispersed Ne bubbles (depicted in brightfield underfocus imaging conditions). The micrograph also suggests that the columnar grain structure has been smoothed out. These features are quite distinct from those observed for the RT implantation case (Fig. 2a). Figure 3c shows the EDX measurement for Ne distribution as a function of distance in the film. The depth distribution of the Ne content is very similar to the one obtained by implantation at room temperature (Fig. 2c).

Figure 3a, b shows that, for implantations at 250 8C, a significant fraction of the amorphous structure has been transformed into randomly oriented crystallites with sizes with diameters from 5 to about 20 nm. This phenomenon is also evidenced by the dense ringlike distribution of diffraction spots in the SAED pattern presented in Fig. 4a. The ring pattern is well fitted using the hexagonal structure of the CrSi 2 crystal lattice (space group P6 2 22, lattice parameters a = 4.42758 A ˚and c = 6.36805 A ˚, Wyckoff positions 3d (Cr) and 6j (Si)) [START_REF] Dasgupta | Structural, thermal, and electrical properties of CrSi2[END_REF]. The (h,k,l) index of the rings is presented in the figure. Additional rings formed by rather weak spots are observed, showing that other phases have been produced in a minor volume fraction. Figure 4b shows a dark-field TEM micrograph produced by selecting a spot from the innermost ring of the SAED pattern. The dark-field micrograph was taken in the same region as the bright-field ones from Figs. 4a,b.

It seems worth noting that the Ne implantation at 250 8C induced local crystallization reactions at a temperature about eighty degrees lower than the crystallization temperature T = 327 °C expected by thermal treatments [START_REF] Rowe | Thermoelectrics Handbook: macro to nano[END_REF]. This phenomenon is typically expected in ion-implanted/irradiated samples and is usually interpreted as a radiation-enhanced phase transformation process [START_REF] Was | [END_REF]. The formation of a dense array of nanocrystallites is probably connected to the bubbles and the remaining columnar grain boundaries, which may act as inhomogeneous nucleation sites for the amorphous-crystalline phase transition.

Electrical resistivity measurements

Figure 5 shows the electrical resistivity q measurements as a function of temperature S for the as-deposited and implanted samples. The behavior of the q(T) curve for the as-deposited sample is in good agreement with previous measurements in CrSi2 [START_REF] Dasgupta | Structural, thermal, and electrical properties of CrSi2[END_REF][START_REF] Perumal | Effect of co-substitution of Mn and Al on thermoelectric properties of chromium disilicide[END_REF]. The q(T) curves for the RT Ne and Al implantations present a shift toward lower q values. The curves show similar temperature dependence to the pristine deposited film for the measured temperature range from 60 to 330 K.

Al atoms can be considered as acceptor dopants for CrSi 2 [START_REF] Perumal | Effect of co-substitution of Mn and Al on thermoelectric properties of chromium disilicide[END_REF]. If these atoms become active dopants within the amorphous lattice, a charge carrier concentration is expected to increase within the valence band and, therefore, decrease film resistivity. This behavior is indeed in qualitative agreement with our experimental results. On the other side, as an inert gas atom, Ne should not behave as a dopant. Nevertheless, the decrease in q values is more pronounced for the Ne than for the Al implantation case. This behavior seems consistent with the concept that small defect clusters may become electrically active and cause the formation of charge carriers as observed for crystalline semiconducting lattices. This argument has been previously presented, for example, by Saji et al., that report a decrease in q observed in Bi 2 Se 3 samples after H and He irradiation [START_REF] Saji | Effects of doping, electron irradiation, H? and He? implantation on the thermoelectric properties of Bi2Se3 single crystals[END_REF]. Bala et al. also consider that the reduction in q for Agimplanted PbTe samples is caused by electrically active defects produced during ion implantation [START_REF] Bala | Enhancement of thermoelectric power of PbTe thin films by Ag ion implantation[END_REF]. They also show that the decrease in q depends on the implantation fluence, which correlates with the buildup of the defect concentration.

Furthermore, Banwell et al. performed Ne, Ar and Xe irradiations in CrSi 2 and obtained a considerable decrease in q [START_REF] Banwell | Effects of ion irradiation on conductivity of CrSi2 thin films[END_REF]. They also explained the result in terms of an increase in free carrier concentration that arises from completely ionized defects produced by the irradiation. As presented in [START_REF] Desalvo | Doping of amorphous silicon by potassium ion implantation[END_REF] and [START_REF] Show | The Effects of defects on the electrical properties of amorphous carbon layer formed by ion implantation into CVD diamond films[END_REF], the concept of implantation-induced defects affecting the carrier concentration in crystals can also be generalized to an amorphous lattice considering the formation of dangling bonds. We can use the same arguments to explain the resistivity reduction observed for both the Ne and the Al implantation cases. Since the damage level for the RT Al implantation is approximately 100 times lower than the Ne implantation, the corresponding resistivity decrease seems to be more affected by the Al doping effect. On the other side, the dangling bonds produced by the RT Ne implantation reduce the resistivity despite the bubble system effect. However, if the defects produced by ion implantation/irradiation processes do not affect the carrier concentration, the opposite behavior can be expected. For example, in a recent study, Tureson et al. have analyzed the influence of room temperature Mg implantation on the thermoelectric properties of ScN films [START_REF] Tureson | Effect of ionimplantation-induced defects and Mg dopants on the thermoelectric properties of ScN[END_REF]. They observed an increase in q and a decrease in j. The increase in q was explained considering that the defects induced by the implantation process contribute to scattering the charge carriers, causing a significant reduction in their mobility. However, in the case of amorphous lattices, atomic displacements induced by ion implantation may not produce lattice defects acting as efficient scattering centers for charge carriers.

The electrical resistivity values for the Ne implantation at 250 8C are about 70% larger than the asdeposited film value (Fig. 5). Figure 4 shows that the Ne implantation causes the formation of a dense array of randomly oriented nanocrystalline structures. These nanocrystallites introduce interfaces that can selectively scatter carriers. The increase in electrical resistivity in partially crystallized CrSi 2 thin films was also observed by Moll et al. [START_REF] Moll | Influence of Nanostructuration on the Vibrational, Electronic, and Optical Properties of CrSi2 Thin Films[END_REF]. Its occurrence is also credited to an additional interface scattering mechanism produced by the presence of nanocrystallites. We must also consider that the implantation temperature increases the mobility of defects produced by Ne ions and can induce their agglomeration. These defects can reduce the charge carrier mean free path with a consequent increase in electrical resistivity. In addition, considering that, if the average size of the nanocrystals is comparable to the de Broglie wavelength of the carriers in the material, the crystallite may effectively act as scattering centers [START_REF] Novikov | Grain boundary scattering contribution to the thermopower in Cr-Si nanocrystalline films[END_REF]. For CrSi 2 samples, the de Broglie wavelength is k = 5 nm, comparable to the average crystallite size produced by the 250 8C Ne implantation. In the present case, we assume, however, that it is the lattice disorder generated by a high number of nanocrystallites' interface regions the main cause for the increase in electrical resistivity. The same explanation applies to the Al 250 8C case, which presents similar electric resistivity values (Fig. 5).

Thermal conductivity measurements

Figure 6 shows the effective thermal conductivity (j eff ) measurements obtained with SThM for the ionimplanted samples. j eff could not be successfully determined for the as-deposited sample because the SThM technique presents significant uncertainties with increasing thermal conductivity values. Therefore, following the results from Ref. [START_REF] Dasgupta | Structural, thermal, and electrical properties of CrSi2[END_REF], we assume j eff & 10 W/mK as a reference value for the as-deposited film. For each sample, ten point measurements were taken at five different locations. The effective thermal conductivity of thin films and layered compounds may be considered a logical extension of the thermal conductivity, such as for solid, liquid or gaseous materials [START_REF] Adelberg | Effective Thermal Conductivity and Multilayered Insulation[END_REF]. In our case, the effective thermal conductivity j eff is the thermal conductivity of the film-on-substrate system (not just CrSi 2 ); its value includes a contribution from the 25 nm SiO 2 film, that from the Si substrate and the thermal contact resistances between the layers.

