Introduction

Moreno Andreatta, Emmanuel Amiot, Jason Yust

To cite this version:

INTRODUCTION

This special issue focuses on some geometrical and topological aspects of the representation and formalisation of musical structures and processes. There are six words in the previous sentence which have been emphasized since they could work as explanatory guidelines for the present issue. More generally, they seem to offer to the "working" musicologist and music-theorist some conceptual and operational principles for approaching contemporary "mathemusical" research from a more philosophical and epistemological perspective. Shedding some light to this double underlying dimension may constitute a fruitful exercise, so we hope, for the reader of the Jour-nal of Mathematics and Music and of the present issue in particular. Let us start with geometry, one of the oldest fields of mathematics and whose history accompanied the discussions on the relations between music and mathematics since the time of Greek philosophy. Geometry has a central place in the Pythagorean tradition, and particularly in Plato's philosophy, which is well expressed by the famous quotation that was supposed to be engraved at the door of the Plato's Academy in Athens: "Let no one ignorant of geometry enter." In Plato's perspective, geometry is surely far from what it should be according to its etymological meaning, which reduces the scope of the discipline to an "earth measure." Plato himself found such a name totally ridiculous, as he says in the Epinomis, the short dialogue that serves as an appendix to Plato's Laws. 1 As it has been pointed out by several scholars and commentators of Plato, starting from the great and influential music theorist Aristoxenus of Tarentum, geometry is first of all a science of abstrac-tion enabling to transcend the level of sensible experience and reach the world of Ideas. One is immediately reminded of the experiment described in the Meno, where the slave boy discov-ers the existence of truths via geometrical thinking, as in the case of doubling the surface of a square. 2 Geometry is therefore intrinsically linked to the visual way in which concepts are repre-sented and, eventually, drawn, as it is largely shown by Euclid in his Elements. 3 The influence of this geometrical thinking goes until the twentieth century, as one may observe by considering, for example, David Hilbert's debt to Euclid as documented in his Grundlagen der Geometrie (1899).

It can be easily shown that this emphasis on the axiomatization of geometry, which is prevalent in Hilbert's work, has a direct influence in the way in which twentieth century composers, starting from Ernst [START_REF] Krenek | Über Neue Musik. Sechs Vorlesungen zur Einführung in die theoretischen Grundlagen. Vienna: Universal Edition. Wien[END_REF], approached the problem of the theoretical foundations in music via formalization. 4 A turning point towards a full axiomatic thinking in geometry, with extremely significant musical consequences, is symbolized by the way in which geometry is progressively approached via an algebraic perspective. Starting from Felix Klein's Erlangen Program (Klein 1893), the way of conceiving the geometry of a space is in fact intrinsically linked to the way in which transformation groups act on such a space. With the entry of abstract group theory into play we assist to a progressive shift from the purely representational nature of geometrical entities to their truly operational power which has a big influence on the way in which music theorists conceptualize the musical phenomena. 5 This little historical digression may offer some elements to understand the complex link one can establish between the geometrical properties of musical structures and their underlying formalization which accompanied the evolution of mathematical music theory from a simply representational to an operational discipline capable of approaching in a powerful way the dynamic character of musical processes. Topology -let us now focus on this term -precisely enters the picture as a subsidiary discipline when the focus on musical structures is substituted or at least accompanied by a new attention of music theo-rists and analysts towards musical processes, i.e. theoretical constructions in which time plays a fundamental role. The extensive use of topological techniques in the mathematical modeling of musical structures and processes is a relatively new phenomenon, although one may already find many examples of genuine topological constructions in Guerino Mazzola's Geometrie der Töne (1990). The foundational basis of the concept of "global compositions," as opposed to the "local" ones (originally considered as finite non-empty subsets of a module over a ring), is the mathe-matical structure of Riemannian manifolds allowing the music theorist to make use of several topological constructions. The reader familiar with this approach in mathematical music theory will easily recognize some traditional visualizations of musical structures, such as the Möbius strip 6 and, more generally, the nerve of a n-dimensional simplicial complex as representing the complexity of the specific covering that the analyst will associate to a given global composition. Interestingly, there is already in this approach an interplay between geometry and combinatorial topology which shows that there is a shift from the simple characterization of (static) musi-cal structures to (dynamic) musical processes, such as modulation, whose properties are now described in a topological way. 7 More recently, it became clear to music theorists that some of most popular constructions in neo-Riemannian and, more generally, transformational music anal-ysis do intrinsically possess a topological structure. The most celebrated example is probably the Tonnetz, originally mostly studied from a graph-theoretical and algebraic perspective, and suc-cessively apprehended from a more topological way, enabling the complete characterization of their different instances (or Tonnetze) via the study of the associated algebraic invariants, such as Betti numbers [START_REF] Bigo | Building Topological Spaces for Musical Objects[END_REF][START_REF] Catanzaro | Generalized Tonnetze[END_REF][START_REF] Bigo | Représentation symboliques musicales et calcul spatial[END_REF]. This interplay between geometry, algebra and topology provides nowadays a better understanding of the way in which generalized musical spaces can be represented and formalized from a very elegant and computationally powerful perspective. 8 Each of the three contributions of the present issue can be considered as offering to the reader a specific perspective on this complex interplay starting from different but intrinsically linked music-theoretical problems.

"Why Topology?" by Dmitri Tymoczko makes use of topological techniques to formalize voice leadings as paths in n-dimensional geometric spaces. The approach is an extension of the conceptual constructions that constitute the foundation of Tymoczko's geometrical theory of music as presented and discussed in details in his book A Geometry of Music [START_REF] Tymoczko | Geometry of Music: Harmony and Counterpoint in the Extended Common Practice[END_REF]. Previous work by the author also uses topological techniques but without the detailed argumentation for the value of topology in contemporary mathematics and music research. 9 Higher dimensional geometric spaces whose points are generic n-notes chords are orbifolds in a technical mathematical sense, i.e. quotient spaces obtained by the action of the permutation (or symmetric) group of n! elements on the n-dimensional torus and possess therefore nontrivial topological properties. The present paper focuses on the interpretation of voice-leading properties via the fundamental group and can therefore offer new insights to the analyst to fully understand the author's previous music-theoretical constructions. Moreover, it shows that not only voice leading but, more generally, transformational constructions in music analysis are linked to the fundamental concept of homotopy equivalence, which again puts algebra -and group theory in particular -into play. Although homotopy theory has become very popular in recent years thanks to Guerino Mazzola's gesture theory, the approach described in Tymoczko's paper is fundamen-tally different since it provides the theoretical background to fully understand the topological properties of different harmonic spaces. While Tymoczko's voice-leading geometries have led to many significant analytical and music-theoretic applications, as is in evidence of Geometry of Music, to some extent there has always been a disconnect between the higher dimensional orbifolds defined in Callender, Quinn, and Tymoczko 2008 and music analysis and composition, because as the spaces get more com-plex we lose the benefit of direct spatial intuition. In previous work (such as Tymoczko 2011) spatial intuition has often been recovered by simplifying chord spaces to lattices on a limited number of chord-types, which renders their more interesting topological features irrelevant. In the present work, Tymoczko instead uses homotopic equivalence to simplify the spaces. Remark-ably, he is able to reduce these topological features to simple two-dimensional representations, making them easily surveyable and available to analytical intuition, regardless of the chord cardi-nality and resulting complexity of the original space. He introduces two kinds of simplifications: one is the circle of pitch-class sums, which has the topology of an annulus or circle, regardless of cardinality. The homotopy of this space corresponds to total voice-leading ascent and descent. The other simplification is a homotopic classification of set-class spaces (which factor out transposition or transposition or inversion). The analytical applications in Tymoczko's article begin to reveal the wealth of musical interest in the interactions between features of a voice leading that we usually are tempted to "reduce out," such as voice crossings and transpositions along the chord. The article also introduces a principled generalization of neo-Riemannian transformations to any chord type, as voice leadings to its inversions.

The following paper by Jason Yust, entitled "Generalized Tonnetze and Zeitnetze, and the Topology of Music Concepts," also focuses on the interplay between the topological struc-ture of the generalized Tonnetz seen as a n-dimensional simplicial complex and its possible graph-theoretical and geometric realizations. The notion of a generalized n-chord Tonnetz can be articulated in fact at different levels of abstraction, ranging from the traditional presentation as a lattice of chords relating by maximal intersection to more sophisticated representations, embedding this structure on the powerful theory of Fourier phase spaces. This further interpreta-tion adds a new subfield of mathematics to the overall picture, also showing the natural bridges between geometry, algebra, and (mathematical) functional analysis. Many music-theoretical problems, from the construction of rhythmic tiling canons to the classification of homometric musical structures, 10 have been shown to be deeply rooted into Fourier theory, but adequate understanding of the topological implications of such a formalism still constitutes an open research area in mathematics and music. A first crucial achievement has been the characterization of the geometrical structure of generalized Tonnetze as toroidal representations in the spaces of Fourier phases, as proposed by Emmanuel Amiot (2013) and further developed by Jason Yust in his critical discussion on the relations between the geometric extensions of Generalized Tonnetze and the Fourier phase Spaces [START_REF] Yust | Geometric Generalizations of the Tonnetz and their Relation to Fourier Phase Spaces[END_REF]. The present article pushes the previous conclusions further by focusing on the implications of geometrical Tonnetze interpreted as embedding a sim-plicial Tonnetze in some geometric space. Yust's article complements Tymoczko's in showing how a different theoretical emphasis, on shared pitch-class content between chords rather than the distances moved by voices, leads in the direction of different kinds of topologies. Despite the different mathematic foundations of their spaces (Fourier phase vs. voice-leading orbifold), the two theories arrive at surprisingly similar constructions in places, with some of Yust's foldings producing spaces topologically equivalent to Tymoczko's pitch-class sum annuli. Despite appear-ances however, a closer examination reveals that a path or homotopy-class of paths in the two kinds of space actually have distinct musical meanings. The two approaches also lead to inter-estingly different generalizations of neo-Riemannian transformations, with Tymoczko focusing on the idea of inversional voice leadings while Yust emphasizes maximal intersection.

