D Pribat

C S Cojocaru

M Gowtham

B Marquardt

T Wade

J E Wegrowe

B S Kim

Carbon nanotube electronics / Électronique à nanotubes de carbone Organisation of carbon nanotubes and semiconductor nanowires using lateral alumina templates

Keywords: Carbon nanotubes, Silicon nanowires, Device applications, Controlled growth and organisation, Templates, Porous anodic alumina

Carbon nanotubes and semiconductor nanowires have been thoroughly studied for the future replacement of silicon-based complementary metal oxide semiconductor (CMOS) devices and circuits. However, the organisation of these nanomaterials in dense transistor arrays, where each device is capable of delivering drive currents comparable with those of their silicon counterparts is still a big challenge. Here, we present a novel approach to the organisation of carbon nanotubes and semiconductor nanowires, based on the use of porous lateral alumina templates obtained by the controlled anodic oxidation of aluminium thin films. We discuss the growth of nanomaterials inside the pores of such templates and show the feasibility of our approach. Our first results point to further work on controlling the synthesis of catalyst nanoparticles at the bottom of the pores, these particles being necessary to nucleate and sustain the growth of carbon nanotubes or semiconductor nanowires. .

Résumé

Organisation de nanotubes de carbones et de nanofils semiconducteurs en utilisant une membrane latérale en alumine nanoporeuse. Depuis quelques années, les nanotubes de carbone et les nanofils semiconducteurs sont étudiés pour le remplacement des composants CMOS à base de silicium. Des caractéristiques de transistors à effet de champ impressionnantes ont été obtenues en utilisant ces nanomatériaux. Le problème toutefois est d'organiser les nanotubes et les nanofils en réseaux de transistors avec des densités comparables à celles de leurs homologues en silicium. Nous présentons ici une approche nouvelle, dans laquelle les nanotubes et les nanofils sont organisés par croissance dans des membranes latérales nanoporeuses obtenues par oxydation anodique de films minces d'aluminium. Nous discutons la croissance des nanotubes et des nanofils dans les pores et démontrons la faisabilité de l'approche proposée. Nos premiers résultats indiquent que le contrôle du dépôt, au fond des pores, des nanoparticules catalytiques nécessaires à la nucléation et à la croissance des nanotubes et des nanofils est essentiel pour permettre l'obtention de réseaux denses et organisés, compatibles avec les standards de la microélectronique.

Introduction and context

As the scaling down of the silicon metal oxide (MOS) transistor is reaching physical limitations, researchers are concentrating on finding new materials or paradigms that could enable the semiconductor industry to continue following the pace of Moore's law. Within this context, single wall carbon nanotubes (SWNTs) have been thoroughly investigated over the past few years and devices based on individual specimens have shown breathtaking characteristics [START_REF] Dai | For recent reviews on CNT-based devices[END_REF]. Semiconductor nanowires (s-NWs) are another class of nanomaterials also studied within the frame of MOS replacement [START_REF]Emerging Research Devices[END_REF].

However, when dealing with MOS replacement, one must be well aware of the magnitude of the challenge. The microelectronics industry has produced an incredible and unprecedented level of organisation, where more than 10 9 transistors are crammed onto ~ one cm 2 of a silicon substrate [3], each individual device exhibiting a lithographycontrolled gate length of ~ 30 nm (corresponding to the so-called 32 nm node). This spectacular miniaturisation has been made possible by constant progress in photolithography and micromachining (top-down approach), a paradigm that applies well to bulk or thin-film type "continuous" materials.

Actually, carbon nanotubes represent a new type of nanomaterial that does not exist in traditional bulk form. The same holds for semiconductor nanowires. Random carbon nanotube networks [START_REF] Snow | [END_REF] are the "thin-film" equivalent form of CNTs. Although devices made from such films exhibit stunning characteristics [5] (especially considering their relatively simple processing conditions), such characteristics (carrier mobility, subthreshold slope, . . .) are still degraded compared to those of counterpart devices obtained with individual SWNT specimens (compare device results from Ref. [START_REF] Dai | For recent reviews on CNT-based devices[END_REF] with those of Ref. [5]). Also, such random films are not amenable to ultra large scale integration (ULSI)type manufacturing methods because they are not continuous at the nanometric scale. Hence, new paradigms and fabrication schemes have to be found, in order to allow the use of such nano-materials (or rather collections of such nano-objects) and be able to compete with the organisation and complexity levels reached by silicon MOS technology.