When compared to the as-deposited sample (j eff & 10 W/mK) [START_REF] Dasgupta | Structural, thermal, and electrical properties of CrSi2[END_REF], all the implanted samples present a significant decrease in the thermal conductivity. The samples implanted at RT present a less pronounced reduction in thermal conductivity than their 250 8C counterparts. In the case of the Al implantation at RT, even though the implanted atomic concentration is very low compared to Ne at RT (see Table 1), the formation of defect clusters can play a dominant role in controlling thermal conductivity behavior. It has been shown, for example, that even very low irradiation doses (as low as 0.00032 dpa-displacement per atom) decrease the thermal diffusivity of self-irradiated tungsten due to the formation of sub-nanometric defects [START_REF] Reza | Thermal diffusivity degradation and point defect density in self-ion implanted tungsten[END_REF]. Furthermore, the introduction of isoelectronic elements preserves the crystalline electronic structure while enhancing electrical conductivity and creating a mass contrast that disrupts the path of the heat carriers [START_REF] Rowe | Thermoelectrics Handbook: macro to nano[END_REF][START_REF] Snyder | Complex thermoelectric materials[END_REF]. In this sense, Perumal et al. showed that the introduction of Al and Mn as co-substitutors in CrSi 2 induces a decrease in thermal conductivity due to a mass fluctuation effect from the atomic weight difference between the solute and solvent atoms [START_REF] Perumal | Effect of co-substitution of Mn and Al on thermoelectric properties of chromium disilicide[END_REF].

In the case of the samples implanted with Ne, the critical feature is the presence of Ne bubbles. Bubbles represent a discontinuity on the solid structure of the matrix and may act as efficient scattering centers for the phonons. Therefore, the phononic component of the thermal conductivity decreases, explaining the reduction observed on the j eff measurements of these two samples compared to the non-implanted CrSi 2 sample. The samples implanted with Ne and Al at a temperature of 250 8C present a more significant reduction in j eff than the RT Ne-and Al-implanted samples. In these cases, we can assume that the nanocrystallites formed during the 250 8C implantations are effective scattering centers for the phonons and contribute to a more pronounced effect on j eff . Similarly, Novikov et al. show that nanocrystal interfaces in stoichiometric CrSi 2 thin films largely contribute to the increase in the electrical resistivity, which also happens in our case for the 250 8C Neimplanted CrSi 2 sample (see Fig. 5) but also to the decrease in the thermal conductivity [START_REF] Novikov | Grain boundary scattering contribution to the thermopower in Cr-Si nanocrystalline films[END_REF]. For the Al 250 8C case, the combination of the nanocrystallites and the mass fluctuation effect discussed above could explain the significant decrease in the thermal conductivity.

Conclusions

This paper shows that ion implantation induces substantial modifications on the microstructure, electrical resistivity q and thermal conductivity j of amorphous 260-nm-thick CrSi 2 films. Al and Ne implantations performed at room temperature induce a decrease in both q and j. The decrease of q is attributed to the introduction of dopants (Al) or of electrically active irradiation-induced defects acting as dopants. She decrease in j results from the formation of Ne bubbles acting as scattering centers for phonons or by creating a mass contrast effect by incorporating Al in the amorphous matrix. We also demonstrate the increase in q for Ne and Al implantations at 250 8C. This is attributed to the formation of a dense array of nanocrystallites that interfere on the mobility of charge carriers. The presence of such a dense array of nanocrystallites also causes the decrease of j.

In a more general way, this work demonstrates that electrical and thermal conductivity behavior of amorphous structures can be tailored via ion implantation or irradiation techniques. This is done by exploring doping, amorphous to crystalline phase transitions and irradiation-induced electrically active defects, providing new possibilities for applications.

Figure

  Figure 2a, b shows cross-sectional TEM micrographs of room temperature (RT) Ne-and Al-implanted CrSi 2 samples. The Ne implantation to a concentration of &1 at.% triggered the formation of a dense array of Ne-filled cavities (i.e., bubbles) distributed

  represent, respectively, the EDX signals for Ne and Al as a function of distance along with the film. The dash dotted lines correspond to the Ne and Al concentration-depth profiles predicted by SRIM simulations [39]. The EDX measurements indicate that the concentration-depth profile of the implanted Ne ions did not diffuse much into the CrSi 2 layer due to the trapping of Ne by bubbles. Differently, Al has diffused along the film due to implantation-enhanced diffusion and knock-on by successive ions. The corresponding SAED patterns of Ne-and Al-implanted samples (not shown) remain similar to the one in Fig. 1b. These measurements indicate that the RT implantations preserved the amorphous structure of the film.

Figure 2

 2 Figure 2 Cross-sectional TEM micrograph for the CrSi 2 sample implanted at room temperature with a Ne and b Al. EDX spectrum appears superimposed with SRIM ion concentration-depth profile simulation for c Ne and d Al implantation.

Figure 3 a

 3 Figure 3 a Cross-sectional TEM micrographs of the CrSi 2 sample implanted with Ne at 250 8C; b same sample at a higher magnification; and c Ne EDX profile as a function of distance in the film.

Figure 4 a

 4 Figure 4 a SAED pattern superimposed with the theoretical planes of crystalline CrSi 2 . b Dark-field contrast micrograph from the same area shown in Fig. 3a of the CrSi 2 sample implanted with Ne at 250 8C.

Figure 5

 5 Figure 5 Electrical resistivity q measurements as a function of the temperature of as-deposited and implanted CrSi 2 samples. The lines serve as a guide for the eyes. Measurements error bars are within the symbol size.

Figure 6

 6 Figure 6 Thermal conductivity measurements of the implanted CrSi 2 samples. The error bars represent the standard deviation of the measurements, and the solid black circle represents the mean j eff value of each sample.

Table 1

 1 Ion implantation parameters

	Ne		Al
	Room temperature and T = 250 8C	Room temperature		T = 250 8C
	Energy (keV)	Fluence (at/cm 2 )	Energy (keV)	Fluence (at/cm 2 )	Energy (keV)	Fluence (at/cm 2 )
	250	1 9 10 16	200	1 9 10 14	200	8 9 10 15
	120	7 9 10 15	90	4.5 9 10 13	90	3.6 9 10 15
	50	4 9 10 15	40	1 9 10 13	40	8 9 10 14
	20	1.5 9 10 15	30	1 9 10 13	30	8 9 10 14
	Concentration (at.%)
	* 1.0		* 0.008		* 0.64
	Figure 1 a Cross-sectional
	TEM micrograph showing the
	as-deposited CrSi 2 thin film,
	where the S vector points to
	the surface of the film. The Si
	substrate and the SiO 2 layer
	are also indicated in the image.

b Selected area electron diffraction (SAED) pattern of the as-deposited CrSi 2 sample shown in (a). c RBS spectrum of the as-deposited CrSi 2 film.

Acknowledgements

The authors would like to acknowledge the use of the facilities from the Laborato ´rio de Implantac ¸a ˜o Io ˆnica-UFRGS, from the Plateforme MEA-Universite ´de Montpellier and from the Centrale de Technologie en Micro et nanoe ´lectronique (CTM)-Universite ´de Montpellier. They also thank D. Troadec (IENM, Lille, France) for the FIB sample preparation. This study was partially financed by Conselho Nacional de Desenvolvimento Cientı ´fico e Tecnolo ´gico-Brazil (CNPq), by the Coordenac ¸a ˜od e Aperfeic ¸oamento de Pessoal de Nı ´vel Superior-Brazil (CAPES)-Finance Code 001 and partially supported by the French RENATECH network and the Chinese Scholarship Council (CSC). The authors declare no conflict of interest or competing interests.


Simone Galimberti

Maria Potop-Butucaru

Game Theoretical Analysis of DAG-Ledgers Backbone

Keywords: Blockchains,, Game Theory,, Directed Acyclic Graph. 3 DAG-Based Ledger Backbone Protocol with Obedient Users

We study the rational behaviors of participants in DAG-Based Distributed Ledgers. We analyze generic algorithms that encapsulate the main actions of participants in a DAG-based distributed ledger: voting for a block, and checking its validity. Knowing that those actions have costs, and validating a block gives rewards to users who participated in the validation procedure, we study using game theory how strategic participants behave while trying to maximize their gains. We consider scenarios with different type of participants and investigate if there exist equilibria where the properties of the protocols are guaranteed. The analysis is focused on the study of equilibria with trembling participants (i.e. rational participants that can do unintended actions with a low probability). We found that in presence of trembling participants, there exist equilibria where protocols properties may be violated.