There is much music-theoretical and mathematical potential in using this new geometric approach, as the reader will see by following Yust's systematic presentation of a panoply of gen-erated collections. These include dyadic Tonnetze (that reduce to the traditional chromatic and fifths-generated circular representations of the equal-tempered system), trichordal Tonnetze and higher-dimensional Tonnetze. All these geometric structures are first defined in toroidal spaces before applying higher-dimensional foldings which shed a new light on different n-chord gen-erated spaces. To this family belong, for example, the Tonnetz of chromatic tetrachords and of different 4-note chords and also Tonnetze of ninth chords, which are rare in the literature. Spe-cial attention is given to Tonnetze of diminished seventh chords, which enables the author to stress the fact that an object that is trivial from a graph-theoretical and simplicial perspective can still be interesting from a geometrical and music-theoretical perspective. In addition to its elegant and powerful character, taking phase spaces as a geometrical realization of Tonnetze has dramatic consequences in the way in which traditional neo-Riemannian harmonic constructions can be generalized and applied to other musical parameters, and in particular to rhythm. Build-ing bridges between harmonic and rhythmic spaces has a long tradition in music theory and composition, as the history of serialism clearly shows. Many other attempts at providing a gen-eral theory on the structural relations between harmonic and rhythmic domains have marked the development of mathematical music theory, from Mazzola's formalization of the two domains as local/global compositions based on the same underlying module structure to more perceptual-based approaches, such as Jeff Pressing's foundational studies on the cognitive isomorphism between pitch and rhythm in ethnomusicology [START_REF] Pressing | Cognitive Isomorphisms Between Pitch and Rhythm in World Musics: West Africa, the Balkans, and Western Tonality[END_REF]. In defining a "Zeitnetz" in anal-ogy with the Tonnetz, the paper also suggests new very interesting connections between research in mathematical music theory and problems raised by scholars working on different areas, in particular cognitive and empirical musicology.

The final paper by Mattia G. Bergomi and Adriano Baratè, entitled "Homological persistence in time series: an application to music classification", provides a third perspective on the inter-play between geometry, topology and algebra focusing on the problem of automatic stylistic music analysis. Based on persistence homology as a specific domain in the field of topological data analysis, the paper shows not only the possibility but, for some extend, the necessity of establishing a fruitful dialogue between two apparently orthogonal research tradition in the field of application of mathematical models to music analysis: mathematical music theory and music information research (MIR). Despite efforts to bring together researchers belonging to the Society for Mathematics and Computation in Music with those working in the area of area of the Society for Music Information Retrieval, it is not difficult to recognize that there are deep methodologi-cal and theoretical differences between these two major orientations in the field of computational musicology. 11 The paper by Bergomi and Baratè begins to fill the gap by applying a panoply of topological, geometrical, and algebraic techniques to the domain of automatic stylistic music classification. Providing models and tools for the automatic stylistic classification is probably one of the most active areas in the field of music information research. Moreover, the authors' focus on time series suggests that topology not only applies to the formalization of musical struc-tures but clearly deals with the modeling of musical processes (or, what they call "time-varying systems"). Using persistence homology as the main conceptual and theoretical framework for approaching style analysis is a relatively recent and very promising approach. The approach orig-inates in one of the authors' dissertation [START_REF] Bergomi | Dynamical and Topological Tools for (Modern) Music Analysis[END_REF] which gave rise to different research directions exploring the interplay between geometry, topology, and, more recently, category the-ory. 12 The present paper clearly shows the interplay between static topological representations of musical structures and dynamic geometric formalizations of these topological features. In the former case, the authors use persistent homology as the main topological framework to represent the inner structure of musical object via a collection of persistent diagrams. These static repre-sentations are then studied dynamically by formalizing their temporal evolutions as time series that can be compared through dynamic time warping. This very rich mixture of topological and geometrical techniques in the representation and formalization of musical structures and pro-cesses, together with their implementation in computer-aided analytical systems, also enable the authors to offer a new and complementary computational perspective on the generalized Tonnetz with respect to the one presented in the first two articles. It also starts with the same simplicial complex interpretation of the Tonnetz as described in Yust's article, but instead of focusing on their subcomplexes generated by a sequence of pitch classes (as extensively studied in [START_REF] Bigo | Représentation symboliques musicales et calcul spatial[END_REF][START_REF] Bigo | Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes[END_REF]) it proposes to take into account additional music information in order to break the isotropic character of the simplicial complex. The new geometric space in which musi-cal structures and processes are represented is therefore a deformed Tonnetz, i.e. a non-isotropic Tonnetz exhibiting variable geometric features which can be retrieved via ad-hoc filtration func-tions. This enables the authors to approach automatic stylistic music classification in a geometric structural way, which is very original in the field of music information research, where statistics and machine learning constitute the main theoretical framework for retrieval purposes. The tests that have been performed by the author on different data sets, and which are discussed in the final part of the article, show very promising results that shed new lights on the nature of musical style.

We hope that this short introduction provides the reader with enough elements to appreciate not only the singular character of each of the three papers published in this special issue but their deep and fruitful intersections. We insisted several times on the necessity of building bridges between different musictheoretical and computational approaches; but before establishing connections between research groups having different aims and methodological criteria, it is important to reinforce the synergies within the members of the same community. Research carried out by members of the Society for Mathematics and Computation in Music has spread in a multitude of different directions, as the readers of our Journal can easily see. Whereas each of the three contributions offers distinct perspectives on the same musictheoretical constructions, all of them together provide a comprehensive picture of the interplay between geometry and topology which is surely more than the sum of the three local perspectives. Probably the time for a synthetic view on all these different approaches has not yet come, but the reader has the opportunity to build his or her own conceptual bridges between geometry and topology and their mutual contribution to music representation and formalization.

NOTES

1 This and other very interesting aspects of the role of geometry in Plato's dialogues are developed by Athanase Papadopoulos in his forthcoming essay devoted to the relation between music, mathematics and astronomy starting from Plato's Timaeus (Papadopoulos 2020).

2 There are several editions available of Plato's dialogues, including the two mentioned ones. See, for example, Plato's collected work edited by John M. Cooper and Douglas S. Hutchinson (Plato 1997).

3 The thirteen books of Euclid's Elements are available online at Clay Mathematics Institute Historical Archive (https://www.claymath.org/library/historical/euclid/) 4 See [START_REF] Andreatta | Méthodes algébriques dans la musique et musicologie du XX e siècle: aspects théoriques, analytiques et compositionnels[END_REF] for a first attempt at reconstructing this legacy from both an European and American perspective.

5 This dialectical principle between the objectal and operational nature of the reality has been deeply studied by the French philosopher and epistemologist Gilles-Gaston Granger who took this duality as the foundational basis for the very notion of "concept" in philosophy [START_REF] Granger | Pygmalion. Réflexions sur la pensée formelle[END_REF][START_REF] Granger | Pygmalion. Réflexions sur la pensée formelle[END_REF].

6 Which the author also calls "Harmonisches Band" to emphasize its truly music-theoretical origin, that he bases on Arnold Schoenberg's writing on the theory of harmony [START_REF] Schoenberg | Harmonielehre[END_REF].

7 There are other examples that could be considered as genuinely "topological" within the same approach in mathemat-ical music theory. We only mention two of them, both dealing with the formalization of musical motifs as combinatorial and topological structures: [START_REF] Netske | Paradigmatic Motivic Analysis[END_REF] and [START_REF] Buteau | Motivic Analysis According to Rudolph Réti: Formalization by a Topological Model[END_REF].

8 We could also add to the picture the categorical perspective which is intrinsically linked to the interplay between geometry and algebra and which can also shed some new light on existing topological approaches not only in music the-ory and analysis but also in performance studied like in the case of a category and topos-based theory of musical gestures. See in particular the third volume of the revised version of Topos of Music [START_REF] Mazzola | The Topos of Music III: Gestures. Musical Multiverse Ontologies[END_REF]. The reader who might be interested in going more deeply into some philosophical and epistemological aspects of the categorical perspectives on the interplay between geometry, algebra, and topology, will find a good starting point in Jean-Pierre Marquis' monograph From a Geometrical Point of View [START_REF] Marquis | From a Geometrical Point of View[END_REF]. 9 See, in particular, [START_REF] Callender | Generalized Voice-Leading Spaces[END_REF] for a more detailed presentation of the technical aspects of a geometrical approach in music theory, also including the introduction of orbifold structures into music formalization.