The problem now is to find means to organise SWNTs and s-NWs in dense arrays that will allow device densities comparable to those of ULSI circuits. For instance, the current-carrying capability of a well engineered (strained) Si MOS transistor is around 1 mA/µm of gate width. 1 On the other hand, a carbon nanotube field-effect transistor (CNTFET) built with an individual semiconductor SWNT specimen will deliver several µA of on-current [START_REF] Dai | For recent reviews on CNT-based devices[END_REF] (say 10 µA/tube for the sake of being quantitative [START_REF]Ballistic devices that can carry higher currents are not considered here[END_REF]). On this basis, by using a scaling law that takes into account a partial screening of the gate field as SWNTs are brought closer, drive currents well over 2 mA/µm of gate width can be calculated for CNTFETs based on parallel organised, compact arrays of SWNTs [7,8]. Thus CNTFETs are potentially capable of much higher drive currents per unit width than Si MOSFETs.

The problem, however, is that so-far, no one has been able to organise SWNTs in dense enough arrays, exhibiting several hundreds SWNTs (~ 700 for a 1.4 nm diameter) per µm and capable of delivering currents well over 2 mA/µm. Many techniques of organisation have been studied, whether put in practice during SWNT growth [9] or following their solubilisation in liquid suspensions [10]. However, to the best of our knowledge, the highest in-plane SWNT density reported so far is around 50 µm 1 [11], obtained by direct growth on (ST cut) quartz surfaces and this density is far from being homogeneous, which seems to preclude any scaling of the method over large areas and large device quantities. The most recent papers present drive currents of 20 µA/µm of gate width for CNTFET devices and circuits [12], far below the extrapolated values of Refs. [7] and [8].

For s-NWs, the best reported in-plane densities are even smaller, below 10 µm 1 [13]. With such NW densities, it takes about 100 µm of gate width for a transistor to reach a drive current of ~ 1 mA (i.e., 100 times larger than the corresponding bulk Si MOSFET, see Fig. 5(b) of Ref. [13]).

In this article, we present a novel approach to the organisation of SWNTs or semiconductor NWs, based on the use of template structures. Such templates can yield very dense assemblies of nano-objects as will be detailed below.

Porous anodic alumina templates

In recent years, various templates have been used for the organisation of nano-objects such as nanowires (metallic or semiconducting) or carbon nanotubes (see Ref. [14] for a recent review). The most widespread templates are ion track-etched membranes and porous anodic alumina membranes. Ion-track etched membranes are usually synthesized by high energy ion irradiation of polycarbonate films coated on a conducting substrate [15]. Because of the irradiation process, the pores are randomly organised. Moreover, being made of polycarbonate, such membranes are not refractory and cannot withstand a high temperature treatment such as the growth of SWNTs or s-NWs by chemical vapour deposition (see below).

On the other hand, porous alumina membranes (PAA) are synthesised by anodic oxidation of aluminium films/substrates at room temperature. Anodic oxidation is an electrochemical process by which aluminium is transformed into its oxide at the anode of an electrochemical cell. Under certain oxidation conditions (i.e., when the resulting alumina is soluble in the electrolyte), the alumina film turns out to be porous (instead of continuous) and the pores can be self-organised into a hexagonal structure [START_REF] Wade | Nanoporous alumina templates for nanowire electron devices[END_REF][START_REF] Masuda | [END_REF][18]. The electric field-assisted oxide dissolution, together with the oxide volume expansion of ~ 1.2 (that generates an interface stress) are at the origin of pore nucleation. Pore growth is the result of an equilibrium process between oxide dissolution at the interface with the electrolyte and oxide formation. A two step anodic oxidation process yields highly self-ordered, honeycomb-like hexagonal pore arrays, with a circular pore at the centre of each hexagon [START_REF] Masuda | [END_REF].

The pore diameter, Φ p , and the inter-pore distance, D int , are proportional to the anodization voltage U . Actually,

Φ p = k 1 U and D int = k 2 U ,
where k 1 ~ 1.29 nm/V and k 2 ~ 2.5 nm/V at room temperature [18]. For low anodization voltages (e.g., 2-3 V) in sulphuric acid electrolytes, high pore densities can be obtained, of the order of several 10 12 cm 2 , corresponding to more than 100 µm 1 on a linear scale.