Introduction

Blockchain and distributed ledger technologies, [START_REF] Anta | Principles of Blockchain Systems[END_REF], have emerged as one of the most revolutionary developments in recent years, with the goal of eliminating centralised intermediaries and installing distributed trusted services. They facilitate trustworthy trades and exchanges over the Internet, power cryptocurrencies, ensure transparency for documents, and much more. Traditionally, blockchain systems maintain a continuously-growing list of ordered blocks that include one or more transactions that have been verified by the members of the system, called miners. Blocks are linked using cryptography and the order of blocks in the blockchain is the result of a form of agreement among the system participants. After the releasing of the most popular blockchains (e.g., Bitcoin or Ethereum) with a specific focus on economical transactions their huge potential for various other applications became evident. However, quickly after their release blockchains reached their limits in terms of throughput, blocksize and unforeseen behaviors. Therefore, non academic research further concentrate in proposing alternatives by improving some performance aspects but with non zero costs either on security or throughput. One of these solutions extends the blockchain to a DAG overlay and provide an ordering over all blocks and transactions, and outputs a consistent set of accepted transactions. In the research along this line transactions are still organized on blocks. All these DAG-based systems structure blocks in a Directed Acyclic Graph. Each block can include several references to predecessors. The objective of this paper is to analyse from the game theoretical point of view a generic backbone of that captures the characteristics of existing DAG-based blockchains (e.g. Spectre [START_REF] Sompolinsky | Spectre: A fast and scalable cryptocurrency protocol[END_REF], GhostDAG [START_REF] Sompolinsky | PHANTOM GHOSTDAG: a scalable generalization of nakamoto consensus: September 2[END_REF], Phantom [START_REF] Sompolinsky | PHANTOM: A scalable blockdag protocol[END_REF], IOTA [START_REF] Popov | The tangle[END_REF] etc) into a unified framework.

It should be noted that the analysis of strategic behaviors in blockchains (see [START_REF] Amoussou-Guenou | Governing the commons in blockchains[END_REF] for an extended state of the art). To the best of our knowledge, no work has been dedicated to analyze or discuss the rational behaviors among participants in DAG-based blockchains. The only work proposed in the context of the IOTA DAG ( [START_REF] Popov | Equilibria in the tangle[END_REF]) studies the attachment preferences.

In this paper we extract the backbone of a DAG-based blockchain we analyze the behavior of rational participants. We show the different equilibria that exist given different scenarios where other type of participants are present. We analyze if the equilibria do satisfy the DAG-backbone invariants. Additionally, we used the notion of "trembling hand". The trembling hand can be viewed as a failure of rational participants. With low probability, the player can tremble and do an unintended action.

Model and Problem Definition

We consider a system composed of a finite and ordered set Π of n players, of synchronous sequential players denoted by their index, namely Π= {1, . . . , n}.

Each participant has an internal state and proceeds in rounds, it is important to note that users are not synchronized. Every user has an internal clock that is different from the others so in every phase the transmission of blocks or informations may have delay compared to the sending time of other participants in the network. At the end of each round, the participant goes to the next round and so on. A round is composed of three phases:

-A send phase: It is during the send phase that a participant can send a block to the other participants. The block to send should be prepared during the previous rounds. After the send phase, the participant goes to the delivery phase. -A delivery phase: during this phase, a participant collects from the network blocks previously sent. We consider that participants have an unbounded memory, i.e., they can store all blocks they collect. When a participant effectively collects a whole block and can process it, we say that it has delivered the block, otherwise, even if part of the block is collected, it has not delivered the block. The delivery phase has a finite and positive duration. The duration represents the time the participant plans to wait for collecting messages sent. After its delivery phase, the participant goes to the compute phase. -A compute phase: In compute phase, a participant uses all blocks it delivered to update its local DAG and to prepare the block for the next round if it will issue one. After the compute phase, the participant goes to the send phase of the next round.

Participants communicate by sending and receiving messages through a synchronous network. We assume that each participant proceeds in rounds. A round consists of three sequential phases, in order: the send, the delivery, and the compute phases. The delivery phase has a fixed duration that allows collecting all the messages sent by the participants. At the end of a round, a participant exits from the current round and starts the next one. We assume the existence of a reliable broadcast primitive (see Appendix for the formal definition).

When a participant i delivers a message, it knows the participant j that created the message. When a participant sends a message, other participants do not deliver the block instantaneously, there is a message (transmission) delay or a transmission latency. Let ∆ ≥ 0 be the maximum message delay between any two participants, i.e., ∀i, j ∈ Π, if i sends a message to j at date (time) t, then at date t + ∆, j has delivered the message.

-Synchronous Communication: In synchronous communication systems, ∆ is finite and known by the participants. If a participant broadcasts a block at a time t, then at time t + ∆, all participants have delivered the block.

Since the maximum delay ∆ is known, participants can use that information in their execution; typically, they can match the duration of their delivery phase with ∆ to ensure the reception of all blocks sent at the beginning of the round. In a blockchain system, all participants do not necessarily have the same behaviour. The first two types of participants are the classical types we can find in the distributed systems' literature: the correct and the Byzantine participants. Both correct and Byzantine participants do not care about the costs of their actions, nor the global execution of the system. For the sake of clarity and consistency, we use the terminology obedient instead of correct.

A participant is obedient with respect to a protocol A if always follows protocol A.

A participant is Byzantine with respect to a protocol A if she can arbitrarily deviate from protocol A.

A participant is rational if she is self-interested. It does an action instead of another action if and only if doing so increases her expected gain.

A rational participant has preferences and takes actions to have her most preferred execution of the system, taking into account what the other participants do. She does not necessarily want to hurt the system; she wants to benefit from it. We now introduce two refinements of the rational participants: the strategic, and the malicious participants. Having a property on the system's executions, one can know whether a given execution satisfies that property. In particular, a property defines a partition on the executions of the system; in one hand, the executions that satisfy the property and on the other hand, the executions that do not satisfy the property.

A participant is strategic with respect to a property P ifshe is rational, and she prefers executions that satisfy P, i.e., she assigns a positive value to executions where P is satisfied, and a negative value to executions where P is not satisfied.

A strategic participant prefers the executions where the system achieves its properties, but think about her own interest in the first place.

A participant is malicious with respect to a property Pif she is rational, and it prefers executions that do not satisfy P, i.e., she assigns a positive value to executions where P is not satisfied.

A malicious participant's objective is to hurt the system. A malicious participant takes all actions she can to increase the chance of having a global execution that does not satisfy the properties of the system. Problem definition. We study the behaviour of strategic participants in a DAG-based ledger. Each node in the network maintains locally a data-structure Directed Acyclic Graph which is an extension of the traditional linear sequence of blockchains. Each node of the DAG data-structure is a block containing one or more transactions (in the case of IOTA for example each node in the DAG is a single transaction). A block B ∈ contains the (i) identifier of the user w hich built B, (ii) a transaction tr, (iii) a couple of hashes of parents blocks in the DAG structure.

A DAG-based ledger verifies the following properties:

-DAG-Growth. For every user i given two different instants of time t 2 > t 1 it follows that DAG t1 i ⊆ DAG t2 i , which means DAG t1 i is a prefix of DAG t2 i ; -DAG-Uniformity. For every two users i, j, in the absence of new blocks appending, eventually DAG i = DAG j ; -DAG-Liveness. If a correct user, i, appends a correct block B to DAG i , all correct users, j, eventually append B to their own DAG j ; -DAG-Validity. For any correct user, j, each block in DAG j is valid; it satisfies a predefined validity predicate;

A selection tip algorithm is run by the function select tips(DAG i ) (line 1 ) to select the tips to approve in the local DAG i , i.e., the protocol IOTA perform a random walker algorithm to select the best two tips, this procedure is computational expensive compared to a randomized choice but the former has many advantages: (i) Discourages lazy behaviour and encourages approving fresh tips, (ii) continuously merges small branches into a single large branch, thus increasing confirmation rate.(iii) In case of conflicts, kills off all but one of the conflicting branches. Issuer i embedded (line 2 ) the block B with her own id, the transaction tr and the hashes referring to the parents.

Finally the user updates her own DAG i (line 3 ) and broadcast (line 4 ) the block to other users and wait a time ∆ to be sure that anyone receives the block. We note that in this simply version of the protocol the user receive a block, without checking the validity since it is guaranteed, then she updates her own local DAG i . The following lemma proves that Algorithms 1 and 2 verify the properties of a DAG-based ledger. Proof. -DAG-Liveness: DAG-Liveness property is straightforward. Indeed, every the user who receives a block, she will be append it since it is valid. -DAG-Validity: Since there is no trembling issuer neither malicious participant each block is valid by construction. -DAG-Uniformity: Assume by contradiction that DAG i = DAG j this means that one the two user has a block in her own DAG that is not included in the other tree. This is not possible, since we have a unique issuer who issues the same blocks and broadcast to all users, so every users receive a valid block and adds it to the DAG. -DAG-Growth: Given two different instants of time t 2 > t 1 we have two cases:

no block is issued between t 2 and t 1 , or some blocks are issued. In the first case DAG t2 i = DAG t1 i , while if blocks are issued these are added to the DAG since they are valid blocks and so DAG t1 i ⊆ DAG t2 i . Remark 1. Note that in this simplified version we can not talk about Nash Equilibrium since there are no strategic players, there is no game but only users who follows prescribed instructions.

