Mathias Rapacioli

Fernand Spiegelman

Nathalie Tarrat

Evidencing the relationship between isomer spectra and melting : the 20-and 55-atom silver and gold clusters cases †

Evidencing the relationship between isomer spectra and melting: the 20-and 55-atom silver and gold cluster cases

Introduction

One of the originalities of clusters, as finite systems, is their melting behaviour. Early studies evidenced the size evolution of the solid to liquid melting and showed that the melting temperature decreases with size in the nanoparticle regime 1,2 . Moreover, while in macroscopic bulk, transitions are sharp and occur at a well defined temperature, phase transitions in finite systems occur over a finite range of temperatures. These mechanisms of phase transitions in clusters were investigated some years ago by several authors [3][4][START_REF] Wales | Energy Landscapes: Applications to Clusters, Biomolecules and Glasses[END_REF] , who demonstrated the inequivalence of the thermodynamical ensembles for finite systems, in analogy with other finite systems such as nuclei and the major role of phase coexistence at the transition. A number of these pioneering studies were conducted using model Lennard-Jones potentials 4,[START_REF] Weerasinghe | [END_REF]7 . In the case of metal clusters, several experimental [8][9][10][11][12][13][14][15][16][17] and theoretical [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] investigations were achieved to evidence the specific features of finite size melting with a description of the interatomic forces more realistic than the simple Lennard-Jones potentials.

a Laboratoire de Chimie et Physique Quantiques LCPQ/IRSAMC, UMR5626, Université de Toulouse (UPS) and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France. Tel: 33 5 6155 8318; E-mail: mathias.rapacioli@irsamc.ups-tlse.fr b CEMES, Université de Toulouse, CNRS, 29, rue Jeanne Marvig, 31055 Toulouse, France. Tel: 33 5 6752 4347; E-mail: nathalie.tarrat@cemes.fr † Electronic Supplementary Information (ESI) available: [Details on the parameterization of the DFTB method for gold and silver]. See DOI: 00.0000/00000000.

While a continuous behaviour of their properties can be expected for relatively large sizes and in the nanoparticle size domain (N>100-1000 depending on the species), a discrete evolution is exhibited for small sizes down to the microcluster domain. Such finite size effects clearly affects the structure and energetics of clusters with the occurrence of the so-called magic numbers. It also affects the thermodynamical properties 28,29,[34][35][36][37][38] . The pioneering experimental study on small sodium clusters in the range N=20-147, reported by Haberland et al. 8 , showed particular large melting temperature for specific size around N=55 and N=147, and strong finite size effects were also observed on the latent heats. The underlying mechanisms of phase transitions were unraveled by a number of authors 12,24,25,27,30,31,34,36,[38][39][START_REF] Adekoya | Band Structure, Morphology, Functionality, and Size-Dependent Properties of Metal Nanoparticles[END_REF][START_REF] Alarifi | [END_REF] , detailing the melting process in finite systems to initial isomerisation and finally melting as the number of accessible isomers at a given energy/temperature increases yielding a fluctuating state of the system. Thus, the density of isomers becomes a key property in describing melting at the microscopic level. While the equilibrium geometries in themselves are not supposed to influence melting, the potentials energy landscape is obviously determinant. Nevertheless, the gaps between the lowest energy equilibrium geometry and the lowest isomers can be related to the particular stability of the former, e.g its magic character. It has been shown in a number of simulations that clusters with a magic character often present a large gap to isomerisation, and therefore a higher melting temperature with also a sharper peak in the caloric curve than its immediate non-magic neighbors 8,42 . Obviously, the magic character of a given size can stem either from geometric stability or electronic stability, as it is the case in metal clusters. Both types of stabilities can also interfere and even combine, like in the specific clusters Au 20 (see 32 and references therein) or Al 32 particularly stable and exhibiting sharp caloric curves 17,31 . Not only size but also charge can influence the caloric properties of relatively small metal clusters 17,31,32 , since charge changes the filling of the electronic shells. We will hereafter focus on gold and silver clusters. Their structural and melting properties have been investigated using various experimental setups 34,[START_REF] Adekoya | Band Structure, Morphology, Functionality, and Size-Dependent Properties of Metal Nanoparticles[END_REF]43,44 and theoretical approaches 19,20,[24][25][26][27][28][28][29][30]33,35,36,[START_REF] Alarifi | [END_REF][45][46][47][48][49] , mostly with empirical potentials. The most stable structures reported for different cluster sizes exhibit either very regular shape (e.g. Au 20 or Ag 55 ) or amorphous patterns 50 . As an example, it has been reported that Au 55 could present cavities below its outer shell 33,48,51 and a competition between ordered and disordered geometries has been evidenced in the case of Au 147 33,52 . Beyond the search for the most stable isomer, studies have been conducted to identify structural excited isomers 26,45,47,51 . In small clusters, a premelting involving transition towards these secondary minima has been reported [24][25][26]35,46,49 . For large clusters, another possible premelting mechanism has been suggested in the literature involving a melting of the cluster shell prior to the the cluster core 19,36,[START_REF] Alarifi | [END_REF] .

The melting temperature was reported to increase with the metal nanoparticle size for large clusters 27,34,43 , confirming the pioneer observation of Buffat and Borel 2 . For small clusters, the melting temperature evolution with size is non-monotonous 36 and the addition or removal of an atom can lead to drastic variation of the melting temperature [28][29][30]45 . Moreover, for such small sizes, the melting temperature may also depend on the charge state 32 . The present work is devoted to a detailed comparative study of melting between isovalent noble metal gold and silver clusters for the intermediate sizes 20 and 55, which can be considered as the low limit of the transition between clusters and nanoparticles. It extends a previous work restricted to the influence of charge on the melting of Au 20 32 . While isovalent, silver and gold exhibit quite different properties due to the strong relativistic influence in the electronic structure of gold 53,54 . The size choice was made because because N=20 is a double magic number for gold (geometrically a symmetric pyramid associated with a 20 electron closing of its outer electronic shells 55 ), while N=55 is not magic for gold. In contrast, Ag 55 was found in many theoretical studies as an icosahedron with two complete geometrical shells. Note that as far as we are aware of, the only investigations of the caloric curves for gold and silver cluster clusters with size N=55 were done in the framework of EAM potentials 26,56 . We analyze the different thermodynamical behavior of the silver and gold clusters, based on the correlation between the density of isomers, the isomerization gaps and the characteristic of the solid-to-liquid transition. The influence of the charge of the cluster on the density of isomers for such intermediate sizes is also illustrated. The multiple histogram method to obtain the melting curves from DFTB simulations involving Parallel Tempering Molecular Dynamics is now standard and briefly summarized in the section 2. Section 3 presents the caloric curves for Au

( ,0,+) 20 , Ag ( ,0,+) 20
, Au

( ,0,+) 55

and Ag

( ,0,+) 55

and the discussion about the underlying microscopic aspects of the transitions. Finally a conclusion is given in section 4.

Computational details

In the present work, we have used the Density Functional based Tight Binding (DFTB) scheme 57,58 , in its second order formulation (also known as Self-Consistent Charge DFTB 59 ), to compute the potential energy surfaces for neutral, anionic and cationic gold and silver clusters. In a previous work 60,61 , we adapted (see supplementary materials for more details) existing gold 62 and silver parameters 63 and validated their use to properly model these metals from cluster to bulk sizes. The exploration of the PES has been done using a Parallel-Tempering Molecular Dynamics (PTMD) scheme 64 , as implemented in the deMonNano code 65,66 . In this scheme, several trajectories are run in parallel in the canonical ensemble at different temperatures. In practice, 60 trajectories were computed, corresponding to 60 temperatures following an exponential distribution between 50 K and 2000 K. The trajectory time lengths were progressively increased in order to ensure convergence of the heat capacity curves (between 1 and 6 ns). The timestep was 1.5 fs and the temperature was controlled by a Nose-Hoover chain 67,68 of 5 thermostats coupled to the system through a frequency of 80 cm 1 . In order to increase ergodicity, exchanges of temperatures between the different trajectories were attempted each 1.2 ps using a metropolis criterion. Although heat capacities can in principle be build from the potential energy distribution obtained for the different simulated temperatures, we have used the multiple histogram approach, as formulated by Labastie and Whetten 4 which reduces the statistical noise and extrapolates heat capacity values for temperatures not included in the PTMD simulation.

