Ranim Mohamad

Jun Chen

Pierre Ruterana

*Graphical Abstracts (for review)

In 0.17 Al 0.83 N, lattice matched to GaN, is an attractive solution for applications such as high electron mobility transistors, distributed Bragg reflectors, or light-emitting diodes. It has been reported that InAlN layers lattice matched to GaN on growth along the [0001] exhibit crystallographic degradation when the thickness is increased. In our recent work, it was shown, through theoretical modelling as well as transmission electron microscopy, that the N vacancy (V N ) plays a critical role in this behavior. In this work, we have carried out a detailed theoretical investigation of the possible modification of the properties of this alloy as brought about by the presence of the nitrogen vacancy and the associated formation of the indium rich regions. After testing the ab initio procedures on the AlN and reproducing the lattice parameters and band gap with high accuracy, it is shown that the indium aggregation in InAlN leads to a reduction of the band gap. In contrast to AlN, where acceptor levels can also form, the combination of the nitrogen vacancy and the favored indium aggregation introduced even more levels at the top of the band gap, moreover examining the whole spin configuration we find a strong asymmetry in the occupation of the spin up and down states with a net moment of around 1 µB per supercell. Most importantly, the presence of a nitrogen vacancy changes significantly the top of the band gap: first, the band edges become very sharp, and second, the introduced states originating from local deformation are occupied. This leads to n type doping of the alloy and nicely agree with the use of Ti for generation of N vacancies in the technology of ohmic contacts on nitride materials.

I. Introduction

III-V semiconductor (AlN, GaN, InN) have unique physical properties such as high electron mobility, high saturation velocity, direct and tunable band gaps (6.2 to 0.65 eV), large absorption coefficients and high thermal stability as well as good thermal conductivity. These outstanding properties have allowed them to come at the forefront of optical applications in light emitting diodes (LEDs) [1] and laser diodes (LDs) [2], as well as in microelectronics as high frequency, high power devices (high electron mobility transistors (HEMT)) [3]. Of particular interest is the InAlN alloy, where, close to 18% of In-content, the lattice match to GaN can be achieved with a band gap around 4.2eV and the performances of the corresponding HEMTs are expected to improve by avoiding the interface strain due to the lattice mismatch which is inherent in AlGaN/GaN heterostructures [4]. Moreover, it has been shown that the HEMTs with an InAlN barrier can operate at very highest frequencies (~ 300 GHz) and withstand temperatures up to1000 °C [5]. Still, many challenges need to be addressed, as the performance of the corresponding devices strongly depends on the materials quality of active layers which are grown on foreign substrates. These layers may contain extended defects such as dislocations [6], stacking faults [7,8] and inversion domains [9], but also point defects all of which have a spurious influence on the device performance and stability [10,11,12]. More specifically, in addition to the above-mentioned defects, InAlN layers have been shown to undergo extensive crystallographic degradation with increasing thickness [13,14] and some mechanisms have been proposed in order to explain this behavior. For instance, Chen et al. [13] reported that the threading dislocations (TD) propagating from the underlying GaN layer may be at the origin of a TD governed degradation of the InAlN layers with increased thickness through the formation of V-pits. However, in their work on self-supported low dislocation density InAlN/GaN heterostructures, Perillaz-Merceroz et al. reported a similar degradation of the ternary layers [14] under compressive and tensile strain state. This was then attributed to the particular growth mode of InAlN which is dominated by kinetic roughening and formation of hillocks at the 3 growing surface. The V-defects were then shown to appear above a critical thickness which depended on the growth conditions and they were proposed to correspond to stabilized inclined facets of hillocks. In our current investigations and by carefully examining the results of the literature, we became convinced that probably the coalescence of growth hillocks is not the most predominant origin of the V-defects, as they start to form on the tops of hillock, and not between [14]. Moreover, each crystallographic defect can influence in a typical way the observed degradation of the InAlN/GaN layers [15] and it is not clear that even for the spontaneous degradation, a critical thickness can be accurately determined [16]. As reported more recently, this degradation may also be related to local indium segregation which has been shown to be favorable when relaxing the strain which is introduced in the system by N-vacancies [16].

In this work, we have investigated, in detail, the possible influence of the In aggregation and Nvacancy on the electronic properties of InAlN alloys. Earlier, the role of indium clustering on the band structure of In x Al 1-x N alloys was investigated by Gorczyca et al. [17] by ab-initio methods using supercell geometry with 32 atoms and the effects of varying the composition and atomic arrangements were examined. It was reported that the In clustering influences the band gap which becomes smaller. In the present paper we present the electronic properties in two case, 1-In aggregation in InAlN and 2-In aggregated around a V N situated in the center of supercell, inside the InAlN alloy. To this end, we have first calculated the properties of the bulk AlN, using the Heyd Scuseria Ernzerhof (HSE) functional to verify the robustness of our procedures by comparing with the experimental data. Subsequently, we investigated the effect of V N and aggregation of In on electronic and magnetic properties of the InAlN alloy, in this procedure, their influence on localization of electron was also determined.

II. Computational details

With the objective of investigating the properties of native defects in InAlN and determining how the nitrogen vacancy may affect its electronic properties, we carried out the structural relaxations within the density functional theory as implemented in the Vienna ab initio simulation package (VASP) [18] with the projector augmented wave (PAW) method [19], using supercells of 128 atoms which correspond to 4 4 4 unit cells. In these systems, the (In, Al) and N atoms occupy, respectively, the two hexagonal close packed sublattices of the corresponding wurtzite structure which belongs to the P6 /3 mc space group [20]. The indium concentration of x=17.2%, has been realized by substituting 11 Al atoms by In in the supercell. We investigated two atomic distributions: 1) the uniform alloy in which the positions of the metal atoms were randomly distributed, 2) and the clustering case in which Al atoms in the center of the supercell were substituted by In. The calculations have been performed in two steps, applying two different approximations in order to improve the accuracy as it is well known that the LDA (Local Density Approximation) underestimate the band gap of semiconducting materials. As a first step, the initial atomic coordinates in the supercell were relaxed by minimization of the Hellmann-Feynman forces (smaller than 10 -7 eV. Å -1 ) for each configuration using the GGA (Generalized Gradient Approximation) [21] for the exchange-correlation term. To this end, a cutoff energy of 900 eV for the plane wave basis set was sufficient for attaining converged results and a k-point mesh of Monkhorst-Pack [22] special points was used. The relaxations were performed until the energy on each ion was smaller than 10 -6 eV.

In the second step, the aim was to improve the accuracy of the results from the GGA; to this end, we used the HSE06 (Heyd-Scuseria-Ernzerhof) functional to correct the band gap, and the convergence criteria was that the force and energy were smaller than 0.02 eV. Å -1 and 10 -4 eV, respectively. The accuracy of the HSE functional is determined by the two parameters, α and ω, which are the fractions of semi local Perdew-Burke-Ernzerhof (PBE) functional replaced by a screened nonlocal functional and the inverse screening length, respectively [23]. In this study, α and ω are set at 25% and 0.2Å -1 , respectively, which correspond to the HSE06 functional [24]. For calculating the density of state (DOS) of structures, a k-point mesh of Monkhorst-Pack was used. Clearly, there is a notable difference between the two steps, and as can be seen in table 1, our second step gives values of the band gap for AlN which are close to the published experimental [25] results within 1.8%, whereas the GGA approximation band gap falls away by 34%.

Table 1: Bulk AlN lattice constants ( and ), and electronic band gap ( ), of AlN as calculated using GGA-PBE and HSE06 functional along with the experimental measurements taken from references [25] and [26].

GGA HSE06

III. Results

III.1. AlN

Before addressing the InAlN alloy, we found useful to test our procedures on AlN. Indeed, for this compound many experimental and theoretical data are available in the literature. For that, a supercell with configuration AlN was simulated using the procedure described in section II. The relaxed equilibrium Al-N bond length of 1.91 Å and the lattice parameter of 3.117 Å are close to the experimental value (3.110 Å) [26] and previous theoretical work [27,28]. As can be seen in the electron localization function (ELF) [29] (figure 1), electron rich areas (red to green) are located around the nitrogen atoms (small spheres), meaning that there is a charge transfer from the Al atoms (large spheres)

to N and therefore the bond between the two elements has a strong ionic character. This of course is due to the electronegativity difference between Al and N. As can be seen in figure 4, the 2D ELF contour of AlN underlines the location of the highest density as seen inside the (0001) plane around V N . Considering the 4 Al atoms located at the edges of the tetrahedron centered of V N , figure 4a shows the charge distribution between the three Al atoms of the top base of the tetrahedron (1, 2, 3). The charge density is highest at the barycenter of these three Al atoms, and equally spread towards each of them. Moreover, the distance between these atoms is around 3 Å, which slightly smaller than the 3.1 Å in the case when there is no V N . This charge density distribution along with the shrinking of distance between these Al is a strong indication that these atoms form chemical bonds. Looking now from the bottom, there is another high electron density area on top of the fourth Al atom which means that this Al atom has a dangling bond (Fig. 4b). Figure 5 shows the calculated total DOS (TDOS) of AlN with a V N and as well as the contributions of the valence orbitals of the two atoms (Al and N). First, as related to the bulk AlN, the band gap width has been decreased by the introduction of the nitrogen vacancy, it goes from 6.3 eV to less than 5 eV From the above results, the equilibrium Al-N bond length of 1.91 Å and the lattice parameter of 3.117 Å agree with the experimental values [26]. This also the clear for the calculated the band gap of 6.3 eV which is very close to values reported in the literature of 6.0-6.3eV [30, 31,32,33,34,35], and closer than the theoretical value calculated by HSE functional of 5.71 eV [28] and GGA+U functional of 4.326 [36]. Therefore, in the following, we will use the same procedure to investigate the properties of the lattice matched InAlN alloys which is very rich in Al.

III.2. InAlN

The In 0.17 Al 0.83 N alloy, which is lattice matched to GaN, was then investigated in three configurations (see fig. 6) , first as a random alloy, then the properties of In aggregation as formed in the center of the supercell, and finally the influence of the nitrogen vacancy and its coexistence with the In.

In all the cases, we have kept an identical composition of 17.2% indium in these supercells of 128 10 atoms, meaning that the total number of indium atoms was always 11, and the aggregation of In was realized by exchanging In with the Al atoms closest to the supercell center.