Remark 2. The case where the issuer produces only valid blocks is not interesting from a game theoretical point of view because in that scenario there would not be the risk of updating the structure with invalid blocks. The most challenging scenario is when an invalid block may be produced by every user even if obedient or rational.

Due to space limitations we propose the backbone of a DAG-Based Ledger with obedient participants and a trembling hand issuer in the Appendix.

DAG-Based Distributed Ledger Backbone with Strategic Players

In this section we study the backbone of DAG-Based Distributed Ledgers considering strategic players. In this framework users will try to maximize their own utility, so they may decide not to follow the protocol, i.e., we talk about behaviours of users that may decide to make an action or not. For this reason it is necessary now to introduce a system of reward and penalty to incentivate players to behave correctly.

In the following we assume that when an invalid block is inserted, all participants are punished by incurring in a cost κ. While a reward R, is given to the participants when a block is produced. We already presented before, the cost c check of checking validity, moreover another action will be possible for users which will be able to communicate by paying c send of sending a message in order to coordinate. Additionally, we assume that the reward obtained when a block is produced is smaller than the cost κ of producing an invalid block and the cost of checking is greater than sending. That is,

κ > R > c check > c send > 0.
Note that when a block is produced, in protocols which allow communication only the participants which sent a message are rewarded (and receive R).

Correct Issuer and No Communication between Users

In this section we consider the situation where the issuer is correct and strategic users do not communicate with each other. Strategic participants choice is represented in Algorithm 4 by dedicated variable, namely, action check . The action, initialized at a default value ∅ (line 2 ), can take value from the set {false,true}. For participant i, the value of action check is set by calling respectively the functions σ check i (), returning its strategy (line 5 ). The strategy σ check i () determines if i, the receiving participant chooses to check the validity of the proposal or not, which is a costly action. When the issuer is correct no user will choose to check the validity (line 5 ) paying a cost c check . It is not in her interest paying when they know for sure that the block is valid.

Trembling Issuer and No Communication between Users

In this section we consider the situation where the issuer is trembling and strategic users do not communicate with each other. Algorithm 9 (see Appendix) is the same of Algorithm 4, but in this case the issuer is trembling, so participants will actually choose to check the validity (line 8 ) or not. The choice will depend strictly on the relation between the reward and the penalty, users decide if it is worth paying the cost of checking or not, while when the issuer was correct no one pay for double check the correctness of the proposal. Proposition 1. Participants will not check the validity if and only if κ < c check /p + R.

Proof. First we prove that if the inequality holds players will not check the validity: If a strategic player does not check the validity of a block her average gain is R -pκ, since she will always include the block in her own DAG and receives the penalty of an invalid block with probability p. On the other hand if the user checks, her average gain is (1 -p)R -c check , she will always pay the cost of checking but getting the reward only when the block is valid and so added to the DAG. Not checking is the best strategy if R -pκ > (1 -p)R -c check , solving this inequality we get κ < c check /p + R. Now we prove that if users are not checking the inequality holds: If a strategic player decides not to check this means that the utility is grater compared to the action of checking: R -pκ > (1 -p)R -c check which solved gives κ < c check /p + R. Proof. -DAG-Validity: When a trembling user is present each block may be checked or not, if κ ≥ c check /p + R holds by Proposition 1 only valid block will be appended while invalid block discarded. Otherwise also invalid blocks will be added. -DAG-Liveness: When a block is issued there are two possible scenarios. The users do not check the validity of the block, if κ < c check /p + R holds, otherwise the block is checked. In both cases valid blocks will be appended. -DAG-Uniformity: Note that by hypothesis at time t 0 the local DAG p of each user p is initialized with the same genesis block.

• Base case: Let B be the first block issued by q after t 0 . When the inequality of Proposition 1 does not hold if the block is checked and is valid, then B is appended to the genesis block. Otherwise if the block is checked and it is invalid no user will append it to her own DAG. If the inequality of Proposition 1 holds the block will be appended in any case.

Let DAG p the tree of p at time t 0 + ∆. By ∆ time any users delivered B by the broadcast primitive, hence the DAG of any process is identical to DAG p . It follows that δ time after the first proposal, any two correct processes i and j verify the property: DAG i = DAG j . • Inductive case: Suppose the lemma is true after a sequence of h blocks and prove it holds for the h + 1 block. Let t be the time when the h block is proposed. By inductive hypothesis, we have that by time t + ∆, all users have the same DAG. Let t (t ≥ t + ∆) be the time when the h+1 block is invoked by q. Let DAG q be the local tree of q. By inductive hypothesis, by time t , all the other users have the same local DAG. Two cases have to be considered: * The block proposed at time t is valid : at time t the user checks (or does not check) the validity of the block. Since by assumption of the inductive case they have the same tree as q, then by time t + ∆, B is appended to the DAG. It follows that any two honest users i and j verify the property: DAG i = DAG j . * The block proposed at time t is invalid: in this case if users check the validity, the block B is not appended, otherwise they update the local DAG.

-DAG-Growth: Given two different instants of time t 2 > t 1 we have two cases: no block is issued between t 2 and t 1 , or some blocks are issued. In the first case DAG t2 i = DAG t1 i , while when blocks are issued if they are checked (or not checked and added) and are valid these are added to the DAG and so DAG t1 i ⊆ DAG t2 i .

Trembling Issuer and Communication between Users

Now we add the broadcast of the opinion of users about the validity of the block B. After the checking phase each user will receive the decisions of the others and according to a simple majority rule she will add or not the block to the DAG i . To ensure that the reward covers the costs of checking and sending in this setting we assume that (1 -p)(R -c send ) -c check > 0, that is, the reward covers the costs. We also note that it is better for the participants to send (resp. not to send) without checking than checking and sending (resp. not sending) irrespective to the block validity; that would mean incurring a cost -c check for nothing. It is also not in their best interest to check the validity of the proposal and vote if the proposal is invalid, that would mean increasing the chances of producing an invalid block and incurring a cost -κ.

Strategic participants may decide to check the validity of the block B paying a cost c check (line 8 ), in the same way they can also decide if sending or not the validity of the block paying an additional cost c send (line 11 ). In this version of the protocol users can communicate and for this purpose two more variables are needed. timerVote (line 2 ) is a timer which allows a user to receive every vote expressed by the others, this is strictly required since every participants has his own time clock and maybe some delay is present between votes' arrival. From line 14 to line 17 the receiver must wait ∆ and stores the votes in the vector votes which were initialised as empty.

In the Compute phase each user receives the opinions of the others and insert the block B according to the majority.

Due to space limitation the correctness of Algorithm 6 is proposed in the Appendix. In this section we describe the game, i.e., the possible actions of players,the game tree and finally the utility. Action space. After receiving the proposal block, each participant decides whether to check the block's validity or not (at cost c check ), and second decides whether to send a message (at cost c send ) or not.

Information sets. At the beginning of each round t > 1, the information set of the participant, η t i , includes the observation of the round number t, the participant's own type θ i , as well as the observation of what happened in previous rounds, namely (i) whether the participant decided to check validity, and in that case, she knows the validity of the block, (ii) whether the participant decided to send the validity. Then, in each round t > 1, the participant decides whether to check the validity of the current block. At this point, denoting by B t the block proposed at round t, when the participant does not decide to check validity, the isValid(B t ) function is set to false, while if the participant decides to check, isValid(B t ) is equal to true if the block is valid and false otherwise. Therefore, at this stage, the participant information set becomes H t i = η t i ∪ isV alid(B t ), which is η t i augmented with the validity information participant i has about B t , the proposed block. Strategies. At each round t ≥ 1, the strategy of participant i is a mapping from her information set into her actions. At the point at which the participant can decide to check block validity, her strategy is given by σ check i (η t i ). Finally, after making that decision, the participant must decide whether to send a message or not, and that decision is given by σ send i (H t i ). The decision tree of a participant is depicted in Figure 1. We note that when the participant does not check the validity of the proposal, she does not know if the block is valid or not. We denote by σ = (σ 1 , ..., σ n ), the strategy profile where ∀i ∈ {1, ..., n}, participant i uses strategy σ i , where

σ i (H t i ) is the pair (σ check i (η t i ) ,σ send i (H t i )).
Objective of Strategic Participants. The expected gain of strategic participant i is:

U i = E R × 1 (σ send i (H t i )=true) × 1 (valid block produced) -κ × 1 (invalid block produced) -c check × 1 (σ check i (η t i )=true) -c send × 1 (σ send i (H t i )=true) | η t i
where 1(.) denotes the indicator function, taking the value 1 if its argument is true, and 0 if it is false.