Results and discussion

The structural aspects for neutral, cationic and anionic Ag 20 , Au 20 , Ag 55 and Au 55 within the DFTB approach were discussed in details in a previous work 51 from global exploration (PTMD complemented by quenching). The lowest energy isomers for Au 20 , Au + 20 and Au 20 were found to be very symmetric structures, namely a T d pyramid fort the neutral cluster and a T d pyramid with small Jahn-Teller deviations for the anion, in agreement with previous experimental and theoretical works [69][70][71] . In contrast, the lowest energy structures of Ag 20 and its ions are essentially based on pentagonal arrangements and have lower symmetry, namely C 3 for the neutral and the cation, C s for the anion. For size N=55, the results are interchanged. The lowest energy structures of Ag 55 neutral and ions all result from small Jahn-Teller distorsions of the very symmetric (I h ) double icosahedron (D 2h , D 5d and D 3d for the neutral, the cation and the anion respectively), while the low energy isomers of Au ( ,0,+) 55 have more disordered shapes with cavities inside the cluster. Figure 1 only displays the lowest energy equilibrium structures of the neutral clusters. The equilibrium structures of the charged clusters presenting only small deformations of the neutral ones, the details would not be discernible at the figure scale. For more information, we refer the reader to our previous publication 51 . Note that the presence of cavities in Au 55 gold clusters has been recently confirmed by two other DFT studies 33,48 . In itself, the fact that the structures are symmetric or disordered is not directly influential on the thermodynamical properties which result from averages at a given temperature. However, very symmetric structures often correspond to particularly stable equilibrium structures and may therefore be associated with larger structural excitation energies and higher melting temperatures than the disordered ones. The variability of the isomer spectrum is thus determinant on the thermodynamic properties 21,26,45 and will be analyzed in the discussion.

The heat capacity curves as a function of the canonical temperature are shown in figure 2 in the temperature range of the solid-liquid transition. The case of Au 20 was already discussed with respect to charge effects 32 and is given here for comparison. The first observations concern the temperature widths of the transitions and the heights of the associated peaks. In the case of neutral Au 20 and Ag 55 , the transition is rather well defined, with peak heights around C/N = 8 and 6 k B respectively and with half-widths around 300 K and 180 K respectively (taken as the widths at the peaks half heights). The situation is quite different in the case of Ag 20 which exhibits an extremely smooth and flat transition and in the case of Au 55 for which the width of the transition extends almost over the whole temperature domain investigated with a roughly linear increase, up to the well defined peak maximum. The characteristic temperatures of the heat capacity curves are given in table 1. The transition temperatures T melt are taken as the peak maxima. We also give the inflexion temperatures T in f lex , which are usually interpreted as the starting melting temperatures. However, in some cases, e.g. Ag 20 and Au 55 , the inflexion points of the caloric curves are uncertain, and we have alternatively defined the starting temperature T start as the left temperature at half width. The melting temperatures T melt for neutral Ag 20 , Ag 55 , Au 20 and Au 55 , are 581 K, 763 K, 1101 K and 1154 K respectively. One can observe that while the addition or removal of a single electron may quantitatively change the position of the heat capacity peaks (see for instance Au 20 and its ions), it does not significantly change the qualitative aspects of the heat capacity curves. .

T start T in f lex T melt
In a previous references 32,51 , the structural excitation energies of all isomers found below 1 eV within the PTMD /quenching exploration scheme were reported. Important energy gaps between the lowest energy isomers and the first excited ones were found in the case of the symmetric clusters, namely Au 20 and Ag 55 (0.64 eV for neutral Au 20 and 0.51 eV for Ag 55 for instance). Conversely, in the less ordered systems Ag 20 and Au 55 , small gaps were found (0.12 eV for Ag 20 , almost no gap in the case of Au 55 ). This fingerprints could still be observed in the ions, with finite well established gaps (somewhat reduced) in Au . More thorough insight can be reached from the features of the global isomer spectra given in figure 3. One can observe that the sharpest transitions, corresponding to the most ordered clusters Au 20 and Ag 55 are related to cases where there is a significant isomer gap between the lowest energy structure and the higher ones. Let us note that in the case of Au 20 , the isomer spectrum shows several isomers lying at extremely low energies close to that of the best structure, which are isomers corresponding to slight Jahn-Teller distortions of the symmetric T d structure. Since the present work does not include quantum effect nor vibrational zero point energy effects, we do not discuss those isomers which moreover do not bring any visible features in the heat capacity curve at low temperature. As mentioned above, the gaps are particularly large for Au 20 and Ag 55 , and the isomer spectrum keeps a discrete nature up to 1 eV. This is at strong contrast with Ag 20 for which the isomer spectrum still exhibits lines corresponding to various isomers, however lying close-by. In Au 55 , the isomer spectrum is so congested that it appears as a quasi continuum above 0.15 eV. These high densities of low-energy isomers are associated with flat or very broad transition features. Several oscillations appear in the raise of the caloric curve. We checked that the visited structures do not include a few specific isomers only but a large number of noticeably different amorphous structures, consistently with the almost continuous isomer spectrum. It is thus not possible to assign those features to uniquely defined isomerization. We now consider the ions. In the case of Au 20 and Au + 20 , the gaps are reduced to 0.35 eV and 0.27 eV respectively, and a larger number of isomers appear in the range 0.3-1 eV. This provides an explanation for a broadening of the transition peak and a shift of the melting to lower temperatures, namely 822 and 867 K respectively. Ag 55 is a larger cluster in which the addition or the removal of an electron is less influential. Its isomer spectrum is not significantly changed, characterized by first structural excitation gaps of 0.45, 0.46 and 0.46 eV for the neutral, the anion and the cation respectively. The isomer gap in Ag + 20 is about the same as that of the neutral, while its isomer spectrum becomes strongly congested in the range 0.18-0.38 eV, which may explain the almost flat aspect of its heat capacity curve, the melting temperature indicated in table 1 being to be taken with extreme caution. In Ag 20 , the first structural energy gap is strongly reduced and a very smooth heat capacity is obtained with two shoulders. The isomer spectrum of Au 55 is even more congested than the spectrum of the corresponding neutral cluster. Concerning Au + 55 , a lower density of isomers is observed between 0.1 and 0.3 eV. Finally, an important observation is that, while in Au 20 and Ag 20 the effect of a single charge on the melting temperature may result in shifts in the range 200-300K, this effect almost vanishes for the largest clusters Au 55 and Ag 55 where the shifts are only in the range 30-50 K.

In order to get further insight in the structural order changes along the melting process, we have also analyzed the cluster structures by computing the averaged Radial Distribution functions (RDF) at a given temperature defined as

g 0 (r, T ) =< 1 N N Â i 1 p 2ps e (r 0i r) 2 2s 2 > T (1)
where index 0 stands for the center of mass of the cluster and i a current atom. The average concerns all the cluster configurations at a given temperature. The gaussian smoothing was used (s =0.05 Å) to reduce the noise, especially at low temperature. Temperature-dependent RDFs are reported in Fig. 4 for Ag ( ,0,+) 20 and Au

( ,0,+) 20
, and in Fig. 5 for Ag ( ,0,+) 55 and Au ( ,0,+) 55

. On these figures, the RDF at the melting temperature T melt is depicted as solid black lines and the one at T start as dashed black lines. Clearly, due to their high symmetry, the RDF of Au 20 and Ag 55 at low temperature both have only a few characteristic and rather narrow peaks, corresponding respectively to the pyramidal structure and to the icosahedron. Obviously when the temperature increases, the atomic fluctuations induce a broadening of the RDF peaks. One can observe on the Au 20 spectrum that the evolution of the RDF between T start and T melt is quite significant. At T start , the peak structure in the RDF is still clearly identifiable and still associated to a given structure perturbed by vibrational fluctuations. Alternatively, at the meting temperature, even though the RDF is not flat, the peaks at 1.8 and 4.8 Å have almost disappeared and the RDF has almost completely lost its structuration. A similar situation can be observed in Ag 55 , except that the two-shell structure seems to be still preserved within the two-maxima feature of the RDF around 2.8 and 4.9 Å. Let us note that, above the melting temperature, some distances are now recorded between the two shells in the range 3-4 Å, fingerprinting exchanges between the shells. Again, the situation is significantly different in the case of Ag 20 and Au 55 . In Ag 20 , the peak structure is still slightly visible, however considerably smeared. In Au 55 , the RDF at T start has lost its initial structure, the two peaks of the RDF at 4.8 and 5.4 Å being merged. Moreover, there is no strong difference between the RDFs at T start and T melt , which is clearly related to the temperature width and the smoothness of transition for these clusters. The same differential behaviour of the RDF is seen for the ions Au on the other side (smooth transition). From the above analysis, it appears that the melting temperature can be inferred from the RDF evolution in the case of ordered clusters such as Au 20 and Ag 55 , while the continuous evolution of the RDF in disordered cases prevents such determination.