III.2.1. Lattice parameters

For the first two configurations (random alloy and In aggregation) the equilibrium lattice parameters of the above supercells were calculated. We have three cases, in the random alloy, a nitrogen atom can only be bound at most to one indium. In this work, we address the effect of In by discussing the three cases where the N atom is bounded to two, three and four indium atoms. For each case, the structures were relaxed by minimizing the total energy leading to the optimal atomic coordinates and , parameters of the supercell. As can be seen in Figure 7a, which shows the variation of lattice parameter, it increases with the number of In atoms around one N from 3.164 Å in the random In 0.172 Al 0.828 N alloy to 3.179 Å with an aggregation of 4 In atoms, corresponding to a change of 0.5%. A similar behavior occurs also for (Fig. 7b), it is 5.073 Å (random alloy) and it increases to 5.087 Å when 4 In are brought around the N atom in the middle of the supercell which corresponds to a slightly lower variation of 0.3%.

III.2.2 Electronic structure

We now discuss the electronic properties for the random alloys and when indium atoms are aggregated around the N atom in the middle of the supercell, we typically take into account only 2 and 4 In atoms.

As can be seen in figure 8, the most striking effect is the evolution of the band gap width, indeed band gap is largest for the uniform distribution, 4.4 eV (Fig. 8a) which maybe compared to the published experimental value of optical band gap at the composition of 17-18% (4.3 eV) [37]. When the indium is aggregated around the nitrogen atom, the calculated band gaps are 4.1 eV and 3.8 eV for 2 and 4 In atoms, respectively.

The Fermi level is always at the top of the valence band and the shape of the bands can be seen to change with the largest modification coming from the bottom of the conduction band. Therefore, the aggregation of the In atoms decreases Eg through this modification in the conduction band. Indeed, though we have only one large state at the BCM in the random alloy, three bands have been formed when 4 indium atoms are aggregated. This effect has been reported in reference [17] they conclude that the indium clustering strongly influences the band gap, for a system with 32 atoms. As in the case of pure AlN, in these relaxed structures of In 0.172 Al 0.828 N, the electron distribution shows a predominant localization on the nitrogen atoms (Fig. 8b), this can be explained by the fact that the electronegativity of N (3.04) and larger than that of Al (1.61) or In (1.78). Based on Bader charge analysis, the Al (In) atom loses 2.4 e (1.3 e) to the neighboring N atoms; therefore, the Al-N bond is mainly ionic whereas the In-N bond is of covalent character, along a slightly less ionic character. This is also clearly visible in the ELF image of the InAlN random alloy (Fig. 8b) where the contrast between In and N atoms indicates that the two atoms share electrons.

The total DOS reveals that the spin-up states and spin down states are symmetric, therefore the InAlN with and without In aggregation are nonmagnetic materials.

III. 2. 3. InAlN with N-vacancy

In our recent work, it was shown that the nitrogen vacancy plays an important role on the distribution of indium atoms inside the InAlN alloys [16]. In the following section, we discuss the effect of V N in the middle of the calculation supercell and indium distribution on the properties of lattice matched InAlN alloy.

III. 2. 3. 1. Electronic properties

When the N vacancy is introduced in the alloy, as can be seen in figure 9a, in comparison to the random alloys, the CBM, becomes as well defined as the VBM and clear band gap can be estimated (4.15 eV).

In contrast to the case of AlN, there does not form any acceptor level close to the VBM. However, donor levels are now present just below the CBM, and the first of them is occupied which defines the new Fermi level. Two donor states are clearly visible, they can be attributed to the atoms around the vacancy because they belong only to Al and N, and as no In is present at this location, and its contribution to these states is zero. When Indium atoms are brought to substitute Al around V N , the intensity of states slightly decreases at the Fermi level and more states appear just above, instead of one additional state in the case of the vacancy alone, there appear one state with the spin up, and two states with the spin down in the case of two indium atoms. In the case of 4 indium atoms we have a more complex configuration with the level at the Fermi level still decreasing in intensity; above it, there are now three states with the spin up and 4 levels with the spin down. These levels are now only contributed by indium and N orbitals. As can be seen in figure 10, the 2D ELF crossing (0001) atomic plane shows that the metal atoms around the vacancy establish bonds between them which may explain the changes that take place on the density of states at the top of the band gap. In the case of the random alloy with 4 Al atoms around the V N , two metallic bonds Al-Al were formed between the closest atoms, the distance between 1-2 and 3-4 are 2.92 Å and 2.84 Å respectively; the distance is larger between 1-3, 2-4, 2-3, and 1-4 atoms. This is similar when In atoms are aggregated around the V N , bonds were established between the closest atoms with a length of 2.83 Å (1-3) and 2.92 Å (2-4) for the case of 4 In atoms as this is smaller than the In-In bond length in metallic indium (3.25Å).

III.2. 3.2. Magnetic properties

In comparison with AlN where the V N has been reported to introduce non-symmetric density of states when the spin is considered [35], the above calculations were also carried out with spin polarization for InAlN. As already mentioned above, we do not have symmetry in the additional states that are introduced inside the band gap for spin up and down. Moreover, while the TDOS is completely symmetric when there is no N-vacancy (Fig. 8a), the V N induces up to a total magnetic moment of 1.00 μ B per supercell (Table 2) when it is associated with the indium aggregation. Moreover, the results show 16 that the systems are more stable in comparison with their counter parts when the spin polarization is not taken into account. To better understand the distribution of magnetic moments in the considered systems with V N , the local magnetic moment on each atom is shown in Table 3 where the most contributing atoms have been numbered. We then show in figure 11 the corresponding atoms for InAlN with 4 In around the V N . It is clear that the most contributing atoms are those located around the V N .

IV. Discussion and conclusion

Before investigating the properties of the InAlN alloy, we tested our procedures on AlN which exist in bulk and whose properties are well known [26]. As presented above, our calculations give values which completely agree with the available experimental results (bond length, lattice parameters, band gap width). Interestingly, it is shown that the N vacancy introduces many states in the band gap which are contributed by the atoms closest to the vacancy and this is in agreement with previous reports [36].

These results on AlN are closer to the experimental results that other theoretical values calculated by HSE functional of 5.71 eV [28] or indeed GGA+U functional of 4.326 [36]. As a consequence, we confidently applied the same procedures to the InAlN as lattice matched to GaN which does not exist in bulk, and whose properties have not yet been subjected to extensive investigation with similar accuracy.

For the In (17-18%)AlN alloy lattice parameters, we obtained = 3.165 Å for the random In distribution and up to 3.179 Å for the 4 In atoms aggregation around the central N atom. The corresponding parameters were = 5.073 Å and 5.087 Å, respectively. This is very close to the experimental results as we keep the composition of 18% In where the a parameter is that of GaN and the reported (In(17.7%)AlN) = 5.089 Å [38]. Unfortunately, it is difficult to compare with the values reported in reference [17], where the lattice parameters seem to present a slight decreasing trend, as the so called clustering appears probably to some ordering, indeed the indium atoms were positioned in every 4 hexagonal layer of their supercell. The calculated DOS of random InAlN showed that the band gap width of 4.4 eV is in good agreement with the experimental measurement of 4.3 eV [37]. In addition, the In segregation with 2 In or 4 In atoms reduces the band gap width from 4.4 eV (random) to 4.1 eV (with 2 In), and 3.8 eV (4In). A similar trend was reported by Gorczyca et al. [17] who carried out LDA-LMTO (linear muffin-tin-orbital) calculations on supercells of 32 atoms, for the In x Al 1-x N (x=19%) using three configurations, their random case is similar to ours, and their band gap was 4.2 eV, in the case of 4 indium atoms around a nitrogen one, their band gap slightly decreases to 4.0 eV [17].

In our recent study [16], our theoretical modelling showed that in the presence of nitrogen . Indeed, this falls in the attractive area for an In-rich cluster with a radius smaller than 6 Å as reported recently [16].

More importantly, many donor states form just below the conduction band minimum, in agreement with reports for example on AlN [35,38]. This result is important as it explains why the enrichment of nitride layers in nitrogen vacancies turn them into n+ doped which find application for instance in the mechanism which governs the Ohmic contacts for HEMTs through the use of Ti which captures nitrogen atoms [40,41].

In summary, we have performed a comprehensive study of the properties of lattice matched InAlN to GaN in the presence of the N-vacancy, using first-principles calculations. It is shown that the presence of the N-vacancy modifies the electronic properties of the alloy; it introduces states in the band gap and moves the fermi level from the VBM toward the CBM. Taking into account the total spins shows that the system is in fact a non-symmetric in spin-up and spin down states; this is due to the atoms located around V N and results in a magnetic moment of 1.00 µB. We also investigated the effect of the In aggregation on the electronic properties of InAlN alloy with and without V N . In the absence of the vacancy, the band gap width reduces when In atoms are substituted to Al atoms. In the presence of V N , the top of band gap become well defined in comparison to that of the random InAlN alloy, the number of introduced states in the top of band gap increases with the additional In atoms. Some of the new levels at the top of the band gap are populated and may induce n+ doping. In contrast to the case of AlN where such stated may be attributed to the presence of dangling bonds around the vacancy site, the additional states originate from the local deformation of the bonds that form between pars of metallic atoms.

Figure 1 :

 1 Figure 1: The two dimensional (2D) contour slice of ELF projected on a (0001) atomic plane, Al and N atoms are shown by large and small spheres, respectively

Figure 2 :

 2 Figure 2: DOS of pure AlN, the Fermi level is located just at the top of the valence band (VBM), and the band gap is 6.3 eV (see Conduction band minimum: CBM).

Figure 3 :

 3 Figure 3: ELF 0.73 isosurface for the structure viewed along [11-20], the yellow features mark the high electron density areas and the V N position is shown by the black asterisk *, the Al atoms underlined with a yellow cross are those around the V N .

Figure 4 :

 4 Figure 4: The 2D contour slices of ELF crossing (0001) atomic plane for AlN with a V N . a) Formation of bonds between the three Al atoms, b) The possible dangling bond for the Al atom along the [0001] axis.

( 4 .