10 Summarized in [START_REF] Amiot | Music through Fourier Space: Discrete Fourier Transform in Music Theory[END_REF].

11 As pointed out by Anja Volk and Aline Honingh in their introduction of the special issue of the Journal of Mathe-matics and Music devoted to "Mathematical and computational approaches to music" [START_REF] Volk | Mathematical and Computational Approaches to Music: Challenges in an Interdisciplinary Enterprise[END_REF], this gap was already acknowledged by Thomas Noll and Robert Peck in the first issue of the Journal of Mathematics and Music in 2007 [START_REF] Noll | Welcome[END_REF].

12 See, in particular, [START_REF] Bergomi | Towards a Topological Fingerprint of Music[END_REF] for a topological perspective and [START_REF] Bergomi | Beyond Topological Persistence: Starting from Networks[END_REF] for the first attempts at going beyond a topological approach in the study of a category theorybased persistent homol-ogy. Some extensions of the original framework have also been intensively studied within the SMIR (Structural Music Information Research) project. This research project is a permanent interdisciplinary research axis supported by CNRS and led by Moreno Andreatta at the University of Strasbourg / IRMA (Institut de Recherche Mathématique Avancée), in a formal collaboration with IRCAM. Research topics include Mathematical Morphology, Formal Concept Analysis, Persistent Homology and automatic classification of musical styles, Category theory and transformational (computeraided) music analysis as well as the study of musically-driven open conjectures in mathematics and the epistemological and cognitive implications of contemporary mathemusical research. See [START_REF] Andreatta | From Music to Mathematics and Backwards: Introducing Algebra, Topology and Category Theory into Computational Musicology[END_REF] for a short overview of the different research topics included in the SMIR project.


Hezhi Zhang
email: hez.zhang@dlut.edu.cn

Valerio Piazza

Vladimir Neplokh

Nan Guan

Fabien Bayle

Stéphane Collin

Ludovic Largeau

Andrey Babichev

Francois H Julien

Maria Tchernycheva

Correlated optical and electrical analyses of inhomogeneous core/shell InGaN/GaN nanowire light emitting diodes

Keywords: core-shell nanowire, LED, nitride, EBIC, cathodoluminescence

Introduction

Gallium nitride based Nanowires(NWs) appear to be promising to boost the performance of light emitting diodes (LEDs) [START_REF] Guan | Nitride Nanowires for Light Emitting Diodes[END_REF][START_REF] Robin | Insight into the performance of multi-color InGaN[END_REF][START_REF] Kapoor | Green Electroluminescence from Radial m-Plane InGaN Quantum Wells Grown on GaN Wire Sidewalls by Metal-Organic Vapor Phase Epitaxy[END_REF][START_REF] Schmidt | Direct imaging of Indium-rich triangular nanoprisms self-organized formed at the edges of InGaN/GaN core-shell nanorods[END_REF][START_REF] Yi | Cathodoluminescence and scanning transmission electron microscopy study of InGaN/GaN quantum wells in core-shell GaN nanowires[END_REF][START_REF] Motohisa | Characterization of nanowire light-emitting diodes grown by selective-area metal-organic vapor-phase epitaxy[END_REF][START_REF] Ra | Understanding the p-Type GaN Nanocrystals on InGaN Nanowire Heterostructures[END_REF][START_REF] Bi | InGaN Platelets: Synthesis and Applications toward Green and Red Light-Emitting Diodes[END_REF], particularly for full-color micro-LEDs [START_REF] Nami | Electrically Injected GHz-Class GaN/InGaN Core-Nanowire-Based μLEDs: Carrier Dynamics and Nanoscale Homogeneity[END_REF][START_REF] Hartensveld | Monolithic Integration of GaN Nanowire Light-Emitting Diode With Field Effect Transistor[END_REF]. Indeed, three-dimensional nanowire-based LEDs can improve the material quality and offer new opportunities for strain engineering. Moreover, the core-shell architecture is expected to reduce the efficiency droop at high current by increasing the active area. In nanowires (NWs) having their elongated axis aligned with the [0001] direction, quantum confined Stark effect is not present thanks to the growth of the active region on non-polar m-planes [START_REF] Li | GaN based nanorods for solid state lighting[END_REF][START_REF] Kang | Gallium nitride nanostructures for light-emitting diode applications[END_REF]. Additionally, the use of nanowires, rather than 2D layers, opens new possibilities for the development of multifunctional light sources. For example, flexible nanowire LEDs were demonstrated by transferring the active nanowires to flexible substrates [START_REF] Chung | Growth and characterizations of GaN micro-rods on graphene films for flexible light emitting diodes[END_REF][START_REF] Dai | Flexible Light-Emitting Diodes Based on Vertical Nitride Nanowires[END_REF].

The core/shell LEDs were demonstrated using selfassembled growth [START_REF] Koester | M-Plane Core-Shell InGaN/GaN Multiple-Quantum-Wells on GaN Wires for Electroluminescent Devices[END_REF][START_REF] Jacopin | Single-Wire Light-Emitting Diodes Based on GaN Wires Containing Both Polar and Nonpolar InGaN/GaN Quantum Wells[END_REF] and selective area growth [START_REF] Li | Nitrogen-polar core-shell GaN light-emitting diodes grown by selective area metalorganic vapor phase epitaxy[END_REF][START_REF] Chen | Light-emitting device with regularly patterned growth of an InGaN/GaN quantum-well nanorod light-emitting diode array[END_REF]. Organized growth following the selective area epitaxy procedure provides a better control over the nanowire morphology in comparison with spontaneous growth, since it reduces the composition fluctuations between the NWs and thus facilitates the control of the emission wavelength and the broadening of the spectral emission. However, due to the complexity of the three-dimensional growth it is difficult to control the homogeneity of indium composition inside the quantum well (QW) resulting in compositional gradients with the NW and in the formation of indium-rich regions. In particular, the change of the emission color with the injection current was observed not only in self-assembled NW LEDs [START_REF] Dai | Flexible Light-Emitting Diodes Based on Vertical Nitride Nanowires[END_REF][START_REF] Jacopin | Single-Wire Light-Emitting Diodes Based on GaN Wires Containing Both Polar and Nonpolar InGaN/GaN Quantum Wells[END_REF], but also in organized NW LEDs [START_REF] Tchernycheva | InGaN/GaN Core-Shell Single Nanowire Light Emitting Diodes with Graphene-Based[END_REF][START_REF] Chang | Fabrication and luminescent properties of core-shell InGaN/GaN multiple quantum wells on GaN nanopillars[END_REF][START_REF] Hong | Visible-Color-Tunable Light-Emitting Diodes[END_REF]. The transition from a 2D to a 3D geometry also affects the electrical and optical properties of the LEDs. Previously, charge carrier trapping [START_REF] Tchernycheva | Core-shell InGaN/GaN nanowire light emitting diodes analyzed by electron beam induced current microscopy and cathodoluminescence mapping[END_REF] has been observed at core/underlayer interface of core-shell NWs creating a potential barrier, which may lead to an inefficient carrier injection towards QWs region. This kind of electrical inhomogeneities can critically affect the LED performance, therefore they deserve a deeper analysis.

In this paper, we investigate core-shell InGaN/GaN NWs by means of macroscopic and nanoscale-resolved techniques. A sample which shows a good morphological order, but exhibits inhomogeneities of the electroluminescent (EL) was deliberately chosen. First, the morphology and the internal structure of the NWs is analyzed, together with the EL of the array. Cross-sectional electron beam induced current microscopy (EBIC) is performed, revealing a potential barrier induced by charge carrier traps at the core/underlayer interface. Then, the impact of this potential barrier on the performance of the InGaN/GaN NW LED is statistically analyzed, by correlating top view EBIC, cathodoluminescent (CL) and EL maps. Finally, we developed an alternative processing scheme to overcome this issue by injecting the electrons through n-GaN underlayer thus avoiding the potential barrier. The final device shows a significant improvement in the EL homogeneity of the NW LEDs.

Nanowire growth, device fabrication and preliminary analysis

Organized InGaN/GaN core-shell NWs are grown by Metalorganic Vapour-Phase Epitaxy (MOVPE) on n-type GaN/sapphire substrates coated with a SiNx mask with submicrometer openings. First, n-type GaN core is grown along the c-direction, folowing the lateral overgrowth of n-doped GaN underlayer, which is overgrown with a single InGaN QW. The structure is completed with a magnesium doped ptype cap layer consisting of an AlGaN electron blocking layer and a p-GaN shell. Figure 1(a) shows a schematic of the internal structure of an individual nanowire in the LED. The NWs consist of a base part, which has the form of a hexagonal cylinder defined by m-planes of about 1 µm in height and of a pyramidal top part defined by semi-polar facets. The NW ensemble presents a high morphological wire-to-wire homogeneity as seen from the SEM image of Figure 1(b). However, due to SiN mask imperfections, some of the NWs in the pattern are missing (e.g. an empty place can be seen in the right part of Figure 1(b)). The quantity of missing wires amounts to ~5 -8%, no other morphological defects were observed. These pattern defects are used in the present work to facilitate correlation between different maps as discussed in the following.