A number of previous theoretical studies have determined the caloric curve of gold and silver clusters (mainly neutrals), either for small sizes or in the nanoparticle regime up to a few thousands of atoms 19,20,[24][25][26][27][28][28][29][30]33,35,36,[START_REF] Alarifi | [END_REF][45][46][47]49 . First-principles calculations were used to investigate the thermal properties of small gold clusters up to N⇡24 28,29,45,47 , with a special focus on Au 20 and its neighbours. The thermal stability of small gold cages was examined by Chandrachud et al. 45 . Most of the other studies were conducted using EAM-type potentials. In large clusters and nanoparticles, emphasis was put on the occurrence of a surface premelting 19,36,[START_REF] Alarifi | [END_REF]49 . In gold, specific studies were concerned with a solid-to-solid transformation into metastable icosahedral shape 35,46 . Mainly two studies, both using EAM-type potentials, have investigated the melting of gold and silver clusters for size N=55. Hamid et al. conducted an EAM-type study on several metal clusters emphasizing the crossover between icosahedral I h and f cc O h morphologies and determined the caloric curves 56 . For Au 55 , they found a melting temperature close to 400K, quite far from the one determined in the present study (T melt = 1154K). For reference, the bulk melting temperature of gold is 1337K [START_REF] Kaye | Tables of physical and chemical constants[END_REF] . It is to be noticed that neither the I h nor the O h structures are the lowest energy structures, neither in the EAM study of Hamid et al. nor in the present study. Finally, let us mention that the melting temperature of Au 20 in the present DFTB approach is 1101K to be compared with 770K in the DFT/LDA study of Krishnamurty et al. 29 . This was discussed in a previous article 32 as related to different isomer gaps (0.64 eV with DFTB, 0.75 with DFT/TPSS 71 and 0.44 with DFT/LDA 29 ). In the case of Ag 55 , Mottet et al. 26 found a melting at 570 K, starting from 530 K. Both this latter result and our present melting temperature (T melt = 763K) are significantly lower than that of the bulk (1232 K 73 ). In the two studies, the first isomerisation gaps are similar (⇡ 0. completely sufficient to characterize the transition. Clearly, the differences between the melting temperatures are to be related to the different potentials used and the specific isomer densities above the first gap in the two respective studies. We have addressed here some specific cases, either ordered clusters presenting a large gap between a single lowest isomer and the next one, or disordered clusters presenting almost no gap at all. Clearly, other sizes may exhibit a different energy landscape with two or more low energy isomers, where the temperature evolution would start as a structural isomerization to a hot solid prior to melting. In such case, the isomer spectrum would exhibit two or more well identified low energy lines and might induce a shoulder or a peak in the capacity curve below the main one (see for instance the case of Al + 115 38 , Ga + 80 11 and previous discussions 12,31,38,74 ). Let us mention that the identification of a second isolated isomer well separated from the higher ones does not necessarily induce any premelting feature in the caloric curve, as evidenced in the case of Au 20 in our previous work 32 .

Conclusion

We have determined the heat capacity curves for Ag 20 , Au 20 , Ag 55 , Au 55 and their ions associated with the solid-to-liquid transition. The present work highlights the direct links between the structural excitation spectra and the melting properties of small gold and silver clusters. Clusters exhibiting very symmetric low energy structures, such as Au 20 and Ag 55 and their ions, display large structural excitation energies and a discrete nature of their isomer spectra below 1 eV. They present sharp or relatively sharp solid-to-liquid transitions and rather large melting temperatures, relatively to their respective sizes. A relatively minor evolution of their structure is observed up to the starting melting temperature. Conversely, less ordered clusters, such as Ag 20 , Au 55 and their ions, have small or vanishing structural energy gaps, isomer spectra almost continuous above low energies and they undergo a much smoother transition. Moreover, the evolution of the RDF of Au 55 shows a loss of structuration significantly below the melting temperatures associated with a transition within a broad temperature range. The identification of these very different structural and thermodynamic characteristics for clusters of similar size composed of the isovalent metals gold and silver completes the picture of dissimilar properties for those two elements. The correlation between the thermodynamical behaviour and the isomer spectrum observed in the present work also gives credit to the description of the onset of disorder at phase changes in the spirit developed in the early works of Rose and Berry 75 and concerned with the topologies of the energetic and entropic high-dimensional landscapes of clusters and the role of amorphous basins vs ordered configurations. Note that, relatively to our previous work on the influence of the charge on the caloric properties of Au 20 32 , this study also shows that a single electron default or excess is much less determinant, even quantitatively, for size N=55, than for size N=20.

Fig. 1

 1 Fig. 1 Structures of the DFTB lowest-energy isomers of Ag 20 , Au 20 , Ag 55 and Au 55 .

(

Fig. 2

 2 Fig. 2 Canonical heat capacity curves per atom (C/N in k B ) for neutral (blue), cationic (black) and anionic (red) Ag 20 , Au 20 , Ag 55 and Au 55 clusters.

(

Fig. 3 20 Fig. 4 20 .

 320420 Fig. 3 Structural excitation energies of minimized structures for Ag ( ,0,+) 20 , Au ( ,0,+) 20 , Ag ( ,0,+) 55 and Au ( ,0,+) 55.

Table 1

 1 Starting melting temperatures T

	Ag 20 Ag 20 Ag + 20	359 411 215	-418 -	574 581 398
	Au 20 Au 20 Au + 20	646 978 717	691 1017 763	822 1101 867
	Ag 55 Ag 55 Ag + 55	685 672 665	730 717 685	795 763 724
	Au 55 Au 55 Au + 55	795 808 678	---	1115 1154 1160

start , inflexion point temperatures T in f lex and melting temperatures T melt in K (±3K) for Ag ( ,0,+) 20 , Au ( ,0,+) 20 , Ag ( ,0,+) 55 and Au ( ,0,+) 55

[1][2][3][4][START_REF] Wales | Energy Landscapes: Applications to Clusters, Biomolecules and Glasses[END_REF][START_REF] Weerasinghe | [END_REF][7][8][9][10]

Acknowledgements

This work was granted access to the HPC resources of CALMIP (Grants p18009, p0059 and p1303). It was supported by the CNRS-GDR EMIE and through the EUR grant NanoX n o ANR-17-EURE-0009 in the framework of the "Programme des Investissements d 0 Avenir".

. The temperature color code is indicated on the right. RDF at T start : dashed black line. RDF at T melt : solid black line.

Conflicts of interest

There are no conflicts to declare.


Mohammed Zeghouane

Geoffrey Avit

Yamina Andre

Thierry Taliercio

Pierre Ferret

Evelyne Gil

Dominique Castelluci

Pierre Disseix

Joël Leymarie

Eric Tournié

Yamina André

Joel Leymarie

Agnès Trassoudaine

Morphological Control of InN Nanorods by Selective Area Growth-Hydride Vapor-Phase Epitaxy

Keywords: Hydride Vapor Phase Epitaxy, Indium Nitride, Selective Area Growth, Nanorods

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

Introduction

The III-nitride materials are promising for applications such as light emitting diodes, laser diodes, high power transistors and photovoltaic [START_REF] Ambacher | Growth and applications of Group III-nitrides[END_REF] . In fact, the binary InN material has unique and excellent intrinsic properties with the narrowest band gap of all the III-nitride family, low effective mass and a large drift-velocity, making it an ideal candidate for infrared light emitters and high-speed electronic devices [START_REF] Bhuiyan | Indium nitride (InN): A review on growth, characterization, and properties[END_REF], [START_REF] Mohammad | Progress and prospects of group-III nitride semiconductors[END_REF], [START_REF] Mi | Extending group-III nitrides to the infrared: Recent advances in InN[END_REF], [START_REF] Lye | InN thin-film transistors fabricated on polymer sheets using pulsed sputtering deposition at room temperature[END_REF], [START_REF] Hsu | InN-based heterojunction photodetector with extended infrared response[END_REF] . However, the growth of high-quality InN layers is hampered by the high lattice mismatch with usual substrates [START_REF] Bhuiyan | Indium nitride (InN): A review on growth, characterization, and properties[END_REF], [START_REF] Syrkin | InN-based layers grown by modified HVPE[END_REF], [START_REF] Suihkonen | Growth of InN by vertical flow MOVPE[END_REF], [START_REF] Kumar | Growth of InN layers on Si (111) using ultra-thin silicon nitride buffer layer by NPA-MBE[END_REF], [START_REF] Jain | Effect of polarity on the growth of InN films by metalorganic chemical vapor deposition[END_REF], [START_REF] Wang | Selective growth of N-polar InN through an in situ AlN mask on a sapphire substrate[END_REF] . In an effort to circumvent this limitation, the 1D geometry of nanowires is used [START_REF] Zhao | Chapter Eight -InN Nanowires: Epitaxial Growth, Characterization, and Device Applications[END_REF] . The small footprint on the substrate surface allows to improve the crystal quality and reduce the dislocation density through the effective stress relaxation [START_REF] Calleja | Growth, morphology, and structural properties of group-III-nitride nanocolumns and nanodisks[END_REF] . Furthermore, the nanowire specific surface is larger than that of planar layers which is beneficial to enhance terahertz (THz) wave emission [START_REF] Ahn | Terahertz emission from vertically aligned InN nanorod arrays[END_REF] and increases light output of LED devices [START_REF] Durand | M-Plane GaN/InAlN Multiple Quantum Wells in Core-Shell Wire Structure for UV Emission[END_REF] . The most widely used methods to grow nitride nanowires are Aucatalysed and self-catalysed growths. These growth techniques yield poor uniformity and random growth directions, while deep level defects due to the incorporation of the metal catalyst are observed in the materials [START_REF] Brandt | Sub-meV linewidth of excitonic luminescence in single GaN nanowires: Direct evidence for surface excitons[END_REF], [START_REF] Flissikowski | Suitability of Auand Self-Assisted GaAs Nanowires for Optoelectronic Applications[END_REF] . These issues can be addressed by the catalyst-free selective area growth (SAG) technique. Precise control of the size and the position of wires can be achieved by using openings on a dielectric mask defined by lithography. This approach is based on the difference of the nucleation barrier between the mask and the substrate.