As an illustration, Fig. 1 shows scanning electron microscope (SEM) pictures of PAA membranes synthesised in our group. Note the high degree of in-plane self-organisation (Fig. 1(a)) and the rectitude of the pores (Fig. 1(b)).

Carbon nanotubes [19] (including SWNTs [20]) as well as silicon NWs [21] have been grown inside alumina templates by the so-called vapour-liquid-solid (VLS) mechanism [22]. In the VLS mechanism, a metal (or rather a nanoparticle of a metal [23] in the present context) forming a low temperature eutectic alloy with either silicon [22,23] or carbon [24] is used as a catalyst for growing carbon nanotubes (CNTs) or s-NWs from gaseous precursors such as CH 4 , C 2 H 2 or SiH 4 . When the VLS-type synthesis mechanism is activated, the final diameter of the NW or CNT is controlled by the critical dimension (diameter) of the metallic nanoparticles (e.g., Au for Si growth, see Ref. [23], and Fe for CNT growth, see Ref. [24]). Such metallic nanoparticles can originate from colloidal solutions [23,25] or be prepared in situ, from thin-film fragmentation upon annealing [26]. In any case, clustering and agglomeration of the nanoparticles must be prevented, if a uniform diameter distribution of NWs or CNTs is to be obtained after growth. The advantage of using PAA templates is that catalyst nanoparticles can be electrodeposited at the bottom of the pores of the membrane (after its synthesis) and their possible diffusion (upon exposure to high temperature, e.g., during growth) and agglomeration (by e.g., an Ostwald ripening mechanism [27]) blocked. Moreover, as the pore diameter distribution is usually very uniform and monodispersed, nano-objects with controlled diameters can be obtained [21].

Again as an illustration, Fig. 2 shows Si NWs, CNTs and Ni nanowires grown in our laboratory inside vertical PAA membranes. The Ni nanowires were simply grown by electrodeposition, whereas the Si NWs and CNTs were grown by VLS in a chemical vapour deposition (CVD) reactor using H 2 -diluted SiH 4 and C 2 H 2 respectively as growth nutrients. Prior to CVD-VLS growth, Cu (for Si NW growth) or Ni or Co (for CNT growth) catalyst particles were electrodeposited at the bottom of the pores of the membranes. The Si NWs were grown at 500-550 • C and the CNTs were grown at 600-620 • C. In order to get a uniform catalyst or Ni NW deposition, a special membrane preparation technique is systematically applied, by which the anodization current is decreased according to an exponential law (as a function of time) at the end of the anodic oxidation step [28]. Finally, we note that the CNTs are not single walled, because the pore diameter (and hence the diameter of the Ni or Co catalyst particles) was too large in those samples. SWNTs can be grown in vertical PAA templates by using special tricks to get catalyst particles in the nanometer range (see Ref. [20]).