The NW structure is confirmed by scanning transmission electron microscopy (STEM) as illustrated in Figure 1(c). The growth on the semipolar plane is almost inhibited (cf. Figure 1(d)), so that the thickness of the InGaN QW on the pyramidal facets is below 1 nm and it does not patriciate in the EL emission [START_REF] Zhang | Color control of nanowire InGaN/GaN light emitting diodes by post-growth treatment[END_REF]. However, an indium-rich region is formed at the junction between the m-plane and the semi-polar plane, as confirmed by the Energy-dispersive X-ray diffraction (EDX) measurements shown in Figure 1(d).

In order to test the nanowire array, the sample was processed into standard 300×300 µm 2 size LED mesas. The Indium tin oxide (ITO)-covered mesas were defined by optical lithography and a metal frame (Ti(10 nm)/Au(150 nm)) was deposited on the top surface. To obtain the bottom contact, Inductively Coupled Plasma Etching (ICP) etching was used to remove partially the nanowires in order to access the n-GaN buffer layer from the top where a Ti(10 nm)/Al(30 nm)/Ti(10 nm)/Au(150 nm) metallization was applied.

Under operating condition, the LED produces luminescence starting from a bias of 3.3 V. An example of the EL spectrum and the corresponding optical image of the LED is shown in Figure 1(e)) for electrical power of 1.1 mW. Similarly to our previous report [START_REF] Zhang | Color control of nanowire InGaN/GaN light emitting diodes by post-growth treatment[END_REF], the spectrum presents two contributions: the high-energy peak at 2.6 eV (477 nm) arises from the m-plane QW and the low-energy peak at 2.4 eV (517 nm) is attributed to the emission from the indium-rich region at the m-plane/semi-polar plane junction. For further details on the electrical and optical properties, please refer to our previous publications [START_REF] Tchernycheva | InGaN/GaN Core-Shell Single Nanowire Light Emitting Diodes with Graphene-Based[END_REF][START_REF] Tchernycheva | Core-shell InGaN/GaN nanowire light emitting diodes analyzed by electron beam induced current microscopy and cathodoluminescence mapping[END_REF][START_REF] Zhang | Color control of nanowire InGaN/GaN light emitting diodes by post-growth treatment[END_REF]. In addition, the EL map shows that only a small percentage of NWs are emitting light in operation conditions. Based on high magnification images (Figure 8(a)), this percentage is estimated to be in the range of 20%-28%, depending on the applied external bias (3.3-5.0 V). The spotty nature of the EL map may have different origins such as the difference in the electrical properties changing the local current injection or the wire-to-wire variation of material properties (e.g. some wiredependent non-radiative defects affecting the internal quantum efficiency). In order to understand which factor impacts the most the homogeneity of the array, we performed a multi-technique analysis on the same area by means of EBIC, EL and CL, as described in the following.

Cross-sectional EBIC characterization of individual cleaved nanowires

First, the electrical properties of individual NWs are studied by cleaving the vertical structures as shown in the SEM image of Figure 2(a). The EBIC maps are acquired at room temperature in a Hitachi SU8000 SEM controlled by a Gatan DigiScan system as described in [START_REF] Lavenus | Experimental and theoretical analysis of transport properties of core-shell wire light emitting diodes probed by electron beam induced current microscopy[END_REF]. The experiments are performed under 10 kV -15 kV acceleration voltage and 130 -200 pA impinging current.

As expected, the majority of the cleaved NWs presented an EBIC signal at the p-n junction position in agreement with the previous observations on similar structures [START_REF] Zhang | Color control of nanowire InGaN/GaN light emitting diodes by post-growth treatment[END_REF]. However, several cleaved NWs showed a different behaviour: first, a weak signal appeared at the interface between the n-GaN core and the n-GaN underlayer. An example of this abnormal EBIC signal is shown in Figure 2(b), where it can be well observed as a narrow stripe in the bottom part of the NW, which gradually broadens toward the upper part. Under electron beam irradiation of the wire for about 10 min, the EBIC map was modified: the core/underlayer signal disappeared and transformed into a conventional signal from the p-n junction (Figure 2(c)). The abnormal EBIC signal was observed in several wires. In particular, NWs cleaved almost perpendicular to the c axis also presented this behaviour (cf. Figure 2(d) and (e)). Top view EBIC measurements on NW arrays discussed further have also evidenced the abnormal EBIC signal allowing to obtain a statistical information about the frequency of its occurrence. The transformation of the abnormal signal into the p-n junction signal was permanent: the EBIC activity at the core/underlayer interface could be observed only during first minutes of exposure. When observing the same NW after several hours in the dark, only the p-n junction signal was present.

This kind of behaviour has been previously reported in core/shell NW LEDs [START_REF] Tchernycheva | Core-shell InGaN/GaN nanowire light emitting diodes analyzed by electron beam induced current microscopy and cathodoluminescence mapping[END_REF]. It was tentatively ascribed to a parasitic potential barrier at the core/underlayer interface. One possible origin of this potential barrier may be related to a strong silicon doping of the n-GaN core.

Indeed, in 2D GaN layers, silicon was identified as an antisurfactact which may induce partial masking of the GaN growth surface by Si-N bonds [START_REF] Tanaka | Anti-Surfactant in III-Nitride Epitaxy -Quantum Dot Formation and Dislocation Termination[END_REF] or by Si3N4 islands [START_REF] Rosa | Adsorption and incorporation of silicon at GaN(0001) surfaces[END_REF] or even by a SiGaN 3 monolayer [START_REF] Markurt | Blocking growth by an electrically active subsurface layer: the effect of Si as an antisurfactant in the growth of GaN[END_REF]. In the NW case, for heavily Si-doped GaN NWs grown by MOVPE (doping exceeding 10 20 cm -3 ), a SiN x or SiGa x N y layer on the lateral surface was observed inhibiting the radial growth [START_REF] Markurt | Blocking growth by an electrically active subsurface layer: the effect of Si as an antisurfactant in the growth of GaN[END_REF][START_REF] Eymery | Process for Catalyst-Free Selective Growth on a Semiconductor Structure Patent[END_REF]An ultra-thin contamination layer was also reported in core-shell multi QW wires [START_REF] Mandl | Group III nitride core-shell nano-and microrods for optoelectronic applications[END_REF][START_REF] Yi | Cathodoluminescence and scanning transmission electron microscopy study of InGaN/GaN quantum wells in core-shell GaN nanowires[END_REF][START_REF] Durand | M-Plane GaN/InAlN Multiple Quantum Wells in Core-Shell Wire Structure for UV Emission[END_REF] and it was shown by EDX analyses to be related to Si-enrichment [START_REF] Tessarek | The Role of Si during the Growth of GaN Micro-and Nanorods[END_REF][START_REF] Ren | Nanoscopic insights into the effect of silicon on core-shell InGaN/GaN nanorods: Luminescence, composition, and structure[END_REF][START_REF] Tessarek | Understanding GaN/InGaN core-shell growth towards high quality factor whispering gallery modes from non-polar InGaN quantum wells on GaN rods[END_REF]. This unintentional thin SiGaxNy layer can be due to Si segragation at the surface [START_REF] Kapoor | Role of Underlayer for Efficient Core-Shell InGaN QWs Grown on m-plane GaN Wire Sidewalls[END_REF].

Although the targeted doping level in the n-GaN core part of the present LED was not exceeding ∼1×10 19 cm -3 , we suspect that partial masking of the lateral surface may take place and could be responsible for the formation of a potential barrier at the core / underlayer interface.

Since the resistance of this interface layer may vary from wire to wire, its impact on the homogeneity of the light emission in the array can be significant. Namely, it can be the cause of the low NW EL yield observed on the array LED. In order to verify the impact of the potential barrier on the LED properties, two comparative analyses have been performed: EBIC/CL and EBIC/EL.

EBIC/Cathodoluminescence correlated analyses

Top view CL mapping has been performed on the NW array to analyse the homogeneity of the optical properties from wire to wire. In CL mapping, a spectrum is recorded for each position of the electron beam, hence building a hyperspectral map. By integrating the intensity of the luminescence linked to each pixel in different wavelength ranges, it is possible to obtain spectrally filtered maps [START_REF] Piazza | Investigation of GaN nanowires containing AlN/GaN multiple quantum discs by EBIC and CL techniques[END_REF][START_REF] Chen | Determination of n-type doping level in single GaAs nanowires by cathodoluminescence[END_REF][START_REF] Zagonel | Nanometer scale spectral imaging of quantum emitters in nanowires and its correlation to their atomically resolved structure[END_REF]. The measurements were performed with an Attolight "Chronos" CL microscope with an acceleration voltage of 10 kV and the spectra were recorded on an Andor Newton CCD camera with a Horiba dispersive spectrometer using a 150 grooves/mm grating. Light is collected through an achromatic reflective objective with a numerical aperture of 0.7. The set-up is also equipped with an EBIC module allowing to simultaneously record the CL and EBIC maps.