Until now few studies have been reported on the SAG of InN nanorods due to challenges faced in obtaining a good selectivity with a high crystal quality. It is difficult to obtain a perfect selectivity in the usual temperature range for growing InN. To the best of our knowledge, the SAG of InN by metal organic vapour phase epitaxy (MOVPE) has never been reported. In molecular beam epitaxy (MBE), the growth of InN nanorods with a good selectivity was achieved. However, it is very difficult to control the morphological shape with a well-defined aspect ratio because of the complexity of InN growth kinetics [START_REF] Harui | Transmission electron microscopy characterization of position-controlled InN nanocolumns[END_REF], [START_REF] Kamimura | Dislocation reduction via selective-area growth of InN accompanied by lateral growth by rf-plasma-assisted molecular-beam epitaxy[END_REF], [START_REF] Liang | Selectivearea growth of indium nitride nanowires on gold-patterned Si(100) substrates[END_REF], [START_REF] Wang | Selective-Area Growth of Transferable InN Nanocolumns by Using Anodic Aluminum Oxide Nanotemplates[END_REF] . The perfect selective area growth of vertically aligned InN nanorods with a high aspect ratio was newly demonstrated thanks to the Hydride Vapour Phase Epitaxy (HVPE) process [START_REF] Zeghouane | Selective Growth of Ordered Hexagonal InN Nanorods[END_REF] . The SAG was performed on SiN masked Ga-polar GaN/c-Al2O3 templates with circular openings of 200 nm. It resulted in perfectly hexagonal InN nanorods delimited with (1-100) m-planes, (11-20) a-planes and the top (0001) c-plane. However, it was reported that the optical and electrical properties of 1D nitride structures may heavily depend on their shape [START_REF] Chang | Electrical transport properties of single GaN and InN nanowires[END_REF] . Indeed, the photoluminescence (PL) emission of InN nanorods or nanowires critically relies on their crystal quality and morphology (length-diameter-facets) [START_REF] Ahn | Photoluminescence from InN Nanorod Arrays with a Critical Size[END_REF], [START_REF] Segura-Ruiz | Optical studies of MBE-grown InN nanocolumns: Evidence of surface electron accumulation[END_REF] . That is why the control of the hierarchy of the growth rates of the different nanorod facets and thereby the control of the final shape is of utter interest.

Crystal shape engineering of nitride structures by SAG-HVPE can be carried out by varying the vapor phase composition and the growth temperature. These experimental parameters govern the kinetics involved in the growth process [START_REF] Avit | Crystal engineering by tuning the growth kinetics of GaN 3-D microstructures in SAG-HVPE[END_REF], [START_REF] Lekhal | Selective-area growth of GaN microrods on strain-induced templates by hydride vapor phase epitaxy[END_REF], [START_REF] Li | Computing Equilibrium Shapes of Wurtzite Crystals: The Example of GaN[END_REF] . Understanding how each parameter impacts the crystal growth of InN is important to monitor the form of nanorods. To the best of our knowledge, no study has ever been carried out to understand and ultimately control the SAG of InN nanorods. Therefore, it is timely to conduct detailed and systematic studies of the SAG of InN nanorods which is the aim of this work.

The effect of the growth temperature on the selectivity and the growth rates is first investigated.

Then, the impact of the partial pressure of the V element on the morphology of rods is studied, as well as the evolution of the growth rates over time. Finally, the optical quality of grown InN nanorods is assessed by photoluminescence spectroscopy.

Experiment

The SAG of InN nanorods was carried out in a 2-inch quartz reactor at atmospheric pressure placed in three-zone horizontal furnace. In the first zone, called the source zone, indium trichloride (InCl3) gaseous species were formed in a quartz boat by sublimation of an InCl3 powder conducted by a nitrogen flux (N2) between 450 °C and 500 °C. Ammonia (NH3) was directly introduced into a central zone heated at a higher temperature for an efficient mixing of the gas species. The spot at higher temperature also suppresses the parasitical nucleation upstream the sample. The third zone is the growth zone. Ultra-high purity nitrogen (N2) is used as carrier gas.

C-plane sapphire templates, covered with a 4 µm-thick Ga-polar buffer layer and masked with SiNx, were used as substrates. The pattern was defined with nano-imprint lithography (NIL) and consisted of arrays of circular holes of 200 nm spaced 2 µm apart. Nanorods morphologies, including diameters and heights were studied by Carl Zeiss Supra Scanning Electron Microscope (SEM) using an acceleration voltage of 3 kV. The optical properties were analyzed by Photoluminescence spectroscopy (PL) using an excitation wavelength of 780 nm, focused on nanorods ensemble with a spot of 200 µm. The luminescence signal was detected through a Fourier-transform infrared spectrometer using a liquid-N2-cooled InSb detector.

Results and discussion

The growth morphology of InN nanorods is sensitive to several factors, mainly the growth temperature and the V/III ratio. The growth temperature affects the crystal growth of InN material, both the crystal quality and the growth rates, while insuring at the same time a good selectivity on the mask. Consequently, we first systematically studied the effect of this key parameter by fixing all other growth conditions. The input partial pressures of NH3 (P 0 NH3) and InCl3 (P 0 InCl3) were respectively kept at 1.00 x 10 -1 atm and 1.38 x 10 -3 atm yielding to V/III ratio of about 72. These partial pressures were initially inspired by the work in Ref [22]. The growth temperature was varied from 580 °C to 690 °C in order to define the temperature window of the SAG regime. The choice of this range of temperature was imposed by the decomposition temperature of InN. On the other hand, the partial pressure of NH3 is considered as one of the most impacting parameters on InN growth due to the weak nature of the In-N bond but also to the addition of hydrogen coming from its decomposition. To understand the SAG of InN nanorods, a study as a function of P 0 NH3 was also carried out. The growth temperature was kept at 640 °C and the growth time was 5 minutes. Furthermore, several studies on SAG have shown that the grown nanostructures undergo morphological evolution from the initial to the stable shape driven by free-energy minimization [START_REF] Li | Computing Equilibrium Shapes of Wurtzite Crystals: The Example of GaN[END_REF] . To gain a further understanding of the SAG growth of InN nanorods, additional experiments were carried out by varying the growth time. The growth temperature was 640 °C, P 0 NH3 and P 0 InCl3 were respectively kept at 1.00 x 10 -1 atm and 1.38 x 10 -3 atm. For a better understanding of the impact of each growth parameter, the diameter of the mask holes and the spacing between apertures were fixed for the whole work. § Effect of the growth temperature 1 (a). When the growth temperature is higher than 610 °C, the growth of vertically aligned InN nanorods with a perfect selectivity becomes favorable. Previous reports dealing with SAG of nitrides by MBE and MOVPE, have shown that selectivity is driven by the surface diffusion of adatoms and their probability to reach the mask holes is enhanced at high growth temperatures [START_REF] Gacevic | Formation Mechanisms of GaN Nanowires Grown by Selective Area Growth Homoepitaxy[END_REF], [START_REF] Wang | Growth kinetics and mass transport mechanisms of GaN columns by selective area metal organic vapor phase epitaxy[END_REF] . The special feature of HVPE when it comes to selective epitaxy is the low adsorption of chloride growth species on the dielectric mask. The uniform diameter of the nanorods shows that growth is not supplied with ad-species on the mask. The growth of nanorods results mainly from direct condensation of species from the vapour phase. Growth is then governed by the kinetics of adsorption, desorption, dechlorination and surface diffusion on the growing crystal facets, as we discuss below. The nanorods diameter is larger than the mask hole because of the epitaxial lateral growth, which was already observed in SAG of nitride materials. [START_REF] Lundskog | Morphology Control of Hot-Wall MOCVD Selective Area Grown Hexagonal GaN Pyramids[END_REF], [START_REF] Hersee | The Controlled Growth of GaN Nanowires[END_REF], [START_REF] Bengoechea-Encabo | Understanding the selective area growth of GaN nanocolumns by MBE using Ti nanomasks[END_REF] The axial and radial growth rates of InN nanorods as a function of the growth temperature deduced from SEM images are presented in Figure 1 (f). The relative lengths of InN nanorods grown at 610 °C, 640 °C, 660 °C and 690 °C are 1.53 µm, 2.44 µm, 2.95 µm and 2.68 µm, respectively. The axial growth rate of InN nanorods gradually increases from 51 nm.mn -1 to 98 nm.mn -1 when the growth temperature increases from 610 °C to 660 °C and then decreases to 87 nm.mn -1 at 690 °C. Several studies on InN layer grown by MOVPE, MBE and HVPE shows similar "bell-shaped" curve of the growth rate as a function of the growth temperature [START_REF] Bhuiyan | Indium nitride (InN): A review on growth, characterization, and properties[END_REF] . At temperatures between 610 °C and 660 °C, growth is mainly limited by the adsorption and dechlorination kinetics, and to a lesser extent by the kinetics of surface diffusion of the ad-species on the crystal facets, which all depend on the temperature, dechlorination and surface diffusion being activated upon temperature. Consequently, the axial growth rate shows a strong positive variation with the temperature. At high growth temperature of 690 °C, desorption is favored over adsorption and growth becomes simply limited by the lack of material. At the same time, InN crystal decomposition is enhanced at this temperature due to the weak In-N bond (Figure 1 (e)).