Lateral alumina templates

Another advantage of PAA membranes is that they can be shaped with their pores parallel to the surface of the substrate [29,30]. This can be done by engineering the anodization process of an Al thin film in such a way that the current will flow parallel to the surface of the substrate instead of perpendicular as usually done. Actually, because of the mechanism of their formation, one has to realise that the pores develop parallel to the local electric field lines and their orientation follows the flow of the anodization current inside the Al thin film [START_REF] Pribat | Proceedings of the 2005 SPIE "Quantum Sensing and Nanophotonic Devices II[END_REF]. So, in principle, any pore direction can be obtained, as long as the current can be channelled accordingly. Briefly, PAA lateral structures can be synthesized by using an insulating substrate (e.g., an oxidised Si wafer) and by capping stripes of a deposited Al thin film with an insulating layer (SiO 2 , Si 3 N 4 , . . .) before anodic oxidation, thus forcing the anodization current to flow parallel to the surface of the Si substrate (see Fig. 3 for the principle and Refs. [29,30] for details). After electrodeposition of a catalyst material at the bottom of the pores (in the same way as for vertical pores), such lateral PAA structures can be used as templates for the "planar" growth of NWs and CNTs by the VLS method, again just like in vertical templates. The result is a composite material (see Fig. 4) where nano-objects are packed with a predefined density, corresponding to the pore density (which is adjustable, depending on the anodization voltage and the type of electrolyte) and assuming that each pore is filled. Once growth is performed, and the nano-objects lay in-plane inside the pores, it is quite easy to deposit various metal electrodes in order to build a planar-type, three-terminal device. In fact, the lateral membrane with the nano-objects inside can be shaped and micro-machined by microelectronictype technologies (photolithography and etching) just like any other thin film and there is no need to manipulate individual nanotubes or nanowires to build electronic devices. Moreover, as quoted above, linear densities of pores larger than 100 µm 1 can be obtained (corresponding to an identical potential density of NWs or CNTs), which, as far as the organisation of nano-objects is concerned, represents an advantage over the various methods developed so far. Fig. 5 shows various examples of lateral membranes grown in oxalic acid, using Si 3 N 4 capping layers. In our operating conditions, where the anodic voltage is around 50 V and the electrolytic bath is maintained at 0 • C, the anodization rate in oxalic acid is ~ 45 nm/minute. Although difficult to appreciate at the actual magnification of the pictures in Fig. 5, the typical pore diameter is around 50 nm. For the sake of picture clarity, we have removed the capping Si 3 N 4 layer (by reactive ion etching) as well as the remaining Al electrode at the back of the membrane (by wet chemistry in a HgCl 2 solution) before loading the samples in the SEM. We also point out that the thickness of the starting Al film can be adjusted in order to yield a controlled number of pore rows, down to only one (see Ref. [30]). After synthesis of the lateral membranes to the desired width, catalysts are electrodeposited at the bottom of the pores. For the experiments presented below, we have used CuSO 4 (for Si NW growth) and CoSO 4 (for CNT growth) electrolytes, buffered with H 3 BO 3 (pH ~ 3.8). The electrodeposition configuration used in these experiments is a classical 3-electrode type already reported elsewhere [START_REF] Choi | [END_REF].

After the electrodeposition of the catalysts at the bottom of the pores, the samples are loaded in a CVD reactor and heated to 550 • C under a H 2 atmosphere (typically 15 mbar) for 15 min, in order to stabilise the membranes with the catalyst particles inside them.

For the growth of SiNWs, as quoted above, we have used a mixture of SiH 4 (2.5 sccm) and H 2 (97.5 sccm) at a pressure of 10 mbar and at a temperature of 550 • C. However, when dealing with lateral membranes, one has to realise that the catalyst surface exposed to the growth nutrients is very small, much smaller than in the situation concerning vertical membranes, where the whole top surface of the sample is exposed. As a consequence, and because of the high supersaturation of the vapour phase, SiH 4 tends to decompose on any surface, even those with a poor catalytic activity. Hence, in order to avoid the parasitic deposition of amorphous or ill-crystallized Si on the membrane itself (which leads to a rapid clogging of the pores, thus halting NW growth), the hydrogen dilution gas was passed through a tungsten filament positioned inside the CVD tube, ~ 15 cm upstream of the sample holder. The W filament was heated to ~ 1800 • C (as measured with an optical pyrometer), in order to partially decompose H 2 into atomic hydrogen (H•). The temperature of the sample holder was monitored with an independent thermocouple, in order to take into account any possible additional heating effect originating from the W filament. Actually, H• is well known as a scavenger that preferentially etches amorphous Si. This is the reason why H• is extensively used for the growth of nano-or micro-crystalline Si by plasma enhanced CVD at low temperature [33]. The growth time for the Si NWs was typically set to 10 minutes.