To optimize the top view measurements, the ITO was removed from the top part of the NWs using a wet etch and the uncovered surface of the p-GaN shell was exposed to a CHF3-based plasma to degrade the conductivity of the pdoped layer on top. This process allows for efficient excitation with the electron beam and at the same time preserves the full NW structure including the top part. To ascertain that the additional processing steps do not damaged the LED, its current-voltage curve and the EL spectra were measured as described in the supplementary information (cf. Figure 1S).

First, a CL map of a large area from the LED was recorded. Figure 3 displays the SEM image (a), the corresponding panchromatic CL map (b) and the CL spectrum recorded for the whole area (c). Contrary to the spotty EL, the CL map is quite homogeneous. Indeed, each NW produces a CL signal with relative wire-to-wire intensity fluctuations below 30%, proving that there is no major difference in the non-radiative recombination from wire to wire. Correlated EBIC / CL analyses were then performed at higher magnification as shown in Figure 3(d)-(g) (all measurements were performed under short circuit conditions). All NWs in this region have identical morphology (Figure 3(d)) and produce similar CL intensity (Fig. 3f). However, the EBIC map (Figure 3(e)) shows variations of the induced current. In particular, some NWs produce a weak signal located closer to the NW centre (which is referred to as "abnormal") which arises from the potential barrier at the core/underlayer interface, as it will be discussed in more detail in the next section. The CL properties of NWs with "normal" and "abnormal" EBIC signal were compared by acquiring the corresponding CL spectra. Two nanowires were selected highlighted by pink and red arrows, which exhibit respectively a normal EBIC signal (labelled as "1", in pink) and an abnormal signal (labelled as "2", in red). Despite the apparent difference in the charge collection, the NWs 1 and 2 produce luminescence of a similar intensity and spectral shape (Figure 3(g)). The spectral analysis of the optical emission shows that for all the NWs irrespectively of their EBIC signature the luminescence is characterized by the GaN near band edge emission at 360 nm, the emission from the m-plane QW at 465 nm and the green emission from the indium-rich region at 510 nm (Figure 3(c) and Figure 4(d)). The peak positions are in good agreement with the EL spectrum previously shown (Figure 1(e)).

The correlated CL/EBIC analyses prove that the optical properties of the array are highly homogeneous, while the charge collection properties are different. This indicates that the observed spotty EL is not related to the wire-to-wire fluctuations of the optical properties, but are induced by the difference in electrical properties.

Top view EBIC mapping on "top-free" nanowires

The EL and CL analyses both show the parasitic green emission. We have previously demonstrated that it can be suppressed by dry etching of the top part of the NWs [START_REF] Zhang | Color control of nanowire InGaN/GaN light emitting diodes by post-growth treatment[END_REF]. The current-voltage (I(V)) curves and EL spectra of the LED after etching were assessed as discussed in the supplementary information (cf. Figure 1S). Although etching procedure preserves the EL, it may affect the electrical properties of the NWs and alter the internal potential barrier, object of our investigation. For this reason, top view EBIC measurements were performed on the NW arrays both before and after etching of the top semi-polar facets of the NWs. The SEM images and the corresponding EBIC maps of ITO-free full NWs are displayed in Figures 4(a)-(d), while SEM images and corresponding EBIC maps of etched "top-free" NWs are displayed in Figure 4(e)-(h).

In Figure 4(b) the majority of the NWs present a strong EBIC signal with a hexagonal shape (side of the hexagon ~650 nm) corresponding to charge collection from the p-n junction. However, certain wires exhibit a weak EBIC signal which is located close to the center of the wire. Figure 4(d) shows a detail of the image where the two types of nanowires with the "normal" and "abnormal" EBIC signal are present. For the sake of clarity, all the NWs with an abnormal EBIC signal are marked by red arrows.

The abnormal signal can be better defined in the array where the tops of the nanowires were etched away (Figure 4(e)-(h)). In this case, there is no possible contribution from the semipolar facets. The regular EBIC signal in the "top-free" NWs presents a well-defined hexagonal shape (Figure 4(f)), whereas the abnormal signal manifests itself as a weak circular contrast (cf. Figure 4(h) for a high magnification image). The statistical analysis over several mesas showed that in both cases, there are 7-10% of nanowires with "abnormal" EBIC signal, indicating that the top NW etching does not affect the properties of the internal potential barrier causing the abnormal signal.

It should be noted that the "abnormal" EBIC signal in topview maps in the two cases has a larger diameter (~ 700-850 nm) than that of the nanowire core (~ 250-300 nm). This is contrary to the case of the nanowires cleaved close to their base, where the "abnormal" signal follows the core boundary (cf. Figure 2(e)). We believe that this is due to the projection effect, namely to the broadening due to the interaction volume and the inhomogeneity of the signal along the wire axis. Indeed, for the situation shown in Figure 2(c), the widening of the top view signal can arise from the upper part of the nanowire.

EBIC/Electroluminescence analyses

For all the investigated configurations of the top LED contact the EL was strongly inhomogeneous, i.e. only 20 -28 % of NWs were producing electroluminescence. We note that the percentage of the NWs producing the EL was not degraded by the additional processing steps of ITO removal and top NW etching.

To evaluate the impact of the core/underlayer interface on the electro-optical behavior of the LED, top view EBIC maps were correlated with the optical microscopy images of the "top-free" LED under operation. The selected area was viewed in an optical microscope to visualize the EL from individual nanowires, then the same region was mapped by the EBIC microscopy. This order of experiments was chosen to avoid any possible modifications of the EL due to the electron beam exposure.

Figure 5 shows EL maps, SEM images and EBIC maps for the same area of the LED close to a metal contact frame. To establish the correlation between optical and scanning electron images, we have used the pattern defects (giving missing NWs), which are clearly seen in dark optical microscope images as well as in SEM images. Figure 5(a) shows a large area optical image whereas a higher magnification map of Figure 5(b) corresponds to the region explored in EBIC. As seen, there is a strong inhomogeneity of the EL from wire to wire: the EL intensity strongly fluctuates and some nanowires do not show any emission. The SEM image (Figure 5(c)) of the analysed region shows a regular nanowire morphology without any apparent pathology.

The EL maps in Figures 5(a) and (b) were recorded under low injection conditions (the applied bias of 4.0 V and the injected current density of 0.9 A/cm 2 ). To take into account a possible influence of the injection, similar analyses were performed for higher injection (6.2 V, 4.0 A/cm 2 ) as shown in the supplementary materials (Figure S2). The EL inhomogeneity is found to be almost independent from the injection conditions: the percentage of the active nanowires (i.e. the wires generating electroluminescence) at low injection is ~ 20-25% and it increases only marginally at higher injection to ~ 23-28%.

The EBIC image (Figure 5(d)) presents some wire-to-wire fluctuations of the induced current, however it shows a current generation much more homogeneous than the EL signal. In particular, there is only ~ 10% of nanowires that generate weak or no induced current.

The EBIC map allows to identify the nanowires with "abnormal" EBIC signal (marked with red arrows in Figure 5(d)), which can be then found on the EL map (indicated by red crosses in Figure 5(b)). A systematic analysis of the EL images evidences that each wire with "abnormal" EBIC behaviour shows no EL signal. However, the reverse is not true: a wire having a "normal" EBIC signal does not necessarily show EL.

In general, the number of nanowires producing EL (~ 20-28 %) is well below the number of nanowires having a "normal" EBIC signal ( ≥90 %).

The significant difference in the number of inactive nanowires in EBIC and EL experiments may stem from the difference in the operating conditions in the two measurements. This can be understood based on a simple electrical model of each individual NW as a diode with its individual series resistance. This series resistance represents the internal barrier at the core/underlayer interface. To illustrate the influence of this parameter on the EL and EBIC, simulated I(V) curves with varying access resistance are shown in Figure 5(e), where the EBIC and EL working conditions are highlighted. During EBIC measurements the induced current is collected in short circuit conditions, and therefore the signal dependence on the series resistance of the system is weak in a wide range of resistances. On the contrary, for EL the device is forward biased above the diode opening voltage and, in these conditions, the injected current strongly depends on the series resistance. These considerations applied to the EBIC and EL maps predict, that the EL variations due to internal potential barrier inhomogeneity are expected to be much stronger that the EBIC variations. To inhibit EBIC, a very high series resistance is required.

Front n-contact LED

It is important to notice, that in all the EL experiments discussed so far, the electrons were injected from the n-GaN NW core. In other terms, the electrical path to inject carriers in the QWs and to obtain EL involve the interface between the n-doped core grown in the axial direction and the n-doped GaN underlayer grown in the radial direction, where the potential barrier has been observed. The above-described analyses indicate that the LED operation is perturbed by the potential barrier at the core/underlayer interface, which is at the origin of the injection inhomogeneity. To prove this hypothesis and at the same time to overcome this problem, we developed an alternative LED architecture in which electrons are injected directly in the n-GaN underlayer without crossing the core/underlayer interface ("front-n-contact" architecture). A new LED was processed from a different piece of the same growth wafer.