As a consequence, the axial growth rate decreases. On the over hand, the radial growth rate remains low and varies slightly with the growth temperature. The relative diameters of InN nanorods grown at 610 °C, 640 °C, 660 °C and 690 °C are 0.45 µm, 0.55 µm, 0.58 µm and 0.56 µm, respectively. The morphology of InN nanorods changes from pencil-like shape (Figure 1 The existence of such morphologies is related to the anisotropic growth rates of the different facets [START_REF] Li | Computing Equilibrium Shapes of Wurtzite Crystals: The Example of GaN[END_REF] . The disappearance of the r-planes facets at higher growth temperature of 640 °C is probably due to an increase in their growth rate with respect to the c-plane facet, as will be seen below. § Effect of the input NH3 partial pressure Bird's eye view SEM images of the samples grown for 5 minutes with 5.00 x 10 -2 atm, 1.00 x 10 - 1 atm, 1.50 x 10 -1 atm and 2.00 x 10 -1 atm of P 0 NH3 are shown in Figures 2 (a),(b),(c),(d) respectively. Clearly, selectivity is achieved for all samples whatever P 0 NH3, unlike what was observed in previous MBE studies where selectivity has been shown to depend on the V/III ratio.

For example, Kamimura et al [START_REF] Kamimura | Dislocation reduction via selective-area growth of InN accompanied by lateral growth by rf-plasma-assisted molecular-beam epitaxy[END_REF] showed that the selectivity of InN nanorods grown by MBE was improved under In-rich conditions, allowing indium adatoms to easily diffuse on the mask surface and reach the holes before reacting with nitrogen to yield spontaneous nucleation of InN on the mask. The absence of any InN nucleation on the mask surface at high NH3 partial pressure confirms the low adsorption of III-chlorides species on the dielectric mask in the HVPE process. Also, there is a clear transition in the axial growth rate as a function of P 0 NH3 as plotted in Figure 2 (e) while the radial growth decreases very slightly. For a low P 0 NH3 of 5.00 x 10 -2 atm supplied in the vapor phase (Figure 2 (a)), the axial growth rate is equal to 320 nm.min -1 . An increase of P 0 NH3 enhances the axial growth rate to reach the maximum 362 nm.min -1 at 1.00 x 10 -1 atm (Figure 2 (b)). Then, it decreases for higher P 0 NH3 as seen in Figure 2 (c) and(d). To further explain these observations, the supersaturation parameter is introduced. It represents the state of the growth reaction: InCl3 + NH3 ↔ InN + 3 HCl with respect to its equilibrium. The supersaturation is defined as the ratio of the real partial pressures Pi above the substrate with respect to the partial pressures of equilibrium:

γ "#$ = 𝑃 "#'() 𝑃 $+) 𝐾 𝑃 +'( ) -1 ( 1
)
where K is the equilibrium constant calculated from thermochemical data. Further details on thermodynamic calculation can be found in Ref [START_REF] Zeghouane | Compositional control of homogeneous InGaN nanowires with the In content up to 90%[END_REF] . The supersaturation in the vapour phase above the substrate needed to promote InN crystal growth is plotted in Figure 3. The supersaturation increases with the supply of NH3, until a peak value where it begins to decrease. When P 0 NH3 increases, hydrogen species produced by the dissociation of ammonia in the mixing zone increases and react quasi-totally with InCl3 before deposition according to the following reaction:

𝐼𝑛𝐶𝑙 )(4) + 𝐻 8(4) ↔ 𝐼𝑛𝐶𝑙 (4) + 2𝐻𝐶𝑙 4 (2)
Yet, InCl molecules are not involved in the growth of InN due to their low reactivity towards NH3, as previously demonstrated by Koukitu et al [START_REF] Kumagai | Thermodynamics on halide vaporphase epitaxy of InN using InCl and InCl3[END_REF] . At the same time, this reaction is counterbalanced by the formation of HCl species in the vapour phase. As a consequence, the supersaturation of InN deposition decreases and the axial growth rate decreases at higher P 0 NH3.

Similar findings were observed with respect to the effect of H2 on the growth rate of InN films grown by HVPE [START_REF] Bhuiyan | Indium nitride (InN): A review on growth, characterization, and properties[END_REF], [START_REF] Kumagai | Thermodynamics on halide vaporphase epitaxy of InN using InCl and InCl3[END_REF] . Other studies by MOVPE showed that the effect of NH3 on the growth rate of InN layers is linked to the growth temperature since NH3 decomposition kinetics depends mainly on this parameter [START_REF] Bhuiyan | Indium nitride (InN): A review on growth, characterization, and properties[END_REF] . In the HVPE process, the growth temperature, which is higher than 600 °C, is supposed to be high enough to overcome this limitation. Furthermore, previous studies demonstrated that the radial growth rate of GaN rods grown by HVPE decreases at a high partial pressure of HCl and/or H2 [START_REF] Avit | Crystal engineering by tuning the growth kinetics of GaN 3-D microstructures in SAG-HVPE[END_REF], [START_REF] Lekhal | Selective-area growth of GaN microrods on strain-induced templates by hydride vapor phase epitaxy[END_REF] . These findings are in accordance with the theoretical results dealing with the effect of H2 on the growth rate of non-polar GaN (1-100) surfaces [START_REF] Lymperakis | Adsorption and desorption of hydrogen at nonpolar GaN (1-100) surfaces: Kinetics and impact on surface vibrational and electronic properties[END_REF] . In that sense, we suppose that such decrease of the radial growth rate of InN nanorods can be related to the additional HCl in the vapour phase formed by reaction (2) and/or H2 produced by NH3 dissociation. This supports the fact that the lateral growth rate of nanorods grown in this study remains low whatever the V/III ratio and the growth temperature.

KOH etching in aqueous solution at 40 °C, which is known to etch only N-polar surfaces, reveals that the InN nanorods have In-polarity on the (0001) top facet, indicating that (1-101) r-planes surfaces are nitrogen-terminated. It has been pointed out by several works that hydrogen can easily passivate the N-terminated surfaces [START_REF] Li | Polarity and Its Influence on Growth Mechanism during MOVPE Growth of GaN Sub-micrometer Rods[END_REF] . Under our growth conditions we believe that the rplanes can be passivated by H atoms originating from the NH3 decomposition and forming N-H bonds, which leads to stable (1-101) r-planes facets. However, contrary to what was foreseen, the r-planes facets disappear at higher P 0 NH3 of 2.00 x 10 -1 atm as we can see in Figure 2 (d).

According to Wulff theory of crystal growth, we assume that the disappearance of the From these observations and according to Ref [26], we assume that the growth rate is governed by both mass transport and surface kinetics. At the initial stage of the growth, the growing surface area is small and the depletion of the growth precursors in the vapor phase is very limited.

The growth rates are then at their maximum and can be considered as kinetically-limited. The growth rate of the (0001) c-plane facet is then greater than those of the (1-101) r-planes and (1-100) m-planes facets, this is the reason why the nanorods have a pencil-like shape at 1 minute.

When the growth time increases, the growing surfaces of the nanorods extend gradually due to the positive axial and lateral growth rates. This induces a vapor phase depletion of reacting species near the InN surface that cannot be neglected. The reactive materials that reach the surfaces of the InN rods are then not sufficient. Thus, the growth turns into mass-transport limited. Consequently, the axial growth rate decreases and the top (0001) c-plane facet begins to appear after 5 minutes. Then, the (1-101) r-planes facets, of which growth rate exceeds the growth rate along (0001), disappear (the crystal is now limited by the top c-plane facet) and the hexagonal prism emerges as the dominant structure, as we can see after 15 minutes of growth.