 4 96). The question, what is the origin of this band gap reduction? Looking at the valence band of AlN (Fig.2) and after the introduction of the V N (Fig.4a), it can be seen that the band width does not change, therefore, this reduction of more than 1 eV in the band gap comes from changes at the bottom of the conduction band as brought in by the introduction of the N vacancy. Considering the partial densities of stated, it could be seen that the main contribution at the top of the valence band comes from the N-p orbitals, and that the bottom of the conduction band is slightly dominated by the N-s and Al-s contributions. Interestingly, additional levels are formed inside this band gap where one shallow level with spin up forms at the top of the valence band, while two deep ones are present just above the middle of the band gap. The state closest to the middle of the band gap is contributed by all the considered orbitals. Two spin down states are also present, one just at the top of the valence band the other at the highest position inside the band gap. They are both contributed by N-p and Al-s orbitals, but the highest in the band gap also receives contributions from N-s and Al-s. The calculations show that the state closest to the middle of the band gap with spin up is occupied which means that the Fermi level of the AlN-V N has almost shifted from a neutral semiconductor to one with deep donor states.

Figure 5 :

 5 Figure 5: Density of state curves for AlN with V N : a) The total DOS along with the N and Al contributions, b) The partial DOS of the Np orbitals, with a strong contribution at the top of the valence band, c) The Al-s, N-s and Al-p partial DOS have a slightly lower contribution in the valence and conduction bands, but participate equally as N-p to the deep levels inside the band gap.

Figure 6 :

 6 Figure 6: The three configurations, a) random alloy, b) In aggregation as formed in the center of the supercell, and c) nitrogen vacancy and its coexistence with the In, * shows the position of the vacancy.

Figure 7 :

 7 Figure 7: The equilibrium lattice parameters (a) and (b) for InAlN versus the number of In atoms inside the In-rich region. Four In distributions in the supercell are considered: uniform and 3 aggregates (2, 3 and 4 In around a N atom in the center of the supercell).

Figure 8 :

 8 Figure 8: (a) DOS for the random distribution of In, (b) the corresponding 2D contour slice of ELF crossing (0001) atomic plane, (c) DOS of cell with 2 In atoms around the central N atom, (d) the case of 4 In atoms.

Figure 9 :

 9 Figure 9: The TDOS: a) the random alloy with a V N , b) 4 In atoms around V N .

Figure 10 :

 10 Figure 10: The 2D ELF contour of InAlN around the V N ; a) Random alloy View from the top with bonding of Al (1) and Al (2) (b) view from the bottom of the supercell showing the bonding of Al (2) with Al (4) ; c) 4 In atoms around V N , View from the top with bonding of In (1) and In (3) (d) view from the bottom of the supercell showing the bond between In (2) and In (4)

Figure 11 :

 11 Figure 11: The atoms that most contribute to spin polarization are those around the V N for the case with 4 indium atoms around the vacancy.

  vacancy, indium nitride clusters are energetically stable up to diameters around 1.4 nm. In parallel, Zcontrast TEM (Transmission Electron Microscopy) observations close to the InAlN/GaN interface confirmed that 2-5 nm size indium rich areas form and are systematically connected to the vertical degradation channels. Therefore, the V N can play an important role on this specific crystallographic degradation in lattice matched InAlN. In this work, the modifications of the electronic properties of the InAlN alloys as contributed by the V N have been investigated in detail. It should also be pointed out that our calculations have been carried out considering spin polarization which shows the non-symmetry of spin up and down and it is comes out that the states that are introduced inside the band gap bring about a magnetic moment of 1.00 µB. As seen above, when the V N was introduced in the InAlN ternary alloy, the density of states changes completely, especially at the top of the band gap. As compared to the random alloy or even with indium aggregation cases, the bottom of the conduction band becomes well defined similar to the top of the valence band, and there is no significant trend for the slight change of the band gap width (4.15 eV (random case), 4.04 eV (2 In) to 4.24 eV (4 In)). Around the vacancy, metallic bonds were formed with a length of 2.84 -2.91 Å for In-In and Al-Al which is smaller than the equilibrium distance In-In = 3.25 Å and very close to Al-Al bond (2.86 Å)

Table 2 :

 2 Energy difference between the spin-polarized (E sp ) and spin-unpolarized (E spun ) states (ΔE = E spun -E sp ) and magnetic moments of the supercell M tot .

Table 3 :

 3 Local magnetic moment (µ B ) in InAlN alloy for random and aggregation distribution of indium.

ACKNOWLEDGMENTS

This work has been partly supported by the EU under project OSIRIS contract ECSEL-JU, GA: 662322.

The calculations were carried out in the « Centre Régional Informatique et d'Applications Numériques de Normandie » "CRIANN" (www.criann.fr). We would like to thank H. Lei for fruitful discussions.

The authors declare no conflict of interest

DelarationOfInterestStatement


Ranim Mohamad
email: ranim.mohamad@ensicaen.fr

Marie Pierre Chauvat

Slawomir Kret

Piero Gamarra

Sylvain Delage

Viwanou Hounkpati

Cedric Lacam

Jun Chen

Pierre Ruterana

The critical role of N-vacancy on chemical composition fluctuations and degradation of InAlN layer

Keywords: N-vacancy, InAlN degradation, atomistic modelling, In segregation, energetic stability, TEM, ab-initio. I

Due to its intrinsic properties and the possible lattice match to GaN, InAlN is expected to allow the fabrication of optimal high electron mobility transistors for high power and high frequency applications. However, the crystal quality of InAlN nearly lattice-matched to GaN degrades when the layer thickness is increased, and this is a strong limitation for the fabrication of devices in which thick barriers need to be used. In this work, we have carried out a detailed theoretical investigation of the behavior of indium atoms in the alloy. It is clearly shown that in the presence of nitrogen vacancies, which are common defects in these materials, indium nitride clusters will present excess formation energy up to diameters around 1.4 nm.

In parallel, Z contrast TEM observations close to the InAlN/GaN interface show that 2-5 nm size indium rich areas form and are systematically connected to the vertical degradation channels. This is at variance with published results, which concluded that the observed degradation was due exclusively either to the underlying threading dislocations or to a characteristic three-dimensional growth mode.

The outstanding properties of group III nitride (InN, AlN and GaN) compounds find application in light emitting diode (LEDs) 1 , laser diode (LDs) 2 and high electron mobility transistors (HEMTs) 3 . In particular, the ternary InAlN alloy is a subject of much research effort due to its lattice match with GaN for a composition of indium around 18%, when grown along [0001]. Interestingly, the band gap of InxAl1-xN alloys covers probably the widest wavelength range of known semiconductors (0.67 eV (InN) 4 to 6.2 eV (AlN) 5 ), however this unique property comes along with many challenges for the potential applications. Indeed, although several research groups reported high quality near lattice matched InAlN/GaN heterostructures [6][7][8][9][10] , the growth of InAlN over the full composition range is difficult due to the different physical properties of the binary compounds InN and AlN (bond length, elastic properties, growth temperatures, thermal expansion constants…) 5,[11][12][13] .

Experimentally, the crystallographic degradation of InAlN layers has been reported when the thickness of the layer is increased and some explanations have been reported in the literature 10,14,15 . The first report proposed that the threading dislocations from the GaN could be the main driving force for the alloy degradation through the generation and multiplication of pinholes 10 . Indeed, most of the layers are grown on top of GaN templates which are deposited on sapphire 16.17 , silicon 18 or SiC 19 and they contain high densities of crystallographic defects such as threading dislocations 20 , inversion domains 21 and stacking faults 22.23 . These defects are transmitted into the growing alloy and may generate typical crystallographic degradations 24 . Other authors have proposed that there could be a critical thickness beyond which the alloy undergoes strain relaxation through crystallographic degradation and generation of extended defects 10 . However, this explanation seems difficult to accept because the reported degradations took place at the indium composition of 17-19%, which is very close to the lattice match on GaN. Moreover, as reported by Perillat-Marceroz et al. 14 similar degradations occur also in InAlN layers grown on free standing GaN where the influence of the underlying defects cannot be considered as the dominant mechanism due to their low density. The latter authors suggested that the degradation of InAlN layers is a spontaneous phenomenon due to a specific growth mode which systematically leads to the formation of hillocks. It was proposed then that pinholes naturally form and multiply at the coalescence of hillocks 14 . In addition of the crystallographic degradation that has been mainly reported in the Al rich alloy (In0.18Al0.82N), phase separation was observed by transmission electron microscopy in indium rich alloys such as In0.72Al0.28N 25 .

During last years, numerous modelling works have also been devoted to phase diagrams in ternary and quaternary solid solutions [26][27][28][29][30] to determine the instability zone in this alloy; more recently, we demonstrated that biaxial strain influences its miscibility zone 29 . However, structural defects (hillocks and V-defects) were observed independent of the type of strain in InAlN layers for low In compositions of 13.5% (tensile strain), as well as at 19.7% (compressive strain), suggesting that their origin may not be related only to the strain state, but to the intrinsic properties of the InAlN alloy. Therefore, the mechanisms of this degradation are still unclear and need further investigation. Based on first-principles calculations, the formation energies of native defects in AlN and GaN were reported by Mattila and Nieminen 31 , Gorczyca et al. 32 , Fara et al. 33 and Stampfl and Van de Walle 34.35 . The main conclusion of these investigations was that the self-interstitials and antisites should exhibit high formation energy in wurtzite AlN, whereas the vacancies have the lowest, even for charged defects 36,37,38 . Therefore, it is important to theoretically investigate the possible effects of stable defects in this alloy in order to contribute to a deeper understanding of the mechanism of these degradations.

II. MODELLING PROCEDURE

In this investigation, we have focused on the nitrogen vacancy which is the most stable point defect in the nitride materials 32 and the objective has been to theoretically monitor the behavior of the indium atoms around it. The vacancy is generated by removing a nitrogen atom from the center of a random In0.18Al0.82N alloy in a supercell of 21296 atoms. This size has been previously determined as the low limit of atom number for the minimum dispersion in the energy 29 during our process of relaxation where we use a modified Stillinger Weber type empirical potential, which was previously adapted for the wurtzite nitride compounds 39 . The modification consisted in a parametrization in order to enable efficient atomic simulations of large systems containing wrong, dangling and extra bonds. To this end, the potential was tested through a comparison with ab initio calculations for inversion domain boundaries (IDB), which contained wrongs bonds, as well as a reconstructed IDB and a stacking mismatch boundary (SMB) which could be characterized by four atom rings and dangling bonds, respectively 39 . Using this potential, the calculated elastic properties have also been reported 40 .