The fabrication steps of this "front n-contact" LED are illustrated in Figure 6. First, hole-injecting contact is defined to the NW sidewalls. A 200 nm ITO layer is deposited on the entire surface (Figure 6 The ITO layer forming the contact on the p-GaN shell is metallized on the sample edge to allow for the wire bonding (Figure 6(d)). The sample is then encapsulated into a hydrogen silsesquioxane (HSQ) layer to electrically insulate the contact to p-GaN shells (Figure 6(e)). Excess HSQ is etched with RIE. To contact the n-GaN underlayer, 300 nm ITO layer is deposited on top and patterned by lithography and lift-off processes. Finally, a Ti / Au metallization and lift-off are used to form the top metal contact for the bonding (Figure 6(f)). Panel (g) illustrates the final structure of the front n-contact LED, where the holes are injected through the ITO at lower part of the nanowires while the electrons are injected through the top ITO contact directly in the upper n-GaN underlayer (instead of being injected through the core). First, the electrical properties of the device were tested. We note that a high level of leakage is observed for reverse bias as shown in Figure 7(a). It is possible that the leakage problem arises from a parasitic conduction between the first ITO layer and the n-GaN underlayer. Indeed, after the ICP etching of the upper part of the nanowire, the ITO surpasses the edge of the bottom part of p-GaN shell and it is not mechanically supported by the nanowire. At the time of HSQ encapsulation, the ITO layer can break and touch the n-GaN layer creating local short circuits. This leakage is also present under forward bias, which results in a much higher operation current density in comparison to the previously described EL measurements of the conventional LED.

Despite the high leakage current, the front n-contact LED produces EL starting from an applied voltage of 3.3 V. This number is the same as for LEDs with the electron injection through the n-GaN core. The EL spectra for an increasing current injection are shown in Figure 7(b). As expected, the emission of the front n-contact LED presents a single peak at 2.6 eV (476 nm), which remains stable with increasing current.

Finally, the homogeneity of the EL maps is compared between the standard LEDs where the electrons injected through the core/underlayer interface and the front n-contact LED where the electrons are directly injected in the n-GaN underlayer. As for the previous devices, a statistical analysis is performed on the front n-contact LED operated under a bias of 4 V. The examples of the EL maps for the LEDs discussed in this paper are shown in Figure 8(a)-(c), respectively corresponding to the entire NWs with a full ITO contact (a), to "top-free" NWs (b) and to "front-n-contact" NWs (c). Comparing these maps, a significant increase in the NWs yield can be observe, going from 19%-22% for the first two cases up to 65% in the front-n-contact device. This result demonstrates that the electron injection directly in the n-GaN underlayer strongly improves the homogeneity of the EL, i.e. that the presence of a potential barrier at the core/underlayer interface increases the access resistance when injecting electrons into the active region [START_REF] Tchernycheva | Core-shell InGaN/GaN nanowire light emitting diodes analyzed by electron beam induced current microscopy and cathodoluminescence mapping[END_REF].

Contextually, it can also be noted that front-n-contact architecture is effectively reducing the homogeneity issue at a larger scale, as shown on the EL map taken with a lower magnification in Figure 8(d). It is possible that the EL homogeneity may be further improved by optimizing the fabrication processes. Indeed, the map of Figure 8(d) shows that most of the inactive nanowires are gathered together in localized areas, which may indicate process problems in these regions.

Conclusions

In the present work, we analysed the injection inhomogeneity in core-shell InGaN/GaN NW LEDs. We concluded that in the investigated case it is mainly induced by a potential barrier at the interface between the nanowire n-GaN core and the radially grown n-GaN layer. CL analyses showed that the optical properties of the nanowires are homogeneous. The comparison of EL and EBIC maps on the same region offered a proof that the potential barrier hindered the homogeneous charge carrier injection in different wires and reduced the efficiency of the nanowire LED array. In order to avoid injection through a potential barrier at the core/underlayer interface, we changed the contact architecture and fabricated the "front contacted" LED, for which the electrons are injected directly into the n-GaN underlayer. The "front contacted" LED presented an increased yield of the NWs producing the EL (the yeild was increased from ~20% for standard contact to 65% for front contact). This proves that the front electron injection allows to reduce the impact of the potential barrier at the core/underlayer interface.

Nanotechnology XX (2020) XXXXXX H Zhang et al

Figure 1 .

 1 Figure 1. (a) Schematic of a cleaved NW showing its internal structure. (b) 30° tilted SEM image of the ITO-covered NWs. (c) Cross-sectional STEM image of a NW and (d) the EDX map showing the indium rich region in the QW at the junction between the m-plane and the semi-polar plane. (e) Electroluminescence spectrum of the LED. The insets show a schematic representation of the mesa and an optical microscope photo of the LED under operation.

Figure 2 .

 2 Figure 2. (a) SEM image at Vacc = 10 kV of a nanowire cleaved along its axis. (b) Corresponding EBIC map before transformation. The white arrows show the position of the abnormal signal. (c) EBIC map of the same region after transformation. (d) SEM image at Vacc = 15 kV of a nanowire cleaved almost perpendicularly to its axis. (e) Corresponding EBIC map showing the abnormal signal at the core / underlayer interface.

Figure 3 :

 3 Figure 3: (a) Top-view SEM image and (b) the corresponding panchromatic CL map of the array. (c) CL spectrum of the whole area shown in panel (a). (d) SEM image, (e) EBIC map of a selected area from the array, (f) CL map integrated between 430 and 550 nm. The area contains NWs with a normal and abnormal EBIC signal shown in the inset and labelled as "1" and "2", respectively). (g) CL spectra of the NWs in positions 1 and 2, respectively.

Figure 4 .

 4 Figure 4. (a) SEM image and (b) the corresponding EBIC map of the LED with the ITO etch-back contact on top of nanowires. The crosses indicate the position of missing wires, the red arrows indicate nanowires with "abnormal" EBIC signal. (c) and (d) the same sample as (a) and (b) under higher magnification showing a detail of the "abnormal" signal. (e) SEM image and (f) the corresponding EBIC image of the LED with ICP etching of the NW tops. (g) and (h)the same sample as (e) and (f) under higher magnification showing a detail of the "abnormal" signal. All pictures and maps are acquired at Vacc = 10 kV.

Figure 5 .

 5 Figure 5. EL map of a region of the LED with etched NW tops for an applied voltage of 4 V and injected current density of 0.9 A/cm2. (b) Higher-magnification image corresponding to the rectangle in panel (a). The pink crosses indicate the missing nanowires, the little red crosses indicate the position of nanowires producing an abnormal EBIC signal. (c) SEM image of the region shown with a rectangle in panel (a). The pink crosses show missing nanowires. (c) EBIC map corresponding to the panel (c). The red arrows point to the nanowires with abnormal EBIC signal. (e) Numerical simulation of the I(V) curve of a diode under excitation for different series resistances. The series resistance plays no significant role when measuring the EBIC current, while it strongly affects the carrier injection under forward bias and thus the EL intensity.

  (a)) followed by a partial ITO etching from the upper part of the nanowires (Figure 6(b)). Then the p-GaN layer is ICP-etched on the semipolar facets and on the upper part of the m-plane facets leaving the exposed n-GaN underlayer (Figure 6(c)).

Figure 6 .

 6 Figure 6. Font n-contact LED manufacturing steps: (a) deposition of the ITO for the hole injection. (b) partial etching of the ITO. (c) ICP etching of the p-GaN shell. (d) definition of a metal contact to the ITO layer. (e) encapsulation with the HSQ and its partial etching. (f) lithography and deposition of the ITO and the metal contact for the electron injection. (d) Schematic of the final device.

Figure 7 .

 7 Figure 7. (a) J-V curve of the front n-contact LED. (b) EL spectra of the front n-contact LED for increasing injection.

Figure 8 .

 8 Figure 8. EL maps for LED processes with (a) injection of electrons through the core and a conformal top ITO contact. (b) injection of electrons through the core and the etched upper part of nanowires. (c) front n-contact LED and electrons injected directly in the n-GaN underlayer. (d) Intermediate magnification of EL map showing emission fluctuations over a large area.

Acknowledgements

This work has been partially financially supported by EU ERC project 'NanoHarvest' (grant no. 639052), by the Region Ile-de-France in the framework of C'Nano IdF (the nanoscience competence center of the Paris region), by the European Union (FEDER 2007-2013), and by the Labex GANEX (grant no. ANR-11-LABX-0014) as part of the "Investissements d'Avenir" program managed by the French National Research Agency (ANR). The authors acknowledge O. Kryliuk for providing nanowire samples.