Finally, the appearance of the (11-20) a-planes facets after 30 minutes of growth is the consequence of the decrease of their growth rates with respect to the (1-100) m-planes facets. § Optical properties of InN nanorods

The optical properties of InN nanorods were investigated by photoluminescence (PL) with a continuous laser power of 50 mW. The PL measurements were first performed on nanorod arrays at a temperature of 10 K. Figure 5 exhibits the strong dependence of the PL spectra, in term of intensity, peak energy and linewidth as a function of the growth temperature. For nanorods grown at the lowest temperature (T = 610 °C) only a broad emission centered at 0.85 eV is observed. PL emission of InN nanorods grown at very high temperature (T = 690 °C) has been not detected because of the low quality of the material. The samples grown at T = 640 °C and T = 660 °C, however, exhibit stronger PL emissions centered at 0.77 and 0.76 eV, respectively, with a substantial increase in the PL peak intensity for the sample grown at 660 °C compared to that of sample grown at 640 °C. The nanorods size is higher at 660 °C which may explain the increased PL intensity. These observations are somewhat consistent with previous reported results, where the PL intensity gradually increased with the growth temperature from 440 °C to 525 °C [START_REF] Stoica | MBE growth optimization of InN nanowires[END_REF] .

Moreover, the full width at half maximum (FWHM) of nanorods grown at 640 °C is larger than those obtained at a growth temperature of 660 °C. These observations tend to indicate that InN nanorods grown at 660 °C are of higher quality. that unintentionally-doped InN nanowires emitting at 0.75 eV possess a high electron concentration, around 10 18 cm -3 [START_REF] Stoica | Photoluminescence and Intrinsic Properties of MBE-Grown InN Nanowires[END_REF] . For n-type semiconductors, the increase of electron concentration is known to shift the Fermi level into the conduction band leading to an increase of the emission energy through the band-filling Burstein-Moss effect. This high energy of 0.76 eV indicates that our InN nanorods probably contain a significant density of free electrons. The fact that as grown InN material (nanowires, films) are mostly unintentionally n-type with a high residual doping is not really explained. The reported causes of this unintentional doping are nitrogen vacancies, monatomic hydrogen and other impurities [START_REF] Piper | Origin of the n-type conductivity of InN: The role of positively charged dislocations[END_REF], [START_REF] Janotti | Sources of unintentional conductivity in InN[END_REF], [START_REF] Mahboob | Intrinsic Electron Accumulation at Clean InN Surfaces[END_REF] . As previously shown, nitride materials grown by HVPE contain small amount of Si, coming from the hot quartz wall of the reactor, yielding a residual n-type doping [START_REF] Fujito | Bulk GaN crystals grown by HVPE[END_REF] . On the other hand, several studies evidence the presence of an electron accumulation layer on polar and non-polar planes of both InN films and nanorods [START_REF] Segura-Ruiz | Optical studies of MBE-grown InN nanocolumns: Evidence of surface electron accumulation[END_REF], [START_REF] Stoica | Photoluminescence and Intrinsic Properties of MBE-Grown InN Nanowires[END_REF], [START_REF] Mahboob | Intrinsic Electron Accumulation at Clean InN Surfaces[END_REF], [START_REF] Calleja | Evidence of electron accumulation at nonpolar surfaces of InN nanocolumns[END_REF] . It is assumed that this accumulation layer is due to surface states (density of positive charge), which are intrinsic of InN surfaces, and to doping mechanisms which are thickness-dependent [START_REF] Cimalla | Model for the thickness dependence of electron concentration in InN films[END_REF], [START_REF] Darakchieva | Free electron behavior in InN: On the role of dislocations and surface electron accumulation[END_REF] . Segura-Ruiz et al. have shown that the effect of the surface electron accumulation layer on the optical band gap of InN nanocolumn grown by MBE is more pronounced when the diameter decreases [START_REF] Segura-Ruiz | Optical studies of MBE-grown InN nanocolumns: Evidence of surface electron accumulation[END_REF] . Ahn et al. have revisited these findings through a detailed study comparing the optical properties of InN film and nanorods with different diameters [START_REF] Ahn | Photoluminescence from InN Nanorod Arrays with a Critical Size[END_REF] . This may explain, in part, the high peak energy of 0.85 eV recorded from the InN nanorods grown at 610 °C, which present small diameters.

In order to understand the nature of the InN nanorod PL, temperature-dependent PL experiments have been carried out from 10 to 300 K on the sample grown at 660 °C. It is shown that the emission spectra depend critically on the PL temperature. The temperature-dependent peak energy of InN nanorods plotted in Figure 6 (b) strongly differs from the Varshni-like temperature dependence of the InN bandgap energy. Due to the high free electron density on the lateral surfaces, the carrier redistribution on the electronic states is greatly inhomogeneous and reveals an anomalous behavior of the peak energy and FWHM as a function of temperature. When the PL temperature increases, the generated photoholes in the inner core can move over a deeper region and recombine with electrons in the surface accumulation layer which provides a blueshift followed by a noticeable broadening of the PL peak. The normal redshift induced by the temperature dependence of the band gap, which is always present, can compensate the blueshift, consequently the PL peak energy appears to be independent of the temperature. The variation in the density of the positive charge at the nonpolar surfaces from a nanorod to another can also induce a noticeable broadening of the PL spectra [START_REF] Segura-Ruiz | Optical studies of MBE-grown InN nanocolumns: Evidence of surface electron accumulation[END_REF] .

The thermal quenching of PL peak intensity can be fitted by the following equation:

𝐼(𝑇) = 𝐼 < 1 + 𝐴 exp A- 𝐸 C 𝑘 E 𝑇 F + 𝐵 exp (- 𝐸 H 𝑘 E 𝑇 ) (3)
where Ea and Eb denote the thermal activation energies at low and high temperatures, respectively, A and B are the fitting constants, I0 is the PL peak intensity at 10 K and and kB is the Boltzmann's constant. The Arrhenius plot of I(T)/I0 as a function of 1/T is displayed in Figure 6 (c). Fitting the equation (3) gives Ea and Eb respectively equal to 11 ± 1.5 meV and 31 ± 1.5

meV. The activation energy of 11 meV at low temperature can be attributed to structural defects in our nanorods. The binding energy of the localized hole states considered as acceptor like centers in the n-type region can be extracted from the half-amplitude energy (E1/2) from the PL spectra at low temperature [START_REF] Desheng | Electrical properties and photoluminescence of Tedoped GaAs grown by molecular beam epitaxy[END_REF] . We estimate the binding energy of these acceptor like centers to be 40 meV, in rather good agreement with the value of 31 ± 1.5 meV determined through PL experiments as a function of temperature. Our experimental observation is also very close to previously reported experimental values in the range of ~ 25-50 meV, which are associated to impurities [START_REF] Ahn | Photoluminescence from InN Nanorod Arrays with a Critical Size[END_REF], [START_REF] Segura-Ruiz | Optical studies of MBE-grown InN nanocolumns: Evidence of surface electron accumulation[END_REF] . The n concentration deduced from the analysis of the low temperature spectrum related to the sample grown at 660 °C is about 2.6 x 10 18 cm -3 .

Conclusion

In this work the selective area growth (SAG) of InN nanorods on sapphire substrates with a Gapolar GaN buffer layer was explored. We showed that the selectivity and the nanorods morphology depend strongly on the growth temperature. The effect of the input NH3 on the growth rates and the shape of nanorods was investigated. It revealed that the hydrogen species, generated from NH3 decomposition, have a significant impact on the supersaturation of InN deposition. The effect of the mass transport in the vapor phase on the growth rate is found to be more significant at larger growth time. The nanorod morphology can be simply tuned from pencil to a perfect hexagonal shape by changing the growth conditions. The optical quality of HVPEgrown InN rods is assessed by PL measurements which reveal a good agreement with high quality MBE-grown InN nanowires. These results highlight the potential of the HVPE process to achieve SAG of well-controlled arrays of InN nanorods. HVPE appears to be a relevant promising growth technique for future InN nanorods based-devices.

Figures 1 (

 1 Figures 1 (a) to (e) show tilted-view SEM images of samples grown for 30 minutes at different growth temperatures: T1 = 580 °C, T2 = 610 °C, T3 = 640 °C, T4 = 660 °C and T5 = 690 °C. It is clear that the selectivity and the growth rates are dependent on the growth temperature showing a noticeable effect on the geometry of InN nanorods obtained by SAG. At a growth temperature of

Figure 1 :

 1 Figure 1: (a) -(e) SEM images of InN nanorods grown on Ga polar GaN/c-Al2O3 templates masked by SiNx at growth temperatures of (a) 580 °C, (b) 610 °C, (c) 640 °C, (d) 660 °C and (e) 690 °C. (f) Axial (black) and radial (red) growth rates of InN nanorods as a function of the growth temperature. The selectivity is achieved from 610 °C. The higher growth temperature of 690 °C results in InN crystal deterioration.