Prior to the systematic investigation, we have analyzed the behavior of the SW potential in comparison with ab initio procedures as implemented in the VASP commercial software on systems of 127 atoms (with N-vacancy), which are still not so much time expensive. As clearly shown in the supplementary material, the two procedures have a similar trend during the relaxation. Therefore, we can be confident that the results we obtain on large systems using the SW potentials are relevant. In this instance, the large supercells are relaxed until the forces on all atoms become lower than 10 -15 eV/Å, which makes us choose the final configuration as the most stable. In the relaxed reference supercell containing only the nitrogen vacancy, we then replace the Al atoms around it by In atoms step by step from a radius of 2 to 18 Å, and carry out the SW relaxation in order to until the most stable configuration is reached. Figure 1 shows the initial configuration of an InN cluster of 14 Å radius generated by substituting all the Al atoms by indium.

As all the supercells contain the same number of atoms, the formation energy (E) of the InN cluster is calculated by relating each total energy Ealloy (supercell i) to that of the reference, (∆ ).

III. INDIUM CLUSTERING AROUND THE N-VACANCY

A. Energetic behavior

As soon as an indium atom is brought in to substitute an Al one, the energy of the new system decreases. Of course, in order to have the energy reference, we only consider systems with the same atom number. Therefore, one should bear in mind that the indium concentration in the supercell changes with the cluster size. For instance, for a cluster radius of 18 Å, the total number of indium atoms in the cluster is 991, which brings the indium concentration to 27.3%. Then as can be seen in figure 2, the excess energy of the supercell decreases from 2 Å to 6 Å radius up to a ΔE =0.67 eV. For the largest clusters we have considered, the gain in energy is still present but decreases with cluster size. This saddle point of 6 Å radius is interesting; it means that InN clusters of 1.2 Å diameter should have a high probability to form around N vacancies during the growth of InAlN layers.

B. Strain relaxation

In order to try to understand the above energetic behavior, we analyzed the evolution of the atomic bond lengths in the supercell as well as the variation of the lattice parameter inside the cluster.

B.1. Bond lengths

It is interesting to examine the In-N and Al-N bonds in the supercell as a function of the atom position. Considering first the Al-N bond length, we don't have Al inside the cluster; the bond is highly deformed at the interface with the matrix and it attains rapidly the equilibrium length (1.906 Å) in the alloy (fig. 3a). Taking the reference as the Al-N average bond length in the supercell area of In0.18Al0.82N, the Al-N bonds are 0.45% compressed for the cluster radius of 14Å.

As for the In-N bond length (Fig 3b), in the small cluster sizes (Rc from 2 to 6 Å), it is almost constant, but stretched to reach 2.13 Å inside the cluster slightly more than 2.105 Å in the InAlN alloy. Beyond 6 Å, the compressive strain increases with the cluster size and stabilizes at around 3.5 -3.6% for the largest considered size. It should be pointed out that the minimum energy at a cluster radius of 6 Å is obtained when the In-N bond length has been established to 2.13 Å (Table I), which is close to the bond length in the indium nitride (2.156 Å).

Around the vacancy, for small clusters, the In-In distance (3.68 Å) is larger than the equilibrium distance in the metallic phase of In (3.25 Å), so there is space for the relaxation of the local strain on In-N bonds. This local strain decreases from 2.07% for Rc=2 Å to 1.31% for Rc= 6

Å. Beyond 6 Å, as can be seen in fig. 3, the In-N bond length is compressed; its length is 2.08 Å for 10 Å and 14 Å. At the same time the distance between In-In atoms starts to increase again (Table I), indicating that the four In atoms just around the N-vacancy enter in the compression state.

B.2. Evolution of the local strain versus the cluster size

To estimate the local strain in the cluster, we calculated the relative change of the lattice parameter (strain) for different cluster's diameters with respect to the case of the random distribution of In without the N-vacancy, in a cubic cell of 16 Å 16 Å 16 Å. The local strain as a function of cluster's radius is presented in Fig. 4. As can be seen, the presence of an N-vacancy leads to a local deformation of 0.8%, then when indium atoms are brought in to substitute Al around the vacancy, the strain almost disappears (ε = 0.07%) at the indium nitride cluster radius of 6 Å. For clusters of radius Rc≤6 Å, the strain is positive, and the area is tensile strained with respect to the InAlN matrix. For Rc > 6 Å, the cluster becomes compressed and the deformation goes as high as 4.4%

for Rc = 14 Å.

IV. EXPERIMENTAL OBSERVATIONS

We have also characterized InAlN layers grown on GaN templates deposited on the (0001) sapphire surface by metalorganic vapour phase epitaxy (MOVPE) by transmission electron microscopy. Whereas tens of samples with various growth conditions have been investigated, in the following, we present the results on two typical InAlN layers, S1 and S2. In S1 sample, the InAlN layer has an indium content of 19% close to the lattice match with GaN and its thickness is 330 nm. For S2, the nominal indium content was 32% and the ternary layer thickness was around 150 nm, corresponding to an expected lattice mismatch of around 1.6%. In an ideal case of strain release by plastic relaxation, this should lead to a periodic network of 60° <112 0> type dislocations with a theoretical distance of around 200 nm. For transmission electron microscopy (TEM), the cross sections and plan view samples were thinned down to 10µm by mechanical polishing using a Multiprep®, and the electron transparency was obtained by ion milling at 5 kV and mounted on a liquid N2 cold stage. A gradual final step from 3 kV to 100 V was used to decrease ion-beam damage. Extensive TEM investigation was carried out in the conventional modes in order to identify the crystallographic defects and then through analytical and chemical sensitive highresolution modes in order to determine the local chemistry and strain state.

The microstructures of the two layers exhibit characteristic features as can be seen in the micrographs of figure 5 which are weak beam images recorded using g = 112 0, in addition to strong strain which corresponds to the white contrast visible inside the InAlN layers. In S1 sample (Fig. 5a), the layer is rough, and the strain predominates towards the surface; from the interface with GaN, the irregular bright parts indicate that the corresponding crystallographic degradation occurs all over the layer without any specific critical thickness and can start even at the interface with the GaN template (arrows). Interestingly, the visible threading dislocations (TD) which have an component cross the interface as straight lines, meaning that there is not influence of the misfit strain in agreement with the nominal composition of 19% indium.

In S2, the surface roughness is also present as shown in figure 5b, and as can be seen, the structural degradation is initiated starting at the interface with the GaN template. Moreover, in this sample, the contrast of the dislocation lines disappears at the interface, which is clear confirmation that crystalline degradation of the InAlN layer is high from the start. In plane-view, the observations along the [112 3] zone axis, where all dislocations should be visible 41 ; show only the threading dislocations from the GaN template; they correspond to the visible black line segments (some of them have been underlined by white arrows). In this figure, no single misfit dislocation can be seen inside the InAlN/GaN interface all over this exhibited area of 1.6 µm 2 , although, in this layer, the theoretical misfit strain is more than 1.6%. This observation is in contrast to the growth of InGaN on GaN, indeed, at comparable levels of strain, it is has been reported that a network of misfit dislocations forms at the InGaN/GaN interface 42 . This is a clear indication that the strain relaxation exclusively takes place inside the InAlN alloys, and not at the interface with the underlying GaN template.

Coming back to sample S1, High Angle Annular Dark Field (HAADF) micrographs were recorded along a <101 0> zone axis. This method is highly sensitive to variations in the atomic number of constituents in the sample (Z-contrast images). As can be seen in figure 6, numerous areas, which appear to have a spherical shape, exhibit a brighter contrast (In-rich region) than their surroundings indicative of higher average Z composition. Therefore, even inside this nominal In0.19Al0.81N layer, there form indium rich areas in the range of 1-5 nm size (arrows). Inside such areas, the measured In composition can be as high as 35%, whereas only 11% indium is measured inside the dark areas. Columnar features are also visible in this area and the dark ones which are rich in Al, can be seen to originate from these bright clusters. As these columnar features terminate as V defects at the layer surface, it may be concluded that the degradation mechanism of these alloys is related to them and not mainly to the threading dislocations from the GaN template.

V. DISCUSSION AND CONCLUSION

The above observations demonstrate that the strain inside InAlN/GaN heterostructures is not relaxed by the formation of interface defects such as misfit dislocations. The TEM observations show that the strain relaxation takes place randomly inside the alloy without any critical thickness leading to an intrinsic degradation of the crystalline quality of the alloy. Of course, the threading dislocations, as reported earlier 10 , participate in this degradation, probably through their interaction with the indium as it has been suggested that In atoms could reduce the energy of c-screw dislocations when they are located at in their core 43 . Moreover, the possible three-dimensional growth mode leading to hillocks 14 build up may not be neglected. The above results highlight the role of native defects generated during the layer's growth. In particular, the N-vacancy could play a critical role on the In segregation in In0.18Al0.82N towards the formation of In-rich clusters. The calculated excess energy of In-rich cluster in In0.18Al0.82N environment with different size shows that the In-rich cluster is not thermodynamically stable in the perfect structures without defects 44 .

However, as seen above, the N-vacancy appears to play a critical role for In aggregation, indeed, the excess energy is increased when In substitute all Al atoms around the vacancy to form InN clusters. When the cluster radius reaches 6 Å, the system has the smallest energy with a decrease of 670 meV. Moreover, this energy stays negative up to largest investigated cluster of 36 Å diameter. Therefore, the aggregation of In atoms around the N-vacancy reduces the energy of the whole system by relaxing the bond length. As for the atomic structure, In-N bonds in the cluster are stretched in comparison with the bulk alloy due to the effect of N-vacancy.

From the TEM observations, 1 to 5 nm indium rich clusters are present from the beginning of the InAlN alloy growth. Of course, these clusters participate in the modification of the local chemical uniformity as seen in the Z contrast image. Their connection to columnar features is obvious evidence that they are at the origin of the subsequent degradation of the grown film.