Moreno Andreatta

On Two Computational Models of the Pitch-Rhythm Correspondence A Focus on Milton Babbitt's and Iannis Xenakis's Theoretical Constructions According to a well-established tradition in music theory and systematic musicology, pitch space and the rhythmic domain share the same mathematical underlying structure, i.e. they are ›isomorphic‹. This algebraic relation allows the transfer of pitch-based operations and transformations into the rhythmic domain by interpreting the octave identification as a periodicity in the time axis and the pitch interval as a temporal distance between two successive onsets. As claimed by Godfried Toussaint in his book devoted to the geometric aspects of musical rhythm, »It is well known that there exists an isomorphic relation between pitch and rhythm, which several authors have pointed out from time to time.« 1 Therefore, he observes, there is no need to develop further this relationship, which explains the fact that there is no theoretical statement about the pitch-rhythm correspondence in the entire book. 2 Historically, although one may find some attempts at establishing formal relationships between the pitch and time domains in the theoretical writings by 1 Toussaint 2013, p. xiii. 2 This assumption is far from being universally accepted, as the following quotation by music theorist Justin London clearly shows. In an article entitled Some Non-Isomorphisms between Pitch and Time, he observes that pitch space and the rhythmic domain are essentially nonisomorphic. The lack of isomorphism derives from the observation that »there are no temporal analogs to octave and enharmonic equivalence and that there are no tonal analogs to various limits on our temporal perception and acuity« (London 2002, p. 128). This critical perspective on the pitch-time correspondence offers a nice counterpoint to the main constructions we will discuss in detail in the paper. Our approach follows Jeff Pressing's assumption of the existence of a cognitive isomorphism between pitch and rhythm (Pressing 1983), to which we add an algebraic perspective derived by Milton Babbitt's and Iannis Xenakis's theoretical construction.

For a more advanced mathematical model on the Pitch / Rhythm correspondence, see Anatol Vieru's treatise Cartea modurilor (1980) and its algebraic formalization by Dan Vuza (1985).

Joseph Schillinger (1941), the pitch-rhythm correspondence becomes explicitly an algebraic isomorphism within Milton Babbitt's formalization of the twelvetone system. Babbitt's PhD thesis, submitted in 1946 and finally approved in 1992, contains the germs of this formal correspondence, which is rooted in the set-theoretical as well as permutational character of the system.3 But the idea of a technical correspondence between the pitch and the time domains was formalized for the first time in the early 1960s in an article entitled Twelve-Tone Rhythmic Structure and the Electronic Medium. 4 In order to approach the time dimension from a pitch-theoretical perspective, Babbitt proposes two concepts that make use, in different ways, of the »immanently temporal nature of the twelve-tone pitch-class system.«5 These two concepts express some axiomatic properties that are valid, according to Babbitt, for all temporal relations between two kinds of musical structure: the durational row and the time-point system.

The durational row

Babbit uses this technique in his Three compositions for piano (1947), which is, from the American perspective, the first piece belonging to integral serialism. 6 Example 1 shows the rhythmic pattern P used by the composer in his three compositions. The numerical values assigned to the durational row in Example 1 assume the sixteenth note as the minimal durational unit.

On Two Computational Models of the Pitch-Rhythm Correspondence

Example 1: Rhythmic pattern of M. Babbitt's Three compositions for piano (1947) It is now possible to apply to this rhythmic pattern the three usual twelve-tone operations -that is, inversion, retrograde and inverted retrograde. In doing so, one has to attend to the way in which the analogy is constructed between the inversion operation in the pitch-domain and the rhythmic inversion applied to the durational row. The major difference concerns the value according to which the complement (that is, the difference) is calculated (modulo 5 instead of the usual complement modulo 12, as in the pitch domain). The durational row P=(1, 4, 3, 2) is therefore transformed in the structure (n-1, n-4, n-3, n-2) with n=5, which leads to the inversion I(P) = (4, 1, 2, 3). As in the case of the traditional ›twelve-tone group‹, the operation P plays the role of the identity, so that one can simply write I instead of I(P). Analogously, the retrograde of a durational row reverses the order of their elements, which leads to R(P)=(2, 3, 4, 1), and to the inversional retrograde RI(P) = (3, 2, 1, 4). The four ›twelve-tone‹ forms of the rhythmic pattern P are represented in Example 2.

Example 2: The four ›twelve-tone‹ forms of a rhythmic pattern

As in the case of pitch space, the same algebraic structure (formally known as ›the Klein group‹) acts by permuting the elements of the original rhythmic pattern. It therefore represents an algebraic tool whose structural component, with respect to combinatorial techniques used by composers of the same period, represents a significant progress in the music-theoretical domain. 7 Nevertheless, as pointed out by several music theorists, the durational row technique is not a satisfactory answer to the problem of pitch-rhythm combinatorics since the traditional twelve-tone operations destroy the perceptual character of the temporal intervals. This simple consideration would be enough to justify Babbitt's elaboration of a second concept that will remain one of his most original inventions: the time-point system.

The time-point system

This concept takes its origin from the interpretation of the intervallic distances between different pitches of a twelve-tone row in terms of durations. In the explication given by Babbitt, a »pitch number is interpretable as the point of initiation of a temporal event, that is, as a time-point number.«8 In order to ›project‹ a series of pitches onto the rhythmic domain, Babbitt uses the concept of ›modulus‹, a time-span divided into 12 minimal units representing the twelve classes of time-points.

A juxtaposition of several moduli represents the temporal grid onto which the twelve pitch-classes will be projected. A twelve-tone series can therefore be interpreted rhythmically in different ways, since a pitch-class integer can be projected in different ways within the temporal grid.9 The following musical example shows how a twelve-tone row can be realized rhythmically by taking the thirty-second note as a minimal durational unit.

Example 3: The rhythmic realization of a twelve-tone row via the technique of the timepoint system Note that the time-point system enables the composer to establish a much more natural and perceptually intuitive correspondence between the twelve-tone operations in the pitch realm and their rhythmic analogs. To these considerations, Babbitt adds a remark concerning the possibility of applying combinatoriality to the time-point system, which enables the construction of sometimes very complex temporal patterns, such as rhythmic canons by inversion whose »total rhythmic progression is in disjunct aggregates.« 10 From a theoretical point of view, Milton Babbitt enumerates eleven qualitative properties based on the order relation (indicated with the mathematical symbol < ). In contrast to the modular congruence relation, upon which the twelve-tone system is based, this relation is not, mathematically speaking, an equivalence relation, since neither the reflexive property nor the symmetric properties hold. (In other words, there is no musical event x for which x< x and the relation x< y never implies that y< x. The only property that is shared with the equivalence relation is transitivity, since given three musical events x, y, and z, the two relations x< y and y< z imply x< z.)

The axiomatics proposed by Milton Babbitt reflects a major concern of many music theorists of the 1960s, in particular Michael Kassler (1967) and Benjamin Boretz (1969/1995), who also abandon mathematical equivalence in order to focus on order relations between musical structures. This is exactly Iannis Xenakis's starting point for a musical axiomatics that is inspired by Giuseppe Peano's description of natural numbers and that directly leads to Sieve Theory.11

The pitch-time correspondence in Xenakis's Sieve Theory

In order to understand the algebraic nature of pitch-time correspondence in Xenakis's writings, it is important to provide a short introduction to his algebraic thought. Almost contemporaneously with the emergence of an algebraic perspective in the American set-theoretical tradition, there appear in the 1960s the first signs of Xenakis's consciousness of the algebraic nature of the musical system.

There is an evident ›conceptual‹ proximity between Xenakis's and Babbitt's music-theoretical thought.12 This proximity is particularly striking if one considers the insistence with which Babbitt speaks about the set-theoretical, and at the same time, algebraic character of musical structures. As in the case of Babbitt, Xenakis postulates in his writings the existence of an interplay between settheoretical constructions and algebraic formalization that resides in the abstract nature of modern mathematics. This reflection finds a well-defined place in his book entitled Musiques Formelles (1963/1981), where the search for »new formal principles in music composition« is deeply related to the concepts of abstraction and formalization in mathematics. Following Hilbert's and Peano's axiomatic approach in mathematics, Xenakis proposes to define a new category in contemporary musical thought: ›symbolic music‹. In fact, according to Xenakis, »formalization and axiomatization constitute [ . . . ] a procedural guide, better suited to modern thought.«13 This position is very close to what Milton Babbitt proposes within the American set-theoretical tradition, especially with respect to the relationship between scientific language and music theory, most notably in his article The Structure and the Function of Music Theory. 14 The central notion around which Xenakis defends the axiomatic approach is, as in Babbitt's case, that of the interval. But in contrast to the American music-theorist, Xenakis does not need to use the notion of congruence modulo 12 as a necessary condition for defining the concept of interval. According to Xenakis, the equal-tempered system is just one particular case with respect to a much more general phenomenon that also concerns other musical dimensions such as intensities and durations. This clearly why Xenakis's Sieve Theory, which would be conceived some years later, has a much more generalizing power than Babbitt's interval theory. As stated by Xenakis:

In music, the question of symmetries (spatial identities) or of periodicities (identities in time) plays a fundamental role at all levels, from the sample in sound synthesis by computer to the architecture of a piece. It is thus necessary to formulate a theory permitting the construction of symmetries which are as complex as one might want, and inversely, to retrieve from a given series of events or objects in space or time the symmetries that constitute the series. We shall call these series »sieves«. 15 Interestingly, the point of departure for Xenakis's Sieve Theory is the same as Babbitt's, having to do with the mathematical concept of ›total order‹; such an order obliges any couple of elements a and b of a set to have either a< b or b< a. This is the case, for example, of the set of natural numbers N, of positive or negative integers Z, of rational numbers Q, or real numbers R, with the usual ›total order‹ relation (indicated with the symbol < ). Note that Xenakis makes no attempt to explicitly develop a general algebraic theory based on ordered structures, although one may find some suggestions on how to generalize the sieve-theoretical constructions with the help of partial ordered structures such as trees and networks. As pointed out in the revised and enlarged English-language version of his doctoral thesis entitled Arts / Sciences: Alloys, Sieve Theory is very general since it can be applied to any musical parameter possessing a total order structure (such as intensities, densities of degrees of order / disorder, attacks, etc.). 16 Xenakis also predicts the explosion of interest and applications of this approach in the following years thanks to its computational character, which makes the theory suitable for computer-aided investigations. He also points out the necessary generalization of the total-ordered structures towards partial-ordered structures, such as those which are introduced in the study of timbre and that make use of lattices and networks.