  (b)) to a perfect hexagonal pillar surrounded by the vertical (1-100) and (11-20) facets and (0001) top facet (Figure 1 (c)) by increasing the growth temperature from 610 °C up to 640 °C. The angle for the side edge referring to the (0001) plane in Figure 1(b) is 63 ± 2° which corresponds to the (1-101) r-plane facet of wurtzite InN.

Figue 2 :

 2 Figue 2: (a) -(d) SEM images of InN nanorods grown on Ga polar GaN/c-Al2O3 templates masked by SiNx at different input NH3 partial pressures (a) 5.00 x 10 -2 atm, (b) 1.00 x 10 -1 atm, (c) 1.50 x 10 -1 atm and (d) 2.00 x 10 -1 atm. (e) Axial (black) and radial (red) growth rates of InN nanorods as a function of the input NH3 partial pressure. Nanorods growth rate and morphology depend on the introduced partial pressure of NH3.

Figure 3 :

 3 Figure 3: Plot of the supersaturation parameter for InN deposition as a function of the input NH3 partial pressure. Ptot is the total pressure in the reactor. The supersaturation in the vapor phase to promote InN deposition decreases at higher P 0 NH3.

( 1 -

 1 101) rplanes facets at this high P 0 NH3 is related to the reduced growth rate of the (0001) c-plane facet. This disappearance of the r-planes facets in favor of the c-plane is in agreement with the effect of the growth time as presented below. § Evolution of the morphology of InN nanorods with the growth time The typical morphologies of InN nanorods grown at different growth times: t1 = 1 min, t2 = 5 min, t3 = 15 min and t4 = 30 min are presented in parts (a), (b), (c) and (d) of Figure 4, respectively. The insets show the enlarged tilted view of InN nanorods shape. At the early stage of growth, InN nanorods have a pencil-like shape delimited by the semi-polar (1-101) r-planes and non-polar (1-100) m-planes facets, as shown in Figure 4 (a). As the growth proceeds, the (1-101) facet area decreases and the (0001) facet starts to appear with the exposure time to the vapor phase (Figure 4 (b)). At 15 minutes, InN nanorods growth occurs with well-developed (0001) top c-plane facet and (1-100) side m-planes facets. Longer growth time of 30 min results in the appearance of (11-20) side a-planes facets at the bottom part of the nanorods (Figure 4 (d)). The facets which remain after growth had the slowest development. The growth rate of the (1-100) mplanes is low due to HCl and/or H2 passivation, (1-100) m-planes always limit the final morphology of the rods. Then, there is a competition between the growth rates of the c-plane and r-planes facets.The axial and radial growth rates of the InN nanorods along (0001) and (1-100) directions, respectively, as a function of the growth time are reported in Figure4 (e). During the first stages of the growth, the axial and radial growth rates are very fast, about 938 nm.min -1 and 114 nm.min -1 , respectively, for a 1 min growth time. Then they strongly decrease within a short time of 5 min. Finally, they respectively stabilize around 85 nm.min -1 and 8 nm.min -1 .

Figure 4 :

 4 Figure 4: (a) -(d) SEM images of InN nanorods grown on Ga polar GaN/c-Al2O3 templates masked by SiNx at different growth times (a) 1 min , (b) 5 min, (c) 15 min and (d) 30 min. (e) Axial (black) and radial (red) growth rates of InN nanorods as a function of the growth time. (e) Schematic illustration of InN crystal shape evolution as a function of the growth time. The nanorods morphology changes from a pencil-like shape to a hexagonal shape with the growth time.

Figure 5 :

 5 Figure 5 : Photoluminescence spectra at 10 K of InN nanorods grown on patterned GaN/Al2O3 templates at different growth temperatures under a laser power density of 159 W/cm².

Figure 6 :

 6 Figure 6: Temperature dependence of (a) PL intensity and (b) PL peak energy and PL bandwidths of InN nanorods grown on patterned GaN/Al2O3 templates at 660 °C. (c) Normalized intensity variation of the PL intensity as a function of the reciprocal temperature.

Author contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Acknowledgement

The financial support received from CNRS (PRC1300 CNRS-JSPS) and from GaNeX program France. E-mail: agnes.trassoudaine@uca.fr


Melting and decomposition of orthorhombic B 6 Si under high pressure V Vl la ad di im mi ir r L

Keywords: boron silicide, melting, decomposition, high pressure, high temperature

Melting and decomposition of orthorhombic B6Si under high pressure

Vladimir Solozhenko, Vladimir Mukhanov, Vadim Brazhkin

Introduction

The boron-silicon system -despite of rather long research history -remains not fully understood even at ambient pressure [START_REF]Boron compounds with group IV elements: properties of boron-silicon compounds[END_REF]. There are three main groups of B-Si binary compounds usually referred to as "B 3 Si" (with the stoichiometry ranging from B 2.8 Si to B 4.8 Si); B 6 Si; and boron-rich silicides i.e. B n Si (12  n  50) (for details see [START_REF] Zhang | Predicted structural evolution and detailed insight into configuration correlation, mechanical properties of silicon-boron binary compounds[END_REF] and references therein). Boron silicides attract considerable attention due to superior thermal stability, excellent chemical resistance, promising mechanical and electronic properties that offers potential for their use as advanced engineering and smart functional materials. The boron-silicon phase diagram at ambient pressure is rather complicated and includes a number of peritectic reactions and intermediate phases of different stoichiometries [START_REF] Olesinski | Abbaschian GJ The B-Si (boron-silicon) system[END_REF]. As for the phase relations in the B-Si system at high pressures and high temperatures, they have not been studied at all. B 6 Si is the most explored boron-rich silicide that is used in nuclear technology [START_REF] Mirzayev | Tiep NV Crystal structure changes and weight kinetics of silicon-hexaboride under gamma irradiation dose[END_REF], some nanotechnology applications [START_REF] Kim | boride-coated Si nanoparticles with improved thermal oxidation resistance[END_REF] and as high-temperature thermoelectric material [START_REF] Imam | A review of boron-rich silicon borides based on thermodynamic stability and transport properties of high-temperature thermoelectric materials[END_REF]. It has an orthorhombic unit cell (space group Pnnm) containing 43 silicon atoms and 238 boron atoms forming a dense framework of 18 icosahedra and 4 icosihexahedra [START_REF] Vlasse | The crystal structure of SiB 6[END_REF] (Fig. 1a). According to the predictions made in the framework of thermodynamic model of hardness [START_REF] Mukhanov | Thermodynamic aspects of materials' hardness: prediction of novel superhard high-pressure phases[END_REF], B 6 Si is expected to exhibit hardness of about 35 GPa [START_REF] Mukhanov | Solozhenko VL Thermodynamic model of hardness: Particular case of boron-rich solids[END_REF] comparable to that of polycrystalline boron carbide. High pressure -high temperature behavior of B 6 Si has not been studied so far. In the present work we have performed the first investigation of solid and liquid B 6 Si at pressures up to 8 GPa.

Experimental

Powder (particles size less than 45 μm) of orthorhombic B 6 Si was used as received from ABCR GmbH (Germany). High-pressure experiments in the 2-8 GPa range have been performed using a toroid-type high-pressure apparatus with a specially designed high-temperature (up to 3500 K) cell [START_REF] Mukhanov | Solozhenko VL On melting of B 4 C boron carbide under pressure[END_REF][START_REF] Sokolov | On melting of silicon carbide under pressure[END_REF]. The cell was pressure-calibrated at room temperature using phase transitions in Bi (2.55 and 7.7 GPa), PbSe (4.2 GPa), and PbTe (5.2 GPa). The temperature calibration under pressure was made using well-established reference points: melting of Si, NaCl, CsCl, Pt, Rh, Al 2 O 3 , Mo and Ni-Mn-C ternary eutectic. B 6 Si powder was compacted into pellets and placed in boron nitride (Saint-Gobain, grade AX05) capsules. After isothermal holding time of 60-300 s at desired pressure and temperature the samples were either quenched by switching off the power or stepwise cooled down to room temperature, and then slowly decompressed down to ambient pressure. In some experiments no BN capsule has been used (the sample was in a direct contact with graphite heater), and the appearance of a liquid phase upon heating could be detected in situ by electrical resistance measurements using the method described earlier [START_REF] Mukhanov | Solozhenko VL On electrical conductivity of melts of boron and its compounds under pressure[END_REF][START_REF] Solozhenko | On melting of boron phosphide under pressure[END_REF][START_REF] Solozhenko | On melting of boron subnitride B 13 N 2 under pressure[END_REF]. In a special set of experiments it was found that B 6 Si does not react with BN and graphite in the whole studied pressure-temperature range.