For the growth of CNTs, the temperature was set to 620 • C and we have used a mixture of C 2 H 2 (30 sccm) and H 2 O (10 sccm) at a pressure of 10 mbar. Here, H 2 O is used instead of H• for removing amorphous carbon [34,35]. The typical growth time was 15 min. Fig. 6 shows various SEM pictures of Si NWs emerging from lateral membranes after CVD growth. The effect of H• is clearly seen on the pictures of Fig. 6 (b) and (c). When the W filament is not switched-on (Fig. 6(b)), parasitic Si deposition is observed all over the side wall of the membrane, which tends to block most pores, thus preventing further NW growth. We note that even when H• is used, all the pores are not filled with NWs (Fig. 6 (a) and (c)), despite us taking care to optimize the voltage decrease after synthesis of the membrane [28]. We also note that the NW diameters (~ 20 nm) are smaller than those of the pores they originate from. We suspect that some of the electrodeposited copper has alloyed with Al at the bottom of the pores just before growth (during the thermal stabilization treatment), resulting in either a poisoning (no NW in the pore) or a decrease of the catalyst volume available for growth (smaller NW diameter). This problem will be solved in the future, by carefully adjusting the conditions of the exponential voltage decrease after anodization [28], in order to leave a thin alumina film at the bottom of the pores, which will isolate the catalysts from the Al electrode, acting as a diffusion barrier.

Finally, Fig. 7 shows SEM pictures of CNTs emerging from lateral membranes. Different electrodeposition conditions have been used for the Co catalysts. In Fig. 7(a), electrodeposition was performed with electric pulses having a 6 V amplitude, at 100 Hz frequency and during 60 s, whereas in Fig. 7(b), the deposition time was only 15 s. So the length of the catalyst particles inside the pores was smaller in the later case. As with Si NWs, the shorter Co catalysts have probably been poisoned by the remaining Al at the bottom of the pores, leading to fewer CNTs being grown. Fig. 7 (c) and (d) display details of CNTs emerging from the inside of some pores. In particular, catalyst fragmentation seems to occur in some places, leading to the growth of several CNTs inside the same pore (Fig. 7(d)), a situation never observed for Si NWs. Here again, the CNTs are certainly not single walled (even when fragmentation of the catalyst occurs), since the pore diameter was too large to yield catalysts with diameters around 1 nm. We are currently studying a different way to introduce the catalysts, compatible with the obtainment of nm-sized clusters [36]. We also point out that the membrane's side wall is devoid of any parasitic amorphous carbon deposition, thanks to the scavenging effect of H 2 O during growth.

Conclusions

We have presented some early results concerning a novel way of organising nano-objects such as semiconductor nanowires and carbon nanotubes, based on the use of lateral porous alumina templates synthesized from thin films of aluminium deposited on insulating substrates. Such templates can be tailored with predefined geometrical characteristics (including the pore and the interpore distance down to a few nm), by carefully controlling the thickness of the Al thin film and the anodization conditions (type of electrolyte, anodizing voltage, . . . ). These templates provide a planar-type organisation well adapted to 3-terminal devices and totally compatible with existing tools inherited from advanced microelectronic processes. Pore densities over 100 µm 1 can be obtained, corresponding to an equal density of NWs or CNTs, assuming a 100% filling rate. Work is now in progress in order to stabilize the NW or CNT growth processes inside the pores and increase the filling rate. In particular, special attention will be devoted, in the near future, to the catalyst deposition process, in order to avoid its poisoning by intermixing with Al. Finally, we would like to point out that lateral membranes would also simplify the fabrication process of thin film transistors used in active matrix backplanes for addressing flat panel displays. In particular, only 4 masks would be needed for the fabrication of an active matrix liquid crystal display backplane [37], a mask number competitive with that of the current amorphous silicon technology.

Fig. 1 .

 1 Fig. 1. Scanning electron microscope (SEM) pictures of porous anodic alumina membranes obtained by anodic oxidation of Al substrates in oxalic acid at 55 V. (a): Top view of a large portion of the membrane. (b): Detail of the membrane showing ~ 70 nm diameter pores. Note the quality of the self-ordering highlighted by the white hexagon. Smaller pores (down to ~ 5 nm) can be obtained in sulphuric acid electrolytes and at a lower anodic voltage. (c): Cross section of the membrane showing straight and parallel-aligned pores (~ 1 µm long).

Fig. 2 .