Equilibria with only Strategic Participants

In this section we analyse the different Nash Equilibria which may be possible in the framework of a trembling issuer with strategic players who communicate described in Algorithm 6.

Proposition 2. With only strategic participants and if there is a probability p that the proposer proposes an invalid block, there may be two type of Nash equilibria. In equilibrium either (i) if κ < min R/p -c send /p, R -c send + c check /p , all participants do not check the validity of the proposal and vote it as valid, (ii) if κ > R + c check /p -c send , n/2 +1 participants check the validity and send the vote, since all the players are simmetric we have n n/2 +1 equilibria.

Proof. We prove that the strategy profiles described in the proposition are Nash equilibria.

First, we prove that the strategy profile where all participants do not check the proposal validity and vote is a Nash equilibrium.

(i) The expected gain at equilibrium of any participant is R -c send -pκ. If one participant deviates and does not send the vote, her gain at deviation is 0; if she deviates checking, the gain is (1 -p)(R -c send ) -c check . These utilities are lower than the gain at equilibrium if and only if κ < min R/p -c send /p, Rc send + c check /p , which is our assumption, therefore, the gain at equilibrium is better than the gain if the participant deviates. The strategy profile where all participants don't check the proposal validity is a Nash equilibrium.

We now prove that the strategy profile where n/2 +1 participants check the proposal validity but all users vote is a Nash equilibrium.

(ii) The expected gain at equilibrium of participants who check is (1 -p)(Rc send ) -c check . This gain is positive by assumption and consistency that reward covers the costs, so no users will deviate not sending. If a participant deviates not checking her gain at deviation is R -c send -pκ, which is lower than gain at equilibrium if and only if κ > R + c check /p -c send , which is our assumption, therefore, the gain at equilibrium is better than the gain if the participant deviates. The strategy profile where n/2 +1 participants check the proposal validity and vote is a Nash equilibrium. Remark 3. There are two Nash Equilibria in Proposition 2. In the equilibrium where all participants do not check, if the proposal is invalid, there is no DAG-Validity, however, DAG-Liveness, DAG-Uniformity and DAG-Growth are always ensured. While in the second equilibrium where n/2 +1 participants check we are in the case of Lemma 4 so all invariants hold. Proposition 3. With n/2 obedient participant, if there is a probability p that the proposer proposes an invalid block, if κ > R + c check /p -c send there are n/2 Nash Equilibria where one strategic player checks the validity. (This case can be extended when there are less than n/2 obedient participants, i.e., if there are n/2 -1 obedient the Nash Equilibria are n/2 +1 2 when two strategic participants check the validity).

Proof. We prove that the strategy profiles described in the Proposition 3 are Nash Equilibria. The gain at equilibrium of the strategic participant who checks is (1 -p)(Rc send ) -c check . If she deviates and does not check, her gain at deviation is R-c send -pκ, which is lower than gain at equilibrium if κ > R+c check /p-c send , which is our assumption, therefore, the gain at equilibrium is better than the gain if the participant deviates. The strategy profile where one participant checks the proposal validity and vote is a Nash equilibrium.

Remark 4. In these equilibria we are in the case of Lemma 11 so all invariants hold.

Equilibria with Faulty Participants

We analyse the situation where faulty participants are present. A Faulty participant is a user that crashes and stops following the protocol (the proof is similar to Proposition 2, Proof (i)). Proposition 4. With faulty participants and if there is a probability p that the proposer proposes an invalid block, there may be a Nash equilibrium. If κ < min R/p -c send /p, R -c send + c check /p , in the equilibrium all participants do not check the validity of the proposal and vote it as valid.

Remark 5. In the equilibrium if the proposal is invalid, there is no DAG-Validity, however, DAG-Liveness, DAG-Uniformity and DAG-Growth are always ensured.

Equilibria with Rational Malicious Participants

We consider now a new type of users b rational malicious participants whose objective is harming the network but they incur, differently from malicious ones, in the costs for checking and sending, their utility is defined as following:

Payoff of Rational Malicious Participants. Let ω be an outcome of the game. If ω does not satisfy DAG-Validity, then the malicious participants have a payoff of κV alidity > 0. κ(ω) = κV alidity , if ω does not satisfy DAG-Validity 0, if ω satisfies DAG-Validity

Objective of Rational Malicious Participants. The expected gain of malicious participant i is:

Ũi = E κV alidity × 1 (invalid block is produced) -c check × 1 (σ check i (η t i )=true) -c send × 1 (σ send i (H t i )=true) | η t i .
Proposition 5. When the proposer is trembling rational, there may be a Nash Equilibrium. We indicate with b the number of rational malicious participants and with r the number of strategic ones. If:

κ > R -c send /p + c send -R + c check + P(r > b)[R(1 -p) -c send ] p(1 -P(r > b))
All strategic participants check the validity and send the vote, while rational malicious participants do nothing if r > b, they all send the vote without checking when r < b.

Proof. We prove that the strategy profile described in the proposition is a Nash Equilibrium.

The expected gain at equilibrium of strategic participant is (1 -p)(R -c send )c check . This gain is positive by assumption and consistency that reward covers the costs, so no users will deviate not sending. Since a strategic participant does not know the exact number of the malicious ones, she can only compute the expected reward when deviating not checking, i.e., if r > b, the gain is (1 -p)R -c send , while it is Rp -c send -pκ when r < b. These utilities are lower than the gain at equilibrium if and only if:

κ > R -c send /p + c send -R + c check + P(r > b)[R(1 -p) -c send ] p(1 -P(r > b)) ,
which is our assumption, therefore, the gain at equilibrium is better than the gain if the participant deviates. Regarding malicious rational participants, we can suppose that in the worst case scenario they know how many they are, and so, they will not check neither send the vote if r > b in order to avoid any useless cost. While if r < b, all malicious strategic participants will send the vote without checking with a gain of pκ V alidity -pc send . This is an equilibrium for the malicious rational participants, if one deviates its reward would be 0.

The strategy profile where all strategic participants check the proposal validity and vote is a Nash equilibrium.

Remark 6. We are in the case of Lemma 4 so all invariants hold. However DAG-V alidity property falls if r < b.

Equilibria with Malicious Participants

Finally we consider m malicious participants, these users do not incur any costs, no matter their actions, and their objective is damaging the network by validating invalid blocks. For this reason we can assume that they will always check the block and vote only if this in invalid. Strategic participants are aware of the presence of these other participants but they do not know their precise number.

Payoff of Rational Malicious Participants. Let ω be an outcome of the game. If ω does not satisfy DAG-Validity, then the malicious participants have a payoff of κV alidity > 0, we also assume that malicious participants do not incur any costs, no matter their actions.

κ(ω) = κV alidity , if ω does not satisfy DAG-Validity 0, if ω satisfies DAG-Validity (2)
Objective of Malicious Participants. The expected gain of malicious participant i is:

Ũi = E κV alidity × 1 (invalid block is produced) | η t i .
Proposition 6. When the proposer is trembling rational, there may be one Nash equilibrium. We indicate with m the number of malicious participants and with r the number of rational ones. If:

κ > R -c send /p + c send -R + c check + P(r > m)[R(1 -p) -c send ] p(1 -P(r > m))
All strategic participants check the validity and send the vote.

Proof. For the proof see Proposition 5, Proof. In this scenario malicious participants will always check and send the vote since they have no costs.

Remark 7. We are in the case of Lemma 4 so all invariants hold. However DAG-Validity property falls if r < b.

Considering c send > c check it may happen that sending without checking is no more the best choice for strategic participants, since they would pay more and also incur in the risk of add an invalid block that can be avoided at a less price by checking. As in the previous case to ensure that the reward covers the costs of checking and sending in this setting we assume that (1-p)(R-c send )-c check > 0, that is, the reward covers the costs.

By observing the equilibria presented above it is possible to see that they do not depend on the relation between c send and c check . For this reason even if we consider a cost of sending greater the cost of checking the equilibria will remain the same. However c send > c check may lead in a modification of the values of the thresholds for these equilibria, i.e., in the ones where all users check the validity the corresponding threshold that is in the assumption, will be much easily satisfied. On the other hand the condition for the equilibria where no one checks will be more stricter than the case where c send < c check .

Conclusion

We analyze the behavior of rational players in DAG based protocols. This paper studies the case where there is one proposer at the time. The case of multi proposers is left open. We found that in presence of trembling participants, there exist equilibria where the backbone properties may be violated. The results of our work can be used in improving the design of DAG-based ledgers. Our findings are summarized in Table 5

Appendix

Specification of the Reliable broadcast -Byzantine Reliable Broadcast. It ensures that a message sent by a correct process is received by all correct processes, and that all correct processes eventually receive the same set of messages. The service provides two operations, BRB-broadcast and BRB-delivery; the first broadcasts a message to all processes, and the second delivers a message that was previously broadcast. The service is used by the servers, and from their point of view, the BRB service guarantees the following properties:

• Validity: if a correct process p i BRB-delivers a message m from a correct process p j , then p j BRB-broadcast m. • Integrity: a message is BRB-delivered at most once by a correct server.