The samples recovered from high-pressure experiments have been studied by powder X-ray diffraction (Equinox 1000 Inel diffractometer; Cu Kα and Co Kα radiation). The characteristic diffraction patterns are presented in Fig. 2.

Results and Discussion

Melting of orthorhombic boron silicide B 6 Si has been studied in the 2.6-7.7 GPa pressure range using in situ electrical resistivity measurements and quenching (the results are shown in Fig. 3). The melting curve exhibits negative slope of -31 ( 2) K/GPa that points to a higher density of the melt as compared to the solid phase.

The electrical conductivity of B 6 Si sample before melting is comparable to that of graphite heater at the same temperature, which corresponds to the literature data [START_REF] Cline | An investigation of the compound silicon boride (SiB 6 )[END_REF], taking into account a possible decrease in semiconductor band gap and an conductivity increase at high pressure. Upon melting, a slight (several percent) increase in sample resistance was observed. Melting occurs in a fairly narrow temperature range, comparable to the T-gradient value in the sample. In the samples quenched from the melt at temperatures slightly above the melting curve, no traces of any other phases were found except for orthorhombic B 6 Si, i.e. the melting is congruent. It should be noted that according to the literature, at ambient pressure B 6 Si melts incongruently, and the width of the melting interval differs from one publication to another [START_REF] Olesinski | Abbaschian GJ The B-Si (boron-silicon) system[END_REF][START_REF] Armas | Determination of the boron-rich side of the B-Si phase diagram[END_REF][START_REF] Dirkx | Optimization of thermodynamic data for silicon borides[END_REF]. Extrapolation of the melting line obtained by us to the low-pressure region gives the value of 2110(20) K for melting point at ambient pressure, which is in good agreement with B 6 Si peritectic point (2123 K) according to the assessed equilibrium phase diagram of the B-Si system [START_REF] Olesinski | Abbaschian GJ The B-Si (boron-silicon) system[END_REF].

Quenching experiments with significant overheating of the melt relative to the melting curve resulted in decomposition of B 6 Si into silicon and boron-rich silicides. X-ray diffraction patterns of decomposition products are rather complex (see Fig. 2), but it is obvious that formation of neither B 6 Si and B 3 Si, nor pure boron was observed. The main diffraction lines (in addition to reflections of crystalline silicon) can be attributed to B 14 Si, B 36 Si (Fig. 1b), and B 50 Si compounds and/or their mixtures (note that diffraction patterns of these compounds differ significantly from one publication to another). The onset of B 6 Si decomposition at different pressures is observed at different degrees of overheating relative to the melting point: from 450 K at 2.7 GPa to 150 K at 7.7 GPa (Fig. 3).

The melt decomposition boundary has a large negative slope of -92(3) K/GPa, which indicates that the decomposition is accompanied by a significant volume decrease. When extrapolating the lines of melting and decomposition to the high-pressure region, their intersection is observed at ~10 GPa, which suggests that at higher pressures B 6 Si decomposition into silicon and boron-rich silicides should be observed already in solid state.

The instability of B 6 Si under pressure with respect to disproportionation into silicon and boron-rich silicides can be explained by analyzing the volume effects of two hypothetical reactions

6B 6 Si  B 36 Si + 5Si (1) 2B 6 Si  B 12 Si + Si (2)
For convenience and simplification of the assessment of the second reaction, the B 12 Si stoichiometry rather than B 14 Si was taken which, however, does not affect the final result. Under normal conditions, the densities of the phases involved are as follows: 2.33 g•cm -3 for Si;

2.42-2.44 g•cm -3 for B 6 Si [START_REF] Vlasse | The crystal structure of SiB 6[END_REF]; 2.48-2.51 g•cm -3 for B 14 Si [START_REF] Armas | Pistoulet B Fabrication and study of electrical properties of boron-silicon compounds with high boron concentration[END_REF]; and 2.34-2.37 g•cm -3 for B 36 Si [START_REF] Vlasse | The boron-silicon solid solution: A structural study of the SiB ~36 composition[END_REF].

It is easy to see that under normal conditions reaction (1) proceeds with a significant increase in volume, while in reaction (2) the volume practically does not change, i.e. both reactions cannot be stimulated by pressure.

The situation, however, changes drastically at high pressures and high temperatures. In the temperature range under study thermal expansion leads to an increase in the specific volume (by 3-4% [START_REF] Lugscheider | Quadakkers WJ Das system bor-silicium[END_REF]) for all compounds formed as a result of the considered disproportionation reactions. On the other hand, melting of silicon leads to its density increase by 10%, while the melting of boronrich solids usually occurs with the preservation of B 12 icosahedra as the main structural units, and thus the volume change does not exceed 1-3%. In addition, all considered boron-rich silicides have bulk moduli of 170-190 GPa [START_REF] Zhang | Predicted structural evolution and detailed insight into configuration correlation, mechanical properties of silicon-boron binary compounds[END_REF], which is twice as high as that of liquid silicon † . As a result, at high temperatures reaction (1) occurs with 1-3% volume decrease at pressures above 2 GPa; and reaction (2) occurs with a significant (5-7%) volume decrease in the whole pressure range under study. Thus, the observed large negative slope of the B 6 Si decomposition line is in full agreement with the thermodynamic estimations.

At ambient pressure B 14 Si compound is a semiconductor with a high resistivity, but at high temperatures (~2000 K) it has a semi-metallic conductivity of the order of 10 2 Ohm -1 •cm -1 [START_REF] Pistoulet | The interpretation of electrical properties of B 14 Si compounds using a model of medium range disorder[END_REF].

There is no data on electrical conductivity of rhombohedral B 36 Si, however, it can be assumed that this compound is close in properties to β-rhombohedral boron. The decomposition of B 6 Si in our experiments, as a rule, was accompanied by an increase in conductivity, which is obviously associated with the very high conductivity of liquid silicon.

Note that the assumption about the possibility of solid-state decomposition of B 6 Si at pressures above 10 GPa does not contradict our estimates, since silicon at high (above 11 GPa) pressures passes into the high-pressure phase with a white-tin structure (SiII). This transformation in crystalline silicon is accompanied by ~20% volume decrease [START_REF] Bundy | Phase diagrams of silicon and germanium to 200 kbar, 1000[END_REF], therefore, the thermodynamic stimulus of reactions ( 1) and ( 2) at high pressures can only increase.

Conclusions

Summarizing, we can conclude that in the 2.6-7.7 GPa pressure range B 6 Si melts congruently, and both the solid phase and the melt have a bad-metal conductivity. Melting is accompanied by a small drop in electrical conductivity and a slight volume decrease. At significant overheating relative to the melting line, the decomposition of B 6 Si into silicon and boron-rich silicides is observed, and the onset temperature of decomposition decreases strongly with increasing pressure. In the future, it would be of interest to study B 6 Si up to the megabar pressure range. Apparently, at ultrahigh pressures and room temperature the diffusion processes are strongly suppressed, and decomposition of B 6 Si can occur at nanoscale only resulting in solid-state amorphization.

Fig. 1

 1 Fig. 1 Crystal structures of orthorhombic B 6 Si [6] (a) and rhombohedral B 36 Si [19] (b).B 12 -units are presented by green icosahedral polyhedral; silicon and boron atoms are presented by blue and green balls, respectively.

Fig. 2 X

 2 Fig. 2 X-ray diffraction patterns of pristine orthorhombic B 6 Si (bottom) and samples quenched from melt at different pressures and temperatures. Vertical red ticks correspond to positions of diffraction lines the cubic (Fd-3m) silicon.

Fig. 3 B 6

 36 Fig. 3 B 6 Si melting and decomposition under pressure. Half-filled stars show the melting onset registered in situ by electrical resistivity measurements; blue dashed line is the pressure dependence of melting temperature. The results of quenching experiments are presented by circles (solid, half-filled and open symbols correspond to B 6 Si, B 6 Si + B 12 Si + Si, and B n Si + Si (12  n  50) in the recovered samples, respectively). Red dashed line is the low-temperature boundary of B 6 Si melt disproportionation.

* vladimir.solozhenko@univ-paris13.fr

† In fact, bulk modulus of liquid Si has never been measured. 300-K bulk modulus of crystalline silicon makes 98 GPa, while in vicinity of melting point it drops down to 80-85 GPa[START_REF] Nikanorov | Elastic properties of Si at high temperatures[END_REF]. Since Si melting curve under pressure is linear[START_REF] Bundy | Phase diagrams of silicon and germanium to 200 kbar, 1000[END_REF], one can conclude that at melting temperature bulk moduli of crystalline and liquid silicon should be very close.

Acknowledgements

The authors are grateful to Dr. P.S. Sokolov for his assistance at the initial stage of work. The research was financially supported by the European Union's Horizon 2020 Research and Innovation Programme under Flintstone2020 project (grant agreement No 689279).


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