Are these clusters related to N vacancies? The TEM investigation cannot attest this, but the the above calculations show that this is highly probable. Therefore, considering our observations and associated modelling, it may be suggested that the growth of the InAlN layer proceeds with more and more indium atoms gathering around the N-vacancy, thus depleting the surroundings and leading to possible formation of Al rich columns, associated chimneys, as well as final degraded hillocks and surface pinholes. The next steps will be to take into account larger systems in our simulations and to investigate the kinetics of this intrinsic degradation of the InAlN layers using molecular dynamics procedures. Indeed, at this point, we need to investigate at which stage the degradation takes place, does it start during the growth itself, the cooling: this is still an open question which needs to be addressed in the next steps. The only point we are sure of, is that after cooling, the structure and composition of the final layers is completely set and highly stable.

SUPPLEMENTARY MATERIAL

In this part, we demonstrate the similar behavior of the SW potential in comparison with ab initio energetics when Al atoms are replaced by indium atoms to simulate the initial formation of an indium rich cluster. To this end, 128 atoms supercell are handled within the ab-initio procedures, the similar trend permits us to carry out the modelling on much larger supercells using the SW empirical potential.

■ AUTHOR INFORMATION

Figures captions:

1 .

 1 Figure 1. The R=14Å supercell with an InN cluster embedded in an In 0.18 Al 0.82 N random matrix.

Figure 2 .

 2 Figure 2. Cluster radius dependence of excess energy of In-rich cluster and N-vacancy in the environment of In 0.18 Al 0.82 N for the supercell of 21295 atoms.

Figure 3 .

 3 Figure 3. Profile of Al-N and In-N bond lengths as a function of distance from the cluster's center with radius 2 Å (black), 4 Å (red), 6 Å (green), 10 Å (blue), 14 Å (cyan) and 18 Å (purple).

Figure 4 .Figure 5 .

 45 Figure 4. The local strain on lattice constant a as a function of cluster radius

Figure 6 .

 6 Figure 6. HAADF micrograph recorded along a <101 0> zone axis close to the interface AlInN/GaN in sample S1, bright contrast corresponds to indium rich areas (see arrows).

Table : Table I : The deformation in (%) on In-N bonds around N-vacancy with respect to the equilibrium state 2.156 Å and In-In distance, as function of cluster radius.

 :I

	Rc	In-N	In-In
	(Å)	deformation (%)	distance (Å)
	2	2.07	-
	4	1.98	3.68
	6	1.31	3.25
	7	2.55	3.34
	8	3.25	3.41
	10	3.52	3.52
	14	3.53	3.53

This work has been partly supported by the EU under project OSIRIS contract ECSEL-JU, GA: 662322. The calculations were carried out in the « Centre Régional Informatique et d'Applications Numériques de Normandie » "CRIANN" (www.criann.fr) under project number: 2016009.


David Martínez Martínez

Adam Cooman
email: adam.cooman@ampleon.com

Fabien Seyfert
email: fabien.seyfert@inria.fr

Martine Olivi
email: david.martinez@inria.fr

Stéphane Bila
email: stephane.bila@xlim.fr

Optimal bounds and matching networks of fixed degree for frequency varying impedances

Keywords: antennas, filter synthesis, matching, bounds

In this paper, matching networks of finite degree are computed. Additionally the presented results are compared with the lower fundamental bounds available in the literature. These bounds are used to certify the optimality of the provided matching networks in function of the attained matching tolerance. To illustrate the presented results, two different examples of matching problems are presented.

I. INTRODUCTION

Bounds for the problem of matching have been already computed by numerous authors in the literature. These bounds were first introduced in [START_REF] Bode | Network Analysis and Feedback Amplifier Design[END_REF] where the problem of matching an RC-load is considered as the design of a lowpass filtering network where an RC-element is fixed. In [START_REF] Fano | Theoretical Limitations on the Broadband Matching of Arbitrary Impedances[END_REF] the problem was extended to the case of a generic load by using the Darlington equivalent and reformulated in [START_REF] Youla | A new theory of broadband matching[END_REF] as a complex interpolation problem. Additionally in [START_REF] Helton | Non-euclidean functional analysis and electronics[END_REF] the matching problem was solved optimally by considering the broader class of infinite dimension functions H ∞ and therefore providing hard bounds for the matching problem in finite dimension. Nevertheless the computation of matching networks of finite degree which approach as closely as possible the lower bounds is still a current issue.

In [START_REF] Martínez | Convex optimisation method for matching filters synthesis[END_REF] a method based on convex optimisation was presented. This technique provides us, with hard lower bound for the problem of matching when a matching load of finite degree is considered. These bounds can be considered as an extension of the results computed in [START_REF] Helton | Non-euclidean functional analysis and electronics[END_REF] to the case of a matching network of finite degree.

In this work we provide a study comparing the hard lower bounds to the best matching tolerance attainable for a given load together with the result provided by a rational matching network of a given degree. In this context, two examples of matching network synthesis are presented. In the first one we consider the problem of antenna matching while in the second one we deal with a double matching problem by designing the input and output matching network for a LNA (low noise amplifier).

II. THEORY

In this paper we use a novel algorithm for the computation of matching networks which is based on the pointwise match-ing algorithm introduced in [START_REF] Baratchart | Boundary Nevanlinna-Pick interpolation with prescribed peak points. Application to impedance matching[END_REF]. This algorithm is implemented as part of the matlab toolbox PUMA-HF (see [START_REF] Martinez | Software PUMA-HF[END_REF]). The matching problem considered in this work aims to minimise the reflection of the power transmitted to a given load within a specified frequency band. The load is represented as a 2-port device L in Fig. 1. Usually the power is transmitted to the load through a filter F that rejects out of band signals. Both devices, the filter together with the load compose the global system S.

A. The matching problem

In this section we state the matching problem as the minimisation over the passband I of the magnitude of the global reflection |S 11 | which is expressed as the pseudohyperbolic distance between F 22 and L 11 . We have therefore the following problem Problem 1 (Matching problem).

Find:

ψ opt = min F22 max s∈I F 22 -L 11 1 -F 22 L 11 (1)
Additionally we suppose that the zeros of the functions F 21 and F 12 are fixed as it is customary in classical filter synthesis.

Note that in [START_REF] Martínez | Convex optimisation method for matching filters synthesis[END_REF] hard lower bounds ψ min have been provided for the solution to Problem 1 such that

ψ min ≤ ψ opt (2)
This bound ψ min are the result of a fundamental limitation imposed by load on the global systems that can be realised. Note for instance that in the case of a frequency invariant load we have ψ min = 0, namely any matching level is possible. These lower bounds can now be used to certify the optimality of the computed matching networks.

Load (L) Filter (F) F 22 L 11 F 11 L 22 S 11 S 22
Fig. 1. Global system composed of the cascade of the matching filter with the load and reflection coefficients.

III. EXAMPLES

A. Small superdirective antenna

As a first and simple example, we consider the problem of matching the small super-directive antenna presented in [START_REF] Jonsson | On Methods to Determine Bounds on the Q-Factor for a Given Directivity[END_REF] in the interval I defined as I = [870, 900] M Hz.

(

The reflection L 11 of this antenna is shown in fig. In fig. 3 we show the lower bound ψ min as well as the obtained matching level ψ opt as a function of the degree K of the matching filter from K = 1 to K = 12. These values are also listed in table I. Note the significant improvement of the reflection level around 870 MHz for any value of K obtaining a matching level between -6.5 and -9 dB. It is also interesting to remark the proximity of the obtained level ψ opt to the lower limit ψ min . Indeed in fig. 4 we plot the extremely small optimality gap, which quickly converges towards zero. This fact together with the local optimality of the filter that provides the matching level ψ opt certify the obtained result.

B. Double matching of a low noise amplifier

In this section, we design the input and output matching network for a low-noise amplifier based around the transistor infineon BFP520. We can see in Fig. 6 the ADS schematic of the transistor with the biasing network which has been used to simulate scattering parameters and noise parameters of the transistor.

The schematic of the desired system is shown in Fig. 7. The frequency band considered in this example is mainly limited by the design of the biasing network. In this case we target the band between 2 GHz and 4 GHz: 1) Input matching: First we design the input matching network of degre 1, such that the transistor input sees the reflection coefficient S opt which provides the optimal noise figure for the LNA. We do this by synthesising a matching network to match the reflection S opt . The matching network will show at the output the best approximation of the function S opt (Fig. 8). Thus we obtain a matching network matched at the input to 50Ω and synthesizing S opt at the output. 2) Output matching: Once we have the input matching network, we compute the S 22 parameter of the combination of the input matching with the transistor and compute the matching filter for the total network shown in Fig. 10. Note that this design approach will not work every time. It is possible that stability issues are encountered at this stage, but working with an unconditionally stable transistor+biasing will resolve this issue. Now we compute the matching network for the transistor output by means of Problem 1. We compare in this case the result obtained with a network of degree K from 1 to 6.

I = [2GHz,
Figure 11 shows the level ψ opt obtained with the computed matching network as well as the lower hard limit ψ min , both listed in table II. We obtain a matching level below -20dB in any case. Similarly to the input matching network, we pick a degree K = 1 for the output matching network, obtaining the response shown in Fig. 12. 3) Global system: We can now use the synthesized input and output matching networks, both of them chosen to be of degree 1 to reconstruct the global system of the LNA shown in Fig. 7. We obtain the schematic in Fig. 14 where the input and output matching network have been included. Furthermore we provide in Fig. 13

Fig. 2 .

 2 Fig. 2. Superdirective small antenna

Fig. 3 .1Fig. 4 .

 34 Fig. 3. Bounds for the reflection level

Fig. 5 .

 5 Fig. 5. Global response S 11 and transducer gain obtained with the load in Fig. 2

Fig. 6 .

 6 Fig. 6. Infineon transistor BFP520 with the biasing network.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Schematic of the global system with input and output matching networks.

Fig. 9 .

 9 Fig. 9. Output reflection of the intput matching network.

Fig. 10 .

 10 Fig. 10. Output matching problem.

Fig. 11 .

 11 Fig. 11. Lower bounds and obtained reflection level for the output matching problem.

  the global response of the network shown in Fig. 14. It can be noted the obtained matching level on the parameter S 22 which is below -20dB within the the whole band I = [2GHz, 4GHz]. IV. CONCLUSION In this work a practical algorithm for the computation of matching networks of finite degree is introduced and several examples are presented. The computed matching networks are certified by comparing the provided matching level with the fundamental lower bounds available in the literature. ACKNOWLEDGMENT This work is supported by the CNES (Centre National d'études spatiales) and the DGA (Direction Géneral de l'Armement).