• Termination 1 (local): if a correct process BRB-broadcasts a message, it BRB-delivers it. • Termination 2 (global): if a correct process BRB-delivers a message, all correct processes BRB-deliver it.

Validity relates outputs to inputs. Validity and integrity concern safety. Termination is on the fact that messages must be BRB-delivered; it concerns liveness. It follows that all correct processes BRB-deliver the same set of messages, which includes all the messages they BRB-broadcast.

Notions of Game Theory

One can defined Game theory as a mathematical formalisation of conflicts and cooperation between rational decision-makers in a given environment. Game theory has various domains of application as of computer science, economics, evolutionary biology, mathematics, political science, etc. It is mainly used to model and study rational behaviours between different participants. Using game theory appears to be useful according to our goal of studying rational behaviours in blockchain systems. Formally, a game is a tuple G = (N, (A i ) i∈N , (Θ i ) i∈N , (g i ) i∈N ), where:

-N is a non-empty set of participants; -A i is the set of actions of participant i; -Θ i is the type of participant i; g i is the gain function of participant i; Static Games. At a beginning of a static game, all participants choose simultaneously the actions they will play, without knowing which actions the others will choose. The gain of participant i is of the form g i :

A 1 × • • • × A n → R.
Static games are also called simultaneous game. The most popular simultaneous game is perhaps rock-paper-scissors.

Dynamic or Repeated Games. A repeated game is a sequence of static games where participants may have some information thanks to the previous static games played. We say that the first static game is the game at stage 1, the second is the game at stage 2, and so on. To accommodate the information each participant has, we denote by h t i the information set of participant i at time t. At each time of the game, each participant has an information set that contains information of what happened previously. Let H t i be the set of information sets participant i can have at time t. A pure strategy or simply a strategy of a participant i at time t is a function σ t i : H t → A i . A strategy of the participant is a family (σ t i ) t≥ . However, in the rest of the manuscript, we will write σ i to represent the strategy of participant i, and we will denote by S i the set of strategies of participant i. A vector of strategies for each participant σ = (σ 1 , ...., σ n ) is called a strategy profile. In repeated games, the gain of participant i is of the form g i :

S 1 × • • • × S n → R.
Complete Information Games. In complete information games, all participants know the other participants' information, their types, their gain function, their actions set, etc. At each time in the game, all participants know what actions the other participants did previously.

Incomplete Information Games. When a game is not with complete information, we say that it is an incomplete information game. In incomplete information games, at least one participant is not sure about the type of another participant. We can say that participants have private information that other participants do not know about. For example, it can be known that there is a malicious participant in the system, but except the malicious, no one else knows which participant it is. An initial distribution of the type of participants is set at the beginning of the game, but participants do not know it exactly. Participants only have at each time a probability distribution over the other participants' type; the probability distribution is updated following Bayes' rule and is kept in the information set of each participant. That is why incomplete information games are also called Bayesian games. Let A and B be two events such that P[B] > 0. Bayes' theorem also called Bayes' rule is the following equality:

P[A|B] = P[B|A] • P[A] P[B]
Solution Concept. Now that the game is defined, we can try to predict the behaviour of rational participants. To do so, different concepts exist, the main one being the concept of Nash equilibrium. Intuitively, a strategy profile is a Nash equilibrium when each participant has a strategy that maximises its gain with respect to the other participants' strategies in the strategy profile. A Nash equilibrium can also be seen as a strategy profile where no participant can increase its gain by deviating alone from the strategy profile. Let σ = (σ 1 , . . . , σ n ) be a strategy profile, and let σ i ∈ S i be a strategy of participant i. We denote the strategy profile (σ 1 , ..., σ i-1 , σ i , σ i+1 , ..., σ n ) by (σ i , σ i ).(σ i , σ i ) is the strategy profile where participant i deviates by doing σ i instead of σ i , and all other participants continue with the same strategies. We can now formally define a Nash equilibrium. Definition 1. (Nash Equilibrium). A strategy profile σ is a Nash equilibrium if and only if:

∀i ∈ N, g i (σ) ≥ g i ((σ -i , σ i )).
The definition of a Nash equilibrium can be seen as too restrictive in some games, and the concept of approximate Nash equilibrium is used. An -approximate Nash equilibrium is a strategy profile where if one participant deviates, she can gain at most more than her gain at equilibrium. Formally, a strategy profile σ is an -approximate Nash equilibrium if and only if ∀i ∈ N, g i (σ) ≥ g i ((σ -i , σ i ))-. Approximate Nash equilibria are not stable situations, since a participant can prefer to deviate, even if her gain is just , so we do not consider them in this manuscript. On another hand, and especially in repeated games, the concept of Nash equilibrium allows too many behaviours, and some are not interesting or coherent. The concept of subgame perfect equilibrium is defined to specifically capture equilibria that are coherent throughout the whole game. At any history, the "remaining game" can be regarded as a game on its own. We call such a remaining game a subgame of the game. Note that the whole game is also its own subgame. Definition 2. (Subgame Perfect Nash Equilibrium). A strategy profile σ is a subgame perfect Nash equilibrium if in all subgames, σ restricted to the subgame is a Nash equilibrium. Bayesian Equilibria. In Bayesian games, the concept of Nash equilibrium is also not well suited. In Bayesian game, since at least one participant is unsure of the type of other participants, a strategy profile may give more than one execution, according to the type distributions. An analogous concept to Nash equilibrium, the Bayesian equilibrium is defined for Bayesian games. Contrarily to Nash equilibrium, in a Bayesian equilibrium, each participant's goal is to maximise her expected gain, given her knowledge about the types' distribution; in particular, each participant's beliefs are consistent with Bayes' law when computing probabilities conditional on events that have positive probability on the equilibrium path. In Bayesian games, the gain function g is a probability distribution over the gain of all different executions that correspond to the given strategy profile. Definition 3. (Bayesian Equilibrium). A strategy profile σ is a Bayesian equilibrium at time t if and only if:

∀i ∈ N, E g i (σ)|h t i ≥ E g i (σ -i , σ i )|h t i .
As for Nash equilibria, the concept of subgame perfection exists in Bayesian games, and such equilibrium is called a perfect Bayesian equilibrium. Definition 4. (Perfect Bayesian Equilibrium). A strategy profile σ is called a subgame perfect Bayesian equilibrium if in all subgames, σ restricted to the subgame is a Bayesian equilibrium.

A Comparison of Possible Executions from Participants' Type Let

A be a protocol. We can see from Definition ?? that a rational participant can be viewed as a special subclass of Byzantine participants, rational participants can deviate from the protocol, but only if that deviation is beneficial to them. Let C be a class of rational participants such that their objective is to follow the protocol A no matter what is the global execution of the system and what the other participants do. The class C consists of exactly all the obedient participants with respect to A, and only them. We can consider that some rational participants can behave as obedient, if their gain function is defined in that case. That exhibits that the behaviour of rational participants depends really on their gains. Therefore, the obedient participants are a subclass of rational participants where their objective is to follow the protocol. Usually, a protocol is designed to solve a problem, or equivalently to satisfy a property. It can then happen for strategic participants to behave as obedient. However, if running the protocol is (too) costly, strategic participants may deviate to improve their gain. The class of obedient participants is not a subclass of strategic participants, but both can intersect. As mentioned earlier, malicious and strategic participants are rational participants, but they have opposing objectives, so they do not intersect. A summary of the comparison of possible execution can be viewed in Fig. 2. Now, we assume that there is some negligible probability p for the issuer to generate an invalid proposal, and all participants are aware of the trembling effect. When proposing a value, there is a probability that the hand of the issuer trembles and proposes an invalid block instead of a valid block; i.e., in some sense, we take into account the possibility of making a mistake for the proposal. Algorithm 7 corresponds to Algorithm 1, there is only one difference, the function that generates the block, the new generateTrembling encapsulates the probability of producing invalid transactions which were not present before. Since users may create unintentionally invalid blocks and everyone is aware about, a new action is available to the users, they may pay a cost c check in order to check the validity of the block. Note that checking the validity of a block may be important, there is a risk of producing an invalid block, breaking the DAG-Validity property. The function isValid may be called for this purpose:

-isValid : B →{false,true} is an application-dependent predicate that is satisfied if the given block is valid. If there is a block B such that isV alid(B) = true, we say that B is valid. We note that for any block, we set isValid to false, the validity of the block depends on the blockchain and the application, and isValid is known by all participants.