 2 Fig. 2. Various nanostructures grown inside vertical porous anodic alumina templates. (a): Different SEM views of Si NWs grown from electrodeposited Cu catalysts. From bottom to top: (i) surface of the membrane after VLS growth; for easy observation, the growth time has been purposely made longer, which results in NWs growing out of the membrane, (ii) detail of a NW emerging from a pore, (iii) the membrane has been broken in order to check that NW nucleation occurred at the bottom of the pores, where the Cu nanoparticles are electrodeposited, (iv) Raman spectrum of the nanowires evidencing their good crystalline quality. (b): Different views of CNTs grown from electrodeposited Ni catalysts. From bottom to top: (i) surface of the membrane after VLS growth, (ii) cross section of a membrane showing CNTs inside it, (iii) details of CNTs emerging from the membrane, with a close view on one CNT in inset (~ 40 nm diameter). (c): Ni NWs grown by electrodeposition from a mixture of NiSO 4 , NiCl 2 and H 3 BO 3 (Watts bath). The membrane has been selectively etched in the bottom picture, in order to reveal the Ni NWs which were originally inside the pores.

Fig. 3 .

 3 Fig. 3. Principle of the method developed for synthesising lateral porous alumina templates. (a): Al stripes deposited on an oxidised Si wafer are caped with an insulating layer (e.g., SiO 2 ). (b): The capped Al stripes are made individual by photolithography and cleavage of the substrate. (c): Anodic oxidation of the tip of an individual Al stripe. (d): Schematic structure of the lateral porous alumina membrane at the tip of the Al stripe, after anodic oxidation.

Fig. 4 .

 4 Fig. 4. Schematic principle of the growth of Si-NWs and CNTs inside porous lateral alumina membranes. (a): Schematics of the lateral membrane after synthesis. (b): After electrodeposition of catalysts at the bottom of the pores. (c): After growth of Si NWs. (d): After growth of CNTs.

Fig. 5 .

 5 Fig.5. SEM pictures of lateral membranes grown in oxalic acid. On the top picture, the membrane is ~ 1 µm wide, corresponding to a 22 minutes oxidation time whereas on the bottom picture it is ~ 250 nm wide corresponding to a 5 minutes oxidation time. The capping layer has been removed by reactive ion etching (RIE) and the remaining Al film at the back of the membranes has also been removed by selective etching in HgCl 2 . The small pillars observed on the bottom picture are defects left after RIE of the Si 3 N 4 cap.

Fig. 6 .

 6 Fig. 6. Various SEM pictures of Si NWs grown in porous lateral templates. (a): View of a membrane after growth with H•. The NWs are clearly seen emerging from some of the pores. (b): Another membrane after growth without H• (the W filament was not turned on); the red arrows point to those pores blocked by amorphous Si deposition and the brown one to an emerging NW. (c): Close up of the picture shown in (a). The red and brown arrows have the same meaning as in (b); the blue arrows indicate partially filled or empty pores.

Fig. 7 .

 7 Fig. 7. Various SEM pictures of CNTs grown in porous lateral templates. (a): The membrane is totally masked by CNTs protruding out of it. (b): Different electrodeposition conditions for the Co catalysts: fewer CNTs are produced (see text). (c): Details of CNTs emerging from pores. (d): Some catalysts have fragmented into smaller clusters, yielding several CNTs in one pore. Note the "cleanliness" of the side wall of the membrane, with absolutely no trace of amorphous carbon (H 2 O effect).

At the

IEDM conference (15-17 December), Intel announced 1.55 mA/µm for NMOS and 1.21 mA/µm for PMOS transisors, using their 32 nm platform.

Acknowledgements

This work was partially funded by the NanodiX Chair with Samsung Electronics, the CANDICE EC (FP 6) project and the French ANR project NANOBIOSENSOR.


J.-F Dayen

T L Wade

G Rizza

D S Golubev

C.-S Cojocaru

D Pribat

X Jehl

M Sanquer

J.-E Wegrowe

Conductance of disordered semiconducting nanowires and carbon nanotubes: a chain of quantum dots

A comparative study of the low temperature conductivity of an ensemble of multiwall carbon nanotubes and semiconductor nanowires is presented. The quasi one-dimensional samples are made in nanoporous templates by electrodeposition and CVD growth. Three different structures are studied in parallel: multiwall carbon nanotubes, tellurium nanowires, and silicon nanowires. It is shown that the Coulomb blockade regime dominates the electronic transport below 50 K, together with weak and strong localization effects. In the Coulomb blockade regime, a scaling law of the conductance measured as a func-tion of the temperature and the voltage is systematically observed. This allows a single scaling parameter α to be defined. This parameter accounts for the specific realization of the "disorder", and plays the role of a fingerprint for each sample. Correlations between α and the conductance measured as a function of temperature and voltage, as a function of the perpendicular magnetic field, and as a function of the temperature and voltage in the localized regime below 1 K have been performed. Three universal laws are reported. They relate the coefficient α (1) to the normalized Coulomb blockade conductance G T (α), (2) to the phase coherence length l φ (α), and (3) to the activation energy E a (α). These observations suggest a description of the wires and tubes in terms of a chain of quantum dots; the wires and tubes break into a series of islands. The quantum dots are defined by conducting islands with a typical length on the order of the phase coherence length separated by poorly conducting regions (low density of carriers or potential barriers due to defects). A corresponding model is developed in order to put the three universal laws in a common frame.