Fig. 13 .

 13 Fig. 13. Global response of the network shown in Fig. 14.

Fig. 14 .

 14 Fig. 14. Global schematic of the LNA with input and output matching networks.

  4GHz]

		Degree (K) ψopt dB ψ min dB
		1		-6.5028	-7.0389
		2		-7.4389	-7.7777
		3		-7.9916	-8.1925
		4		-8.3218	-8.4512
		5		-8.5351	-8.6237
		6		-8.6815	-8.7444
		7		-8.7859	-8.8320
		8		-8.8629	-8.8977
		9		-8.9212	-8.9480
		10		-8.9663	-8.98745
		11		-9.0020	-9.0188
		12		-9.0306	-9.0442
				TABLE I
		OBTAINED MATCHING LEVEL VS LOWER BOUND.
		-6
					Lower bound ψmin
					Upper bound ψopt
	Reflection level (dB)	-8 -7
		-9
		2	4	6	8	10	12
		Degree K of the matching network


Ranim Mohamad
email: ranim.mohamad@cnrs-imn.fr

Pierre Ruterana

Indium segregation mechanism and V-defects formation on (0001) InAlN surface: an ab-initio investigation

) and (0001 ̅ ) In (18%) AlN surfaces. First, it was shown that these surfaces are most stable when they contain complex defects. The presence of vacancies causes the In to be strongly bound to the surface with the adsorption energy increasing by 0.11 eV for metal-polar and by 0.78 eV N-polar. In contrast, the adsorption strength of Al to the surface with defects decreases; the corresponding energy goes from 3.96 eV to 2.29 eV (metal-polar) and from 8.30 eV to 5.05 eV (N-polar). Simultaneously, the diffusion of In is reinforced; its energy barrier decreases by 0.74 eV (0.06 eV) for N-polar (metal-polar) InAlN surface, whereas that of the Al adatom increases by 0.32 eV for metal-polar (0.08 eV for N-polar) which should limit its the diffusion on the surface.

Therefore, the indium atoms will tend to migrate towards the complex defects. Eventually, during the epitaxial growth, this aggregation of indium atoms around the defects and the low mobility of Al atoms could be at the origin of the observed V defects, the phase separation and the crystallographic degradation of the InAlN epitaxial layers with increasing thickness.

Introduction

For almost 20 years, InAlN has been attracting a large attention for use as attractive layer in optoelectronic and electronic devices: Bragg mirrors [1] and high electron mobility transistors (HEMT) [2], due to its lattice match with GaN for a composition of indium around 18% when grown along [0001] with a band gap around 4.2 eV. The performances of the corresponding HEMTs are expected to improve by avoiding the interface strain due to the lattice mismatch which is inherent in AlGaN/GaN heterostructures [3]. Moreover, it has been shown that the HEMTs with an InAlN barrier can operate at very high frequencies (~ 300 GHz) and withstand temperatures up to 1000 °C [4].

However, the growth of high quality InAlN films with 18% In content, as needed to avoid the formation of extended crystallographic defects [5], has proved challenging. The size difference between Al and In, in addition to that of Al-N and In-N bond strengths may limit the homogeneous growth leading to theoretically reported spinodal decomposition and phase separation [6,7]. Furthermore, InN is the least stable among the III-nitride semiconductors, with thermal decomposition starting at temperatures as low as 470 °C [8]; this is low compared to that required for the growth of best quality AlN (~1600°C) [9]. Indeed, InAlN layers undergo extensive crystallographic degradation with increasing thickness [10] and mechanisms were proposed in order to explain this behavior [11][12]. For instance, Chen et al. [11] reported that the threading dislocations (TD) propagating from the underlying GaN layer may be at the origin of a TD governed degradation with increased thickness through the formation of V-defects and their multiplication at the InAlN surface. But on self-supported low dislocation density InAlN/GaN heterostructures, Perillat-Merceroz et al. [12] reported a similar degradation of the ternary layers under compressive and tensile strain state, which they attributed to the fact that growth of InAlN takes place by formation of hillocks at the growing surface due to kinetic roughening. The V-defects were assumed to appear above a critical thickness at coalescence of the hillocks where they should correspond to stabilized inclined facets of hillocks [12]. However, a close examination of AFM images presented in reference 12 clearly points out that the V-defects form at the top of the hillocks and not at their coalescence. Therefore, the primary origin of the V defects in AlInN/GaN heterostructures is still an open question.

In this work, after a short discussion of our atomic force microscopy (AFM) and transmission electron microscopy(TEM) results on InAlN/GaN heterostructures, we present the first theoretical investigation of the stability of (0001) and (0001 ̅ ) surfaces of InAlN with and without defects as well as the metal adatom kinetics on these surfaces. Our objective being to shine some new light on the above typical growth mode of InAlN, we carried out a detailed investigation of the key parameters that control the growth rate, the material quality, and the surface morphology in epitaxial growth, namely surface diffusion and adsorption. It is expected that our results will provide critical information towards the explanation of InAlN layers degradation mechanisms through V-shaped defects formation.

II. Simulations details

All the calculations were carried out in the framework of density functional theory as implemented in the VASP code [13]. For the electron exchange-correlation, we used the generalized gradient approximation (GGA) with the parametrization scheme of Perdew Burke Ernzerhof (PBE) [14], and the full-potential projector augmented wave (PAW) method [15] to describe electron-ion interaction.

The plane wave energy cutoff energy is 400 eV was determined as sufficient for the accurate calculation of the properties of AlN and InN in the wurtzite structure as shown in Table 1, along with experimental values, for the lattice parameters and elastic constants [16].

A hexagonal supercell of ( 0001 atoms and 12 Å vacuum are sufficient. For all the structures, the four top layers in addition to the adatom have been allowed to relax. All atomic positions were optimized using the conjugate gradient method until the maximum Hellmann-Feynman force acting on each atom was less than 0.02 eV/Å.

The experimental lattice parameters were used 𝑎 𝐼𝑛(18%)𝐴𝑙𝑁 = 𝑎 𝐺𝑎𝑁 = 3.189 Å and 𝑐 𝐼𝑛(18%)𝐴𝑙𝑁 = 5.089 Å [18].

After relaxation, we determine the stability of the different defects configurations by comparing their formation energies. As the surface stoichiometry is not the same for all the configurations considered here, the relative formation energies depend on the chemical potentials of the excess atomic species.

The relative formation energy is thus:

𝐸 𝑓 = 𝐸 -𝐸 𝑟𝑒𝑓 + ∑ 𝑛 𝑖 • µ 𝑖 𝑖 ( 1
)
Where 𝐸 is the total energy of the supercell containing a relaxed defect (vacancy and vacancy complex), 𝐸 𝑟𝑒𝑓 is the total energy of the same supercell without defects, µ 𝑖 is the chemical potential of species 𝑖 and 𝑛 𝑖 is the number of exchanged particles between the defect supercell and the perfect cell.

The chemical potentials obey the following equations:

µ 𝐴𝑙𝑁 = µ 𝐴𝑙 + µ 𝑁 = µ 𝐴𝑙 0 + µ 𝑁 0 + 𝛥𝐻 𝑓 0 (𝐴𝑙𝑁) µ 𝐼𝑛𝑁 = µ 𝐼𝑛 + µ 𝑁 = µ 𝐼𝑛 0 + µ 𝑁 0 + 𝛥𝐻 𝑓 0 (𝐼𝑛𝑁) (2)
With 𝛥𝐻 𝑓 0 the formation enthalpy of AlN, µ 𝐴𝑙 0 the chemical potential of bulk aluminum, and µ 𝑁 0 the chemical potential of N in the N 2 molecule. From equation (2), we define the allowed thermodynamic range for the chemical potentials, µ 𝐴𝑙 ranges from µ 𝐴𝑙 0 + 𝛥𝐻 𝑓 0 (𝐴𝑙𝑁) to µ 𝐴𝑙 0 and µ 𝐼𝑛 ranges from

µ 𝐼𝑛 0 + 𝛥𝐻 𝑓 0 (𝐼𝑛𝑁) to µ 𝐼𝑛 0 .
To analyze the stability of In, Al adsorbed on InAlN surface, the adsorption energy of an adatom was calculated using:

𝐸 𝑎𝑑 (𝐼𝑛/𝐴𝑙) = 𝐸 𝑠𝑢𝑟𝑓𝑎𝑐𝑒-𝐼𝑛/𝐴𝑙 -𝐸 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 -𝐸 𝐼𝑛/𝐴𝑙 (3)
Where 𝐸 𝑠𝑢𝑟𝑓𝑎𝑐𝑒-𝐼𝑛/𝐴𝑙 and 𝐸 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 are the total energies of InAlN surface with and without In (Al) adsorption, respectively. 𝐸 𝐼𝑛/𝐴𝑙 is the energy of an isolated indium (aluminum) atom. According to the definition, a negative binding energy indicates a favorable exothermic reaction between the InAlN surface and In (Al).

III. The crystalline quality of epitaxial InAlN/GaN layers grown by metalorganic vapor phase epitaxy

Since 2001, the theoretical proposal of attaining highest quality devices [3] has led to extensive research on InAlN/GaN heterostructures. Subsequently, record performances for HEMT devices have been reported in frequencies [19] as well as in power and temperature [4] for the InAlN/GaN HEMTs, however, until now, no such system is commercially available until now. Still, the materials issue for the growth of highest quality InAlN/GaN heterostructures have not been completely solved. Of course, the three nitride compounds (AlN, GaN, InN) exhibit highly different physical properties (Table 1); the growth of their alloys as well as their heterostructures needs particular attention. This has been more successful for InGaN and AlGaN at least for small In and Al concentrations with the production of light emitting diodes and lasers [20]. However, in the case of InAlN, even at the lattice match point to GaN (In~18-19%), increasing the layer thickness leads to crystalline degradation as has been reported [10][11][12]. As can be seen in figure 1, which is a 2x2 µm AFM image, for a layer thickness of 50 nm, the V defects (black dots) density is 3.6*10 9 cm -2 . This slightly smaller than reported in the 100 nm layer grown of a free standing GaN of reference 12, where their density was 4.5*10 9 cm -2 , which was 100 higher than the density of the threading dislocation present inside substrate. In our sample as shown by the arrows, many of the V-defects have already started to coalesce and form trenches. The connection of the V-defects to the threading dislocations is better revealed by TEM, and as can be seen in figure 2, which exhibits weak beam images of a InAlN layer of 336 nm nominal thickness, there are 2 edge, 1 mixed and 1 screw type threading dislocations from the GaN template [21]. From this figure, it can be seen, that in this sample, there are, neither basal [22] nor prismatic stacking faults [23], which would be in contrast in figure 2b. We also eliminate the presence of inversion domains [24], which should be visible in figures 2b and 2c. Obviously, there are no misfit dislocations at the AlInN/GaN interface as they also should be in contrast in figures 2a-c depending on their Burgers vectors. As clearly visible, all the dislocations that cross the interface lead to the formation of V defects at the layer surface (see arrows and V). In this projected area of the surface, at least 30 V defects are present, meaning that the 3 present dislocations contribute only for 10% to their total density. Therefore, the largest majority of the V defects formed inside the AlInN layer in agreement with the AFM observations of reference 12. As can be seen, the underlying GaN template has a uniform contrast; on top, the ternary layer images exhibit multiple changes in brightness, which correspond to local deformation and disorder.