Obedient participants choice is represented in Algorithm 8. Upon receiving a block (line 3 ) they check the validity of the proposal, which is a costly action, and then update the knowledge about the validity of the proposal. Otherwise, the participants keep not knowing if the proposal is valid or not ( validV alue remains false). Note that this value remains false even if the participant is the proposer. This is because we assumed, without loss of generality, that checking validity has a cost and that the only way of checking validity is by executing the isV alid(B) function. Proof. -DAG-Liveness: When a trembling user is present if the block produced is invalid no users will append the block since being obedient they

Algorithm 2 :

 2 Receiver i Block B 1 Upon receiving B : 2 DAGi:= update(DAGi, B)

Lemma 1 .

 1 Algorithms 1 and 2 verify DAG-Liveness, DAG-Uniformity, DAG-Validity and DAG-Growth.

Algorithm 3 :Algorithm 4 :

 34 Issuer i Block B 1 Tips := select tips.(DAGi) 2 B := generate (Tips, tr) 3 Broadcast(< B >) 4 DAGi:= update(DAGi, B) 5 Wait ∆ Strategic Receiver i Block B {true,false} sets the action of // checking or not the validity of the block 6 if action check == false then 7 DAGi:= update(DAGi, B) 8 if action check == true then 9 validValue ← isValid (B) // The execution of isValid (B) has a cost // c check 10 if validValue == true 11 DAGi:= update(DAGi, B)

Algorithm 5 :

 5 Trembling Issuer i Block B 1 Tips := select tips.(DAGi) 2 B := generateTrembling (Tips, tr) 3 Broadcast(< B >) 4 DAGi:= update(DAGi, B) 5 Wait ∆

Lemma 2 .

 2 Algorithms 5 and 9 verify DAG-Uniformity, DAG-Growth and DAG-Liveness. DAG-Validity holds if κ ≥ c check /p + R.

Algorithm 6 : 4 action check :=∅ 5 action send :=∅ 6 validValue

 6456 Receiver i Block B (Strategic Participant) := {∅,...,∅} // Vector storing the votes about // the validity of a block {true,false} sets the action of // checking or not the validity of the block 9 if action check == true then 10 validValue ← isValid (B) // The execution of isValid (B) has a cost // c check 11 action send ← σ send i (validV alue) // σ send i () ∈ {true,false} sets the // action of sending or not the validity of the block 12 if action send == true then 13 broadcast (< validV alue >) // The execution of the broadcast has // a cost c send 14 set timerVote to ∆ 15 while TimerVote not expires do 16 upon receiving < validV aluej > : 17 votesj ← validV aluej // The participants collect all // the opinions on the validity of the block 18 Compute phase: 19 if |< votes >== true |>|< votes >| /2 20 DAGi:= update(DAGi, B) 5 Game Theoretical Analysis for Trembling Issuer and Rational and Byzantine Users

Fig. 1 .

 1 Fig. 1. A Comparison of Participant's Types

Algorithm 7 :

 7 Trembling Issuer i Block B 1 Tips := select tips.(DAGi) 2 B := generateTrembling(Tips,tr ) 3 Broadcast(< B >) 4 Wait ∆

Lemma 3 .

 3 Algorithms 7 and 8 verify DAG-Uniformity, DAG-Validity, DAG-Growth and DAG-Liveness.

Table 1 .

 1 .3. Summary of the Equilibria.

	Participants	DAG-Growth DAG-Uniformity DAG-Liveness Validity
	Only Strategic
	-No one check the validity	Yes	Yes	Yes	No
	-n/2 + 1 users check	Yes	Yes	Yes	Yes
	Strategic & Faulty
	-No one check the validity	Yes	Yes	Yes	No
	Proposition 4
	Strategic & Rational Malicious
	-All users check	Yes	Yes	Yes	Yes
	Proposition 5
	Strategic & Malicious
	-All users check	Yes	Yes	Yes	Yes
	Proposition 6

Tips := select tips.(DAGi)

B := generate (Tips, tr)

DAGi:= update(DAGi, B)

Broadcast(< B >)

Wait ∆

will check the validity of the proposal and decide not to add the block. While if a user receive a valid block after checking it will be appended, and every obedient users will do the same.

-DAG-Validity: When a trembling user is present each block is checked, so only valid block will be appended while invalid block discarded. -DAG-Uniformity: Assume by contradiction that DAG i = DAG j this means that one the two user has a block in her own DAG that is not included in the other tree. This is not possible, since we have a unique issuer who issues the same blocks and broadcast to all users, so every users receive a block and if it is valid she adds it to the DAG. -DAG-Growth: Given two different instants of time t 2 > t 1 we have two cases:

no block is issued between t 2 and t 1 , or some valid blocks are issued. In the first case DAG t2 i = DAG t1 i , while if blocks are issued these are added to the DAG since they are valid blocks and so DAG t1 i ⊆ DAG t2 i .

Remark 8. Note that in this simplified version we still can not talk about Nash Equilibrium since there are no strategic players, there is no game but only users who follows prescribed instructions.

Correctness of Algorithm 6

In the following we are interested about the correctness of Algorithm 6.

Lemma 4. In the trembling issuer case and communication between users Algorithm 6 all invariants.

Proof. -DAG-Validity: When a trembling user is present each block must be voted before added to the DAG, for this reason only valid blocks will be appended while invalid blocks discarded. -DAG-Liveness: When the issuer is trembling invalid blocks may be produced.

When a user receives a block, after the voting system the block will be appended or considered as invalid. When a valid block is issued it will be voted by all the network and each correct user will update her own DAG. -DAG-Uniformity: Note that by hypothesis at time t 0 the local DAG p of each user p is initialized with the same genesis block. • Base case: Let B be the first block issued by q after t 0 . If the block is valid, then B is appended to the genesis block. Otherwise if the block is voted as invalid no user will append it to her own DAG. Let DAG p the tree of p at time t 0 + ∆. By ∆ time any users delivered B by the broadcast primitive, hence the DAG of any process is identical to DAG p . It follows that δ time after the first proposal, any two processes i and j verify the property: DAG i = DAG j . • Inductive case: Suppose the lemma is true after a sequence of h blocks and prove it holds for the h + 1 block. Let t be the time when the h block is proposed. By inductive hypothesis, we have that by time t + ∆, all users have the same DAG. Let t (t ≥ t + ∆) be the time when the h+1 block is invoked by q. Let DAG q be the local tree of q. By inductive hypothesis, by time t , all the other users have the same local DAG. Two cases have to be considered: * The block proposed at time t is valid : at time t the networks votes the block as eligible to be appended, and then by time t +∆ all users delivered B. Since by assumption of the inductive case they have the same tree as q, then by time t + ∆ B is appended to the DAG. It follows that any two honest users i and j verify the property: DAG i = DAG j . * The block proposed at time t is invalid : in this case block B is voted as invalid and it is not appended to the local DAG. -DAG-Growth: Given two different instants of time t 2 > t 1 we have two cases: no block is issued between t 2 and t 1 , or some blocks are issued. In the first case DAG t2 i = DAG t1 i , while when blocks are issued if they are voted as valid these are added to the DAG and so DAG t1 i ⊆ DAG t2 i , otherwise they are discarded and DAG t2 i = DAG t1 i .


Manvi Agrawal
email: magrawal@ntu.edu.sg

Dharmarasu Nethaji

K Radhakrishnan

Christophe Gaquière

Guillaume Ducournau
email: guillaume.ducournau@univ-lille.fr

Marie Lesecq
email: marie.lesecq@univ-lille.fr

Javier Mateos
email: javierm@usal.es

Tomas Gonzalez
email: tomasg@usal.es

Ignacio Iniguez-De-La-Torre

Sergio Garcia
email: sergio_gs@usal.es

Christophe Gaquiere
email: christophe.gaquiere@uniokv-lille.fr

Tomás González

Ignacio Íñiguez

Sergio García

Susana Pérez
email: susana@usal.es

GaN-based SSD structure for THz applications

Keywords: Gunn diode, THz, GaN, SiC, PA-MBE

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

I. INTRODUCTION

The most common solid-state electronics based THz devices are obtained using Schottky diodes for the frequency multiplication of fundamental oscillators such as resonant tunneling diodes (RTDs), impact ionization avalanche transittime (IMPATT) diode, microwave multipliers and Gunn diodes. The oscillation frequency of classical GaAs and InP Gunn diodes (based on the transfer of electrons to the upper valleys) can only be extended to around 300 GHz limited by the scattering mechanism and energy relaxation frequencies [START_REF] Eisele | Submillimeter-Wave InP Gunn Devices[END_REF]. The superior properties of GaN-based materials such as wide band-gap, high saturation velocity and high breakdown field make them suitable candidates for high power, high frequency and high-temperature applications [START_REF] Jain | III-nitrides: Growth, characterization, and properties[END_REF]. GaN-based Gunn diodes can potentially operate at much higher power than traditional GaAs and InP Gunn diodes at very high frequency, above 300 GHz. However, up to now, no clear observations of continuous Gunn oscillations in GaN devices have been achieved, and the realization of a continuous wave GaN-based Gunn oscillator at 300 GHz and beyond has never been reached before. By means of pulse measurements, just the presence of negative differential resistance which leads to a subsequent breakdown of the devices, has been found [START_REF] Ma | Current-controlled negative differential resistance effect induced by Gunn-type instability in n-type GaN epilayers[END_REF].