History has validated Xenakis's predictions in two ways: first, through the development of computer-aided environments in which Sieve Theory has been implemented as a tool for computational music analysis and composition, and second, through the development of partially ordered structures in the last thirty years. In the first case, I would like to mention the use of Sieve Theory by two leading figures of French computational musicology, André Riotte and Marcel Mesnage. They both not only implemented Xenakis's original model in a computational environment, the Morphoscope,17 but they suggested how to generalize the initial pitch-theoretical model in order to obtain complex rhythmic structures. The generalization consists in taking as the underlying support space not only a ›regular‹ rhythmic space, but any kind of rhythmic pattern that will be ›sieved‹ in such a way that the different elementary sieves will produce complex results by the use of classical set-theoretical operations (such as union, intersection, or complementation). Example 4 shows a possible generalization of Xenakis's Sieve Theory starting with an irregular underlying rhythmic space (called the structure métrique).

Example 4: A rhythmic pattern obtained by the union of elementary sieves having as a support space a non-regular metric grid (quoted from the first volume of Riotte and Mesnage 2006, p. 83) Pieces making use of this generalized approach to Sieve Theory include La Bibliothèque de Babel (1985) by André Riotte, for reciting voice, two solo voices, two choirs, a quintet of brass instruments, percussion, and an orchestra of wind instruments; Color (1986) by Claudy Malherbe, for 18 instruments; and Partitionsgouffres (1986), an algorithmic piece by André Riotte for four percussionists. 18 As suggested earlier, Xenakis's introduction of total-order relations in the formalization of musical structure provides a first axiomatization of musical scales, which is nothing less than a translation into the musical domain of Peano's formalization of integer numbers. More precisely, Peano's axioms are translated into the musical domain with the help of three basic concepts: the origin (or stop), the set of elements D (or durations), and the relation ›successor of an element n‹, which is symbolized by the symbol n'. The six ›first propositions‹, as Xenakis calls them, take the following form: 19 As in the case of Milton Babbitt's durational row or time-point system, Xenakis's axiomatics is derived from an underlying formalization that originally applied to the pitch domain. Interestingly, if the set-up of Sieve Theory in the pitch domain goes back to the 1960s (Xenakis, 1965), the transfer of Peano's axiomatics to the rhythmic domain, at least from a theoretical point of view, only occurs in the 1980s. Surprisingly, the theoretical formalization is anticipated in several compositions by Xenakis, such as Persephassa (for a sextet of percussionists) or Psappha (for multi-percussion solo), dating respectively from 1969 and 1975, in which the rhythmic structures are explicitly obtained by application of sievetheoretical constructions in the rhythmic domain. The article in which Xenakis formalizes for the first time the pitch-rhythm correspondence with the tools belonging to Sieve Theory is Redécouvrir le temps (Xenakis 1988/1996), in which Xenakis's previous work on the categories ›on time‹/ ›outside-of-time‹ (en temps/hors temps) provides the general framework for understanding temporality. In Xenakis's words, »any temporal scheme [ . . . ] is an ›outside-of-time‹ representation of the temporal flow within which the phenomena take place.« 21 The of temporal structures allows for a better understanding of the notion of separability between temporal events, which can therefore be »assimilated to attacks-point within the time flow.«22 Note the similarity between Xenakis quotation and Milton Babbitt's time-point system, the only conceptual difference being Xenakis's rejection of the primacy of the octave. This makes Sieve Theory an extremely powerful tool for computational music analysis, as amply demonstrated by André Riotte and Marcel Mesnage in their theoretical and analytical writings. 23 A comparison of attacks-points naturally leads to the notion of distance, which becomes operational precisely through the use of Sieve Theory, provided that the sieve 1 0 is identified with the most elementary rhythmic notion, i.e. the regular rhythm. As in the case of pitch, by using the three main set-theoretical operations which are the union, the intersection, and the complement, »one can build very complex rhythmic architectures which can simulate the pseudoaleatoric distribution of points in a line, if the period is long enough.« 24 This clearly shows the possibility of recovering probabilistic and statistical music processes from a deterministic approach guided by set-theoretical construction, and therefore the primacy, from a theoretical point of view, of the category of ›Symbolic Music‹ with respect to the category of ›Stochastic Music‹. The pitchtime correspondence definitely plays a major role in this structural perspective.

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See Babbitt 1946[START_REF] Xenakis | Formalized Music[END_REF]

[START_REF] Babbitt | Twelve-Tone Rhythmic Structure and the Electronic Medium[END_REF]

See Babbitt 1962, p. 151.

The European perspective prefers to ›ignore‹ this historical antecedent and focus on the central place occupied by Olivier Messiaen and his total-serialized non-dodecaphonic piece Mode de valeurs et d'intensités (1949). This piece had indeed a considerable influence on a generation of composers as the germ of thought in Europe about the generalized series. Nevertheless, as was suggested some years ago, when a French-American Musicological Meeting was organized at IRCAM on the set-theoretical tradition from both American and European perspectives, there is a direct influence by Milton Babbitt on the very conception of this third piece from the Quatre études de rythme. In fact, Milton Babbitt attended Olivier Messiaen's composition classes at Tanglewood in 1948, that is, one year prior the creation of the piece Mode de valeurs et d'intensités, which explains why the first two Etudes de rythme were composed in the United States, as the frontispiece of the partition indicates. The reference »Darmstadt -1949« on the score of Mode de valeurs et d'intensités therefore confirms that the piece was indeed composed after Messiaen and Babbitt met. See[START_REF] Andreatta | Proceedings of the Symposium Around Set Theory: A French-American Musicological Meeting[END_REF] for a discussion of the set-theoretical tradition from a larger and trans-national perspective.

See Fritts 2000 for a short introduction to the concept of group and group actions in Babbitt's compositional techniques, and more generally, in 20 th century music.

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Babbitt 1962, p. 162.

See Mead 1994 for a comprehensive introduction to the music of Milton Babbitt and his timepoint system.

 10 Babbitt 1962, p. 170.

See the article by John Rahn entitled Logic, Set Theory, Music Theory[START_REF] Rahn | Logic, Set Theory, Music Theory[END_REF][START_REF] Rahn | Logic, Set Theory, Music Theory[END_REF]) for a different use of order relation in the formalization of music-theoretical properties.

See Andreatta 2003 for an account of such a ›conceptual‹ proximity starting from a mathematical investigation of algebraic methods in 20 th century music and musicology. See also Schaub 2009 for a critical evaluation of the relations between Xenakis's and Babbitt's theoretical, compositional, and analytical constructions.

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[START_REF] Xenakis | Musiques Formelles[END_REF]Xenakis /1981, p. 212, p. 212. The English translation is quoted fromXenakis 1992, p. 178.

[START_REF] Babbitt | The Structure and Function of Music Theory[END_REF]

Xenakis 1992, p. 268.

See Xenakis 1985, p. 108.

Several articles devoted to this programming language are contained in the two volumes collection of writings by André Riotte et Marcel Mesnage entitled Formalismes et modèles musicaux[START_REF] Riotte | Formalismes et modèles musicaux[END_REF].

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The origin (or stop) is an element of the set of durations D.

If stop n is a duration, then its successor n' is also an element of D.

If stops n and m are elements of D, then the respective successors n' and m' are identical if the two stops n and m are identical.20

If stop n is a duration, it will be different from stop O at the origin.

If elements belonging to D have a special property P, such that the origin also has it, and if, for every element n of D having this property its successor n' has it also, all the elements of D will have the property P.18 For a wider perspective on algorithmic composition, including sieve-theoretical techniques, see[START_REF] Andreatta | Musique algorithmique[END_REF]. 19 The list of six axioms is adapted from Xenakis's Formalized Music(Xenakis 1992, p. 195). 20 By »iff« (read »if and only if«) is denoted the necessary and sufficient condition for a property to be true.

[START_REF] Xenakis | Redécouvrir le temps[END_REF]Xenakis /1996, p. 40. , p. 40.

[START_REF] Xenakis | Redécouvrir le temps[END_REF]Xenakis /1996, p. 41. , p. 41.

[START_REF] Riotte | Formalismes et modèles musicaux[END_REF]

[START_REF] Xenakis | Redécouvrir le temps[END_REF]Xenakis /1996, p. 43. , p. 43.

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