Clearly, the deviations are highest toward the surface of the layer. They start randomly along the

[0001] growth direction and can even start at the interface with GaN. Therefore, this AFM and conventional transmission electron microscopy analysis shows that the spontaneous crystalline degradation of the layers appears to be dominated by a systematic formation of V defects which takes place randomly along the growth direction. This is clearly not due to the coalescence of growth hillock as was previously assumed. It is therefore most interesting to search for other possible mechanisms that could lead to their formation, and this is the main reason to undertake the following theoretical investigation.

IV. Defective surface and stability

We considered the point defects in the {0001} InAlN (N-and Metal-polar) surface area. They include Al (𝑉 𝐴𝑙 ), In (𝑉 𝐼𝑛 ), and N (𝑉 𝑁 ) single vacancies, AlN, InN double vacancies along the c-axis (𝑉 𝐴𝑙𝑁 , 𝑉 𝐼𝑛𝑁 ), as well as vacancy complexes involving the neighbor atoms i.e (𝑉 𝑁𝑀𝑒𝑡𝑎𝑙3 , 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 ), and

(𝑉 𝑁𝑀𝑒𝑡𝑎𝑙4 , 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁4 ). In this way, 10 initial configurations have been generated and relaxed using abinitio calculations, the corresponding most stable ones are reported in figure 3 for the N-polar (0001 ̅ ) case, of course, we have the same defects at the Metal-polar surface. Considering the most stable surface defects as seen in figure 4 (𝑉 𝑁𝑀𝑒𝑡𝑎𝑙3 , 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 for metal-polar and 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁4 for N-polar surface), we now focus on how they can influence the adsorption and diffusion of In and Al on InAlN surfaces.

V. Adsorption of In/Al atom on InAlN surface

At the Metal-and N-polar (0001) surfaces, we considered In and Al above-surface sites: i) a top site above a N atom (𝑇 𝑁 ), ii) above a metal atom (𝑇 𝑚𝑒𝑡𝑎𝑙 = 𝑇 𝐴𝑙 𝑜𝑟 𝑇 𝐼𝑛 ), iii) bridge site above the middle of Al-N bond (B), and iv) the hollow site in the hexagonal center (A), as shown in figure 5. The calculated adsorption energies of In and Al adatoms on N-polar and Metal-polar InAlN surface are shown in Table 2. As can be seen, the In and Al atom in site A is more stable than in the other sites for the N-polar. For the metal-polar, the 𝑇 𝑁 site is the most stable with an energy of -3.12 and -3.96 eV for In and Al adatom, respectively. All the possible positions for the In (Al) adsorption around the point defects were considered.

The absolute values of the adsorption energies of the In (Al) on the defective InAlN surface are increased (decrease) compared with those on the perfect one for the N-polar. This is also the case when there are more than one metal-vacancy (NMetal3-vacancy) for metal-polar surface. For example, the adsorption energy for the In (Al) on the InAlN surface with 𝑉 𝑁𝑀𝑒𝑡𝑎𝑙4 is -5.87 eV (-5.05 eV); this increases by ~0.78 eV (decreases by ~3.25 eV) compared with the -5.09 eV (-8.30 eV) for the In (Al) adatom on a perfect InAlN surface in the case of N-polar, and decrease ~0.11 eV (increase ~1.67 eV) from -3.23 eV (-2.29 eV) with 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 to -3.12 eV (-3.96 eV) for the perfect surface (metal-polar).

Therefore, the adsorption of In atom is more favorable than that of Al on these InAlN defective surfaces. In the presence of the 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁4 , the most stable site for the In and Al adsorption is the A site around the vacancies (similar to the perfect surface) for N-polar surface. The In and Al adatom prefers to be adsorbed on the T site above the N vacancy when the 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 or 𝑉 𝑁𝑀𝑒𝑡𝑎𝑙3 is present, similar to when the In atom is adsorbed on the perfect Metal-polar surface.

After discussing the defects stability and adsorption of In/Al atom on (0001) and ( 0001 As was recently shown, the bond is rather ionic in AlN, whereas it is partially covalent in InN [16]. On the defective InAlN surface, the In adatom (In a in fig. 6b), partially transfers its valence electron (1 e) to the metallic atoms (Al, In s ) located around the vacancies in the first and second layers (the atoms have been underlined with a yellow cross in fig. 6b), the In 2 gains ~0.5 e whereas Al atoms gains on average ~0.3 e. The In adatom becomes positively charged, whereas the defect acquires a negative charge, therefore, this coulomb charging reinforces the adsorption and the possible next step to formation of bonds at the surface.

eV (perfect surface) to 0.2 eV, in the case of In, as can be seen from the energy curves in fig. 7(e). In the case of Al, we still end up with a complex process with two different barriers along the diffusion path, first as 1.4eV and then 1.7 eV.

From these results, we can conclude that the point defects may enhance the diffusion of In, obviously, this is not true for Al atom for which the energy barriers are always high, probably because of its large adsorption energy.

VII. Towards In enrichment

The remaining question for the building up of aggregates is to determine if the In atoms on the surface will be able to naturally diffuse towards the defects. To this end, we moved one, two and three In atoms on the surface from their random position, and made them substitute the Al atoms around the defect. Then, we calculated the formation energy for each case (three cases in all) in order to find out if these three cases are more stable. As can be seen in fig. 8, the formation energies are lowest when the In atoms are brought to settle around the defects, the surface energy decreases more and more with the number of In atoms around the vacancies. The InAlN surface become more stable in the case of three In atoms represented by the light blue, light red and light purple in the figure for MetaN4vacancy (N-polar), MetalN3-vacancy and NMetal3-vacancy (Metal-polar), respectively. We can therefore conclude that In-rich regions may form due to the presence of these surface defects. The same procedure has been carried out in the bulk part of the InAlN alloy; in this case, the energy evolution of the supercell has been followed when we approach an In atom towards a vacancy site, as

shown the figure 9 one In atom has moved according to two paths 1-3 and 2-3. In this case, as well, the energy of the system decreases when the In atom diffuses towards the vacancy by 1.27 eV and 1.24 eV for path 1-3 and 2-3, respectively.

VIII. Discussion

From the AFM and TEM investigations it is now clear that the crystalline degradation of lattice matched InAlN layers on GaN is governed by the generation of V-defects and their rapid multiplication when the layer thickness is increased. On top of low dislocation (4•10 7 cm -2 ) FS GaN substrates the V-defects density reaches two orders of magnitude when the layer thickness was 100 nm [12]. Obviously, from the TEM observations in figure 2, the degradation is independent of the threading dislocations [11]; it takes place randomly all along the InAlN layer thickness. Therefore in contrast to previous reports [12] there is no critical thickness where it is initiated. This is probably different to what has been reported as a critical thickness from which some phase separation has been deduced using Rutherford Backscattering analysis [10], which probably corresponds to some level of thickness where it becomes detectable. Moreover, the AFM clearly show the 3D growth of the InAlN with the formation of hillock, however, the V-shape whose increases steadily with the layer thickness form exactly at the top of the hillocks, and not at their coalescence, so the possible phase separation may not be directly related to the inclined lattice planes of the coalescing growth fronts [12]. Our present work attempts to take into account the intrinsic properties of the various atoms that are interacting during the growth process. Our recent analysis of the InN-AlN phase diagram [6] revealed that even at the lattice match to GaN, the growth of this compound takes place in phase separation zone. For instance, in ultrathin In 0.72 Al 0.28 N layers deposited on AlN [27], and thus first stabilized by the large compressive strain [6], a subsequent thermal annealing leads to a random decomposition of the alloys by formation of 2-3 nanometer size domains which grow larger with annealing temperature and time [27]. This is in agreement with our recent work on the possible role of the nitrogen vacancies in the formation of indium rich areas inside InAlN, as well as the TEM observations which reported cluster of similar size [28]. The above results which show that the {0001} InAlN growing surface should be more stable when it contains defects bring a clear new insight on these observations. The complex defects appear to be the most energetically stable, they are AlN3-, NAl3-vacancy for metalpolar and AlN-4 vacancy for N-polar surfaces, respectively. The above surface diffusion analysis of In/Al adatoms shows that the In atoms should diffuse easier than the Al atom on the defective surface, the diffusion barrier decrease (increase) in case of In (Al) adatom and the adsorption energy for Al is the highest. Moreover, substituting the Al atoms initially located around the vacancy by In atoms minimizes the energy of the system. Therefore, during the growth, the formation of indium clusters around these defects which should be randomly distributed could be a mechanism that predominates in the observed phase separation.