Self-switching diodes (SSDs) produce planar asymmetric non-linear devices and simulations show their potential to create THz waves as a result of Gunn oscillations [START_REF] Balocco | Microwave Detection at 110 GHz by Nanowires with Broken Symmetry[END_REF][START_REF] Drachenko | Compact magnetospectrometer for pulsed magnets based on infrared quantum cascade lasers[END_REF]. In this work, we report on the growth optimization of GaN-based SSD structure on SiC (Fig. 1 (a)), designed using Monte Carlo (MC) simulations, for the fabrication of nanoscale SSDs (Fig. 1 (b)) to reach THz emission.

II. EXPERIMENTAL

The design rules for the growth of the epistructure was provided by means of MC simulations. The designed epistructure was grown on double side polished semi-insulating (SI) 4H-SiC (001) substrate using plasma assisted molecular beam epitaxy (PA-MBE). The growth was initiated by the insitu removal of the native oxide by exposing the surface of the substrate to three cycles of Ga deposition and flash-off. Subsequently, 100 nm AlN nucleation layer was grown at a substrate temperature of 740 °C followed by 1000 nm undoped two-step GaN buffer layer at 710 °C. Initial GaN with a thickness of 100 nm was grown at III/V = 0.85 followed by the growth of the remaining GaN under III/V ≥ 1. Finally, 400 nm silicon doped GaN active layer with a carrier concentration of 2 × 10 18 cm -3 was grown at the same substrate temperature. A set of samples was grown as a function of Ga/N (III/V) ratios to achieve good crystalline quality and smooth surface morphology of the undoped GaN buffer layer. Moreover, Si doping was optimized in the subsequent GaN active layer by varying the temperature of the Si effusion cell. III. RESULTS Fig. 2 (a) and (b) show the AFM images of 5 × 5 μm 2 scan area for GaN grown under nitrogen rich conditions (III/V < 1) and metal rich conditions (III/V ≥ 1), respectively. A rough surface morphology (III/V < 1) with RMS roughness of 6.1 nm while smooth surface (III/V ≥ 1) with an RMS roughness of 0.4 nm is obtained. Spotty reflection high energy electron diffraction (RHEED) pattern was observed during the growth of GaN under III/V < 1, indicative of the 3D growth mode while streaky GaN was obtained under III/V ≥ 1 with a (2×2) reconstruction pattern signifying 2D growth as shown in the inset of Fig. 2 (a) and (b), respectively. Fig. 3 shows the GaN full-width-at-half-maximum (FWHM) along ( 002) and (102) planes as a function of III/V ratio. For the growth performed in nitrogen rich conditions (III/V < 1.0), the lowest FWHM for GaN (102) was achieved at III/V = 0.85, indicating overall good crystalline quality. In the Ga-stable and Ga rich regions (1.0 < III/V ratio < 1.2), the crystalline quality degraded as the value of FWHM increased while it saturated beyond III/V > 1.2.

As can be concluded from Fig. 2 and Fig. 3, smooth surface morphology is achieved for growths with III/V ≥ 1 while good crystalline quality was achieved when III/V < 1. Therefore, a two-step GaN with growth transition from III/V < 1 to III/V ≥ 1 during the initial stage of GaN buffer layer was implemented to obtain reduced dislocation density and smooth surface morphology. Table 1 shows the GaN FWHM along (002) and (102) planes grown using different growth modes. Fig. 4 shows the carrier concentration in GaN with respect to different Si effusion cell temperatures during the growth. As can be seen, a carrier concentration ranging from 0.7 to 6.5×10 18 cm -3 was obtained. Fig. 5 shows the measured resistance versus the separation between the adjacent contacts of the transmission line measurement (TLM) structures on the epistructure using four-point probe technique. The contact resistance (Rc) was extracted from the slope and the y-axis intercept of the corresponding linear fit. The TLM was performed on several locations that resulted in an average Rc of 0.35 Ω.mm. Based on the optimization of the growth conditions for the GaN buffer and the active layers, the complete epistructure was grown on 2-inch SiC substrate. Fig. 6 (a) shows the photograph of the 2-inch epiwafer and (b) shows the three-dimensional (3D) view of the epiwafer-bow of 19 μm obtained by making the measurements in 4 different directions. Fig. 7 shows the mapping of electrical properties of the epistructure across the 2-inch epiwafer. An average (a) sheet resistance of 390 Ω/sq., (b) mobility of 187 cm 2 /V.s (c) and carrier density of 2.14 ×10 18 cm -3 are obtained with the standard deviation of 4.2 %, 6.0 % and 3.2 %, respectively. Fig. 8 (a) and (b) show the FWHM mapping along GaN (002) and GaN (102) planes across the epiwafer, and Fig. 9 shows the photoluminescence (PL) mapping of the epistructure, indicating uniform electrical and optical qualities across the wafer.

Table 2 lists the FWHM of GaN along (002) and (102) planes, RMS roughness of 5 × 5 μm 2 scan area and the average carrier density in the GaN active layer for the four 2-inch epiwafers. As shown, GaN with improved crystalline quality was obtained with a sub nanometer RMS roughness. The SSD fabrication on these epiwafers is underway and the results will be presented elsewhere. IV. SUMMARY The GaN-based SSD structure on SiC substrate, designed using MC simulations was grown by PA-MBE. The crystalline quality of GaN buffer was improved by using 3D-2D growth modes. High carrier density of 2 ×10 18 cm -3 was achieved for GaN active layer with Si doping and a corresponding contact resistance of 0.35 Ω.mm. Crack-free epistructure with sub nanometer RMS roughness and good uniformity across 2-inch was established.

Fig. 1 (

 1 Fig. 1 (a) Cross-sectional schematic of the epistructure and (b) threedimensional geometry of the SSDs.

Fig. 2

 2 Fig. 2 AFM images of 5 × 5 μm 2 scan area for GaN grown under (a) III/V < 1 and (b) III/V ≥ 1. The inset shows RHEED reconstruction patterns of GaN (a) spotty indicating 3D growth and (b) streaky (2×2) indicating 2D growth.

Fig. 3

 3 Fig. 3 GaN FWHM along (002) and (102) planes as a function of III/V ratio.

Fig. 4

 4 Fig. 4 Carrier concentration in GaN layer with different Si effusion cell temperature.

Fig. 5

 5 Fig. 5 Resistance versus separation between the adjacent contacts of TLM structures.

Fig. 6

 6 Fig. 6 (a) Photograph of 2-inch epiwafer and (b) the three-dimensional (3D) view of the epiwafer bow.

Fig. 7

 7 Fig. 7 Electrical properties of the epistructure across the 2-inch epiwafer (a) sheet resistance (Ω/sq.), (b) mobility (cm 2 /V.s) and (c) carrier density (×10 18 cm -3 ).

Fig. 8

 8 Fig. 8 FWHM mapping along GaN (002) (a) and GaN (102) (b) planes of 2inch epiwafer.

Fig. 9

 9 Fig. 9 PL mapping of epistructure.Table2GaN FWHM along (002) and (102) planes, RMS roughness (5 x 5 μm 2 scan area) and the average carrier density for the four 2-inch epiwafers.Wafer ID

Table 1

 1 GaN FWHM along (002) and (102) planes grown using different growth modes.

	Growth	GaN FWHM (arcsec)
	mode	(002)	(102)
	3D	250	1150
	2D	321	1616
	3D-2D	285	1207

ACKNOWLEDGEMENT

This work has been supported by the NRF2017-NRF-ANR003 GaNGUN project. This work has also been partially supported by the Ministerio de Economía y Competitividad (Spain) and FEDER (European Union) through project TEC2017-83910-R and by the Consejería de Educación de la Junta de Castilla y León (Spain) and FEDER (European Union) through project SA254P18.