G(T, V

)T -α = f (eV /kT ).

(3)

 for different samples (CNT, TeNW, and Si NW). The profile, described by

Introduction

The study of electronic properties of nanowires or nanotubes is an active field of research and many questions are still open regarding the reproducibility of the transport parameters from one sample to the other and consequently the ability to massively integrate such objects into efficient nanoelectronic devices. Since a couple of vacancies or a single ion are able to trap electrons, a good single electron transistor should be controlled at the scale of any topological defects and any inhomogeneous charge distribution. However, the disorder is often used in order to create, naturally and randomly, a chain of quantum dots [1]. The reason usually invoked [1,2] is that small defects (topological, chemical, etc.) induce strong charge inhomogeneities. The consequence is that some small highly conducting regions (Q. dots) are separated by poorly conducting ones (tun-nel barriers). Pictures of such chain of quantum dots have been nicely imaged by scanning probe techniques [3][4][5]. The aim of this paper is to report a set of empirical laws observed about transport properties in such nanostruc-tures, and to interpret them within the model of a chain of quantum dots. In disordered nanostructures, the role of defects is not easy to measure since the thermodynamic limit is not necessarily reached for a given sample. Important quantitative variations of the transport parameters are usually observed from one sample to the other even if samples are produced within a well-defined fabrication process [3,[6][7][8][9][10][11][12][13][14][15][16]. Since one sample is a specific realization of a statistical distribution of defects (called "disorder"), the whole distribution can be known by measuring a statistical ensemble of samples. In the study reported here, we use a template synthesis technique [17], in order to produce large enough numbers of samples with different materials. Multiwall carbon nanotubes (CNT), Te nanowires, and Si nanowires are studied in parallel. Some controlled fabrication parameters (sizes, nature of the wire, nature of the contacts) have been modified, and the variations are hence added to the unknown distribution of structural topological and chemical disorder defined over the ensemble of samples. The goal is to access the statistical distribution of the electronic transport properties, in order to extract some general behavior related to disorder.

Fortunately, in the case of carbon nanotubes (see for example [START_REF] Forro | Carbon Nanotubes[END_REF][START_REF] Yao | Carbon Nanotubes[END_REF][START_REF] Dai | [END_REF][START_REF]Understanding Carbon Nanotubes From Basics to Applications[END_REF][22][23][24]), some of the main physical properties have been intensively studied in the last decade. In the present study, these properties (see below) are systematically observed, not only for CNTs, but for all samples of the statistical ensemble. Three main effects are observed and described: (1) Coulomb blockade effects, (2) weak localization like effects and (3) transition to an activated regime at low temperatures.

(1) Due to the small diameter, the disorder and the low carrier density, the screening of the electronic charges is poor, and Coulomb interaction occurs between carriers. This Coulomb blockade effect is observed and quantified experimentally through the so-called "zero bias anomaly" profile of the conductance measured as a function of the bias voltage for various temperatures (Fig. 3). A scaling law is observed (the effect of the bias voltage scales with that of the temperature) that allows a scaling coefficient α to be measured. Each sample is described by a single parameter α, which plays the role of a fingerprint.

(2) The weak-localization effect (or an analogous effect) is also observed systematically in carbon nanotubes and nanowires [14,23,[25][26][27][28][29][30][31][32][33][34][35][36][37]. Weak localization (WL) is a well-known effect [38][START_REF] Altshuler | Electron Electron Interactions in Disordered Systems[END_REF][START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF] due to the interference of an electron with itself during a random walk, and is described in terms of perturbative quantum correction to the Drude conductivity. The main parameter that governs