The above observations may also be used to explain the surface morphology of the InAlN layers, as to the presence of the V-defects. According to Chen et al., threading dislocations are considered to be at the origin of V-defects. Other possibilities for the origin of V-defects have been proposed. The Vdefect density in GaN increases dramatically with an increase in the concentration of impurities in InAlN layers [29]. Based on our observation and on the above results, the V-defects are not predominantly due to threading dislocations or to other structural defects such as a stacking mismatch boundary or a stacking fault [28]. If we consider closely the investigated surface defects, they may also be seen, on a growing InAlN surface, as the nuclei of V-shaped defects. For instance, if we take 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 defect, which is the most stable on a metal-polar InAlN surface (fig. 10a), it may be a starting point for the nucleation of a V-defect as follows. The In-rich region could be formed as shown

IX. Conclusion

In summary, we have carried out a theoretical investigation of the surface structure and adatom kinetics for wurtzite InAlN and their contribution on the crystallographic degradation in nearly lattice matched InAlN (In=18-19%) on (0001) GaN. In agreement with the AFM observation that there is an extensive formation of the V-defects when the layer thickness increases, as well as the TEM observations that show the crystallographic degradation, it is found that the defective InAlN surface is energetically more stable than the perfect one. Moreover, the adsorption energies of In on the defective InAlN (MetalN4-vacancy for N-polar and NMetal3-vacancy for Metal-polar) surface increase compared to those on the perfect one. This means that the In will attach more to such surface which eventually lead to phase separation with In-rich and In-poor regions in the form of initial V-defects.

The presence of vacancies is shown to favor the diffusion of In adatoms towards it as the corresponding energy barrier and the formation energy are decreased. At the growth surface, the degradation started at the stable defect nuclei terminates in the form of V defects, which characterize the final morphology. These findings not only provide insight into the degradation mechanism of lattice matched InAlN to the substrate GaN, but may also be helpful towards the understanding of the growth of In-containing III-nitrides.

  ) and (0001 ̅ ) InAlN was employed to model the point defects and the In/Al adsorption. For calculations involving the metal-polar surface, pseudo hydrogens of charge 0.75 e passivated the N-terminated, N-polar face, while for those involving the N-polar face, pseudo hydrogens of charge 1.25 e passivated the metal-terminated, Al-polar surface. The calculations were made using 4 × 4 × 1 k-point mesh following the work of Monkhorst and Pack [17]. The optimization of the number of InAlN layers and thickness of the vacuum region has shown that 8 layers with 200

Figure 1 .

 1 Figure 1. InAlN/GaN layer grown by MOVPE 50 nm thickness, coalescing V-defects are shown by arrows.

Figure 2 .

 2 Figure 2. Weak beam images of a 336 nm nominal thickness InAlN/GaN layer of 19% indium composition, a) g=112 ̅ 0 image showing a (e) and a+c (m) dislocations, b) g=101 ̅ 0 image which underlines possible stacking faults, c) g=0002 image in which a+c (m) and c (s) are visible

Figure 3 .

 3 Figure 3. The atomistic configurations of the investigated defects shown in the case of the (0001 ̅ ) InAlN surface.

Figure 4 .

 4 Figure 4. The formation energies of the point defects versus aluminum and indium chemical potentials for (a) N-polar and (b) Metal-polar, the corresponding perfect surfaces were taken as references.

Figure 5 .

 5 Figure 5. Schematic top view of the (0001 ̅ ) AlN surface, the blue atoms are Al, and the grey nitrogen, the four initial positions of the Al (In) adatom can be seen (T N(metal) above a nitrogen (metal) atom, A along the 6 3 of the wurtzite structure, and B lies above the saddle point between N-metal).

  ̅ ) InAlN surfaces, the charge transfer at the stable surface defect was investigated by means of Bader charge analysis. For the perfect surface (fig. 6a) with the adsorption of one In atom, the neighbouring nitrogen atoms display 7 e on average which is a gain of 2 e in comparison to the isolated N atom (5 e); the adsorbed atom, In (Al) atom loses 1.5 e (2.3 e). Even with the high electronegativity of N (3.04), which is larger than that of Al (1.61) or In (1.78), there is a clear difference in the behavior of the In-N and Al-N bounds, aluminum nitride is the most stable of the three nitride, and indium nitride is the least stable [25].

Figure 6 .

 6 Figure 6. Bader charge analysis of individual In, Al, N atoms for InAlN N-polar perfect surface (a) and with metalN4vacancy (b).

Figure 7 .

 7 Figure 7. Diffusion of In in perfect In(18%)AlN and with defects. Energy curves for In (red curve) and Al (blue curve) ions diffusion on (a) perfect InAlN surface (metal-polar) and the one with (b) MetalN3-vacancy (metal-polar) (c) NMetal3vacancy(metal-polar), (d) perfect N-polar of InAlN surface, (e) MetalN4-vacancy (N-polar). The corresponding diffusion path (green) is inserted in each case (Al: blue ball, N: gray ball and In: red ball).

Figure 8 .

 8 Figure 8. The formation energies of the most stable defects as a function of aluminum and indium chemical potentials for (a) MetalN4-vacancy (N-polar), (b) MetalN3-vacancy (Metal-polar) and (c) NMetal3-vacancy (Metal-polar) versus number (0-3) of In atoms around the defects. For all the cases, the surface with three In atoms around the vacancy sites exhibits the lowest energy.

Figure 9 .

 9 Figure 9. InAlN structure viewed along [0001], the V N position is shown by the black asterisk *, the numbers 1, 2 and 3 indicate the indium position.

in fig 10

 10 (b) at the V-defect, as the In atoms diffuse easier than Al atoms to the vacancies. The diffusion barrier decreases in case of defective InAlN surface for In, therefore the surface becomes more stable when more In indium atoms replace Al ones around the defects sites. In subsequent layers during the growth, the V-pit forms with the initial concave and convex facets as has been proposed in reference 12. Subsequently, due to the lower growth rate perpendicular to {10-11} surfaces (vs) than along vertical axis (vc), the V-shape defect increases in size with the growth progress, as shown in fig.10(b). With the In segregation at the beginning of V-defects, an In-poor or In-rich InAlN areas may form in agreement with the experimental observation[11,12].

Figure 10 .

 10 Figure 10. Schematic diagram of the growth on InAlN defective surface (a) The early stage of the surface with 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 defect (b) The final stage with a large V-defect formed at the In-rich region. The blue, red and silver balls represent Al, In and N atoms, respectively.

Table 1 .

 1 Relevant properties of III-N compounds.

	Property/compound	AlN	GaN	InN
	Thermal expansion coefficient 300 𝐾 (× 10 -6 𝐾 -1 )	𝑎 ∶ 4.20 𝑐 ∶ 5.30	𝑎 ∶ 5.59 𝑐 ∶ 3.17	𝑎 ∶ 5.70 𝑐 ∶ 3.70
	Band gap (𝑒𝑉)	6.20	3.39	0.67
	Lattice parameter (Å)	𝑎 = 3.11 𝑐 = 4.97	𝑎 = 3.18 𝑐 = 5.18	𝑎 = 3.54 𝑐 = 5.71
	Lattice mismatch, 𝑎, related to InN (%)	13.71	11.32	0.00
	Growth temperature (°C)	1400	1000	500
	Equilibrium bond length (Å)	1.89	1.94	2.15

Table 2 .

 2 Calculated adsorption energies for In and Al adatoms on the N-polar and Metal-polar InAlN surface

			Adsorption energy (eV)
		N-polar		Metal-polar
	Site	In adatom	Al adatom	In adatom	Al adatom
	A	-5.09	-8.30	-2.59	-3.21
	𝑇 𝐼𝑛	-5.00	-7.67	-2.47	-3.29
	𝑇 𝑁	-4.99	-7.48	-3.12	-3.96
	𝑇 𝐴𝑙	-4.34	-7.48	-2.66	-3.29
	B	-4.22	-7.01	-2.12	-2.24

Table 3 .

 3 Adsorption energies for In and Al adatoms on the most stable InAlN surface defects in the two polarity.

		Adsorption energy (eV)
	Defect	In adatom	Al adatom
	𝑉 𝑁𝑀𝑒𝑡𝑎𝑙4 (N-polar)	-5.87	-5.05
	𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 (Metal-polar)	-3.23	-2.29

Acknowledgments

This work has been carried out through regional Normandie project RIN2018 PLACANANO convention N°: 18E01651. The calculations were carried out in the "Centre Régional Informatique et d'Applications Numériques de Normandie 'CRIANN'" (www.criann.fr) under Project No. 2016009.

VI. Diffusion of In/Al adatom on InAlN surface

The diffusion of atoms on a surface is one of the key parameters controlling growth rate, material quality, and surface morphology during epitaxial growth. However, monitoring and analyzing the surface diffusion processes are relatively difficult using experimental methods [26], hence the theoretical investigation of surface diffusion pathways and diffusion barriers may provide important information relevant to the growth of InAlN layers.

In this section, we discuss the role of the most stable defects on the diffusion of In and Al, as well as on the possible aggregation of indium. As seen above, they are, for the metal-polar InAlN surface, 𝑉 𝑁𝑀𝑒𝑡𝑎𝑙3 𝑎𝑛𝑑 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 in In-, N-rich and Al-rich condition, respectively. Indeed, the 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁4 is the most stable for the N-polar InAlN surface. The diffusion pathway of the In (Al) atom was determined by moving the In (Al) atom with a small constant distance on a perfect metal polarity InAlN surface from T N to the next and the total energy was recorded at each step. In the process, the atom was constrained in the direction along the path, and was free to move in the perpendicular directions, which enables it to find the minimum energy position. As shown in insert of fig. 7 For the Al atom, the process is more complex, the migration pathway requires the Al to first go from 𝑇 𝑁 to A site with a diffusion barrier of 1.05 eV (𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 ) and 0.75 eV (𝑉 𝑁𝑀𝑒𝑡𝑎𝑙3 ), respectively. Subsequently, it should diffuse from A to the next 𝑇 𝑁 position (close to the defect) with a barrier of 0.2 eV and 1 eV for 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁3 𝑎𝑛𝑑 𝑉 𝑁𝑀𝑒𝑡𝑎𝑙3 , respectively.

On the perfect (0001 ̅ ) InAlN surface (N-polar), the stable adsorption site for In (Al) adatom is site A (fig. 5d). As shown in the insert, the diffusion of In(Al) goes from site A to the next across 𝑇 𝑁 , and the calculated diffusion barrier is 0.942 eV and 1.936 eV for In and Al atom, respectively. Therefore, for the two adatoms types, the diffusion barrier is larger on N-polar (0001 ̅ ) surface because, as seen above (Table 2], the adsorption energies are higher on this surface. In this case, the corresponding most stable 𝑉 𝑀𝑒𝑡𝑎𝑙𝑁4 defect induces a quite large decrease of energy barrier, from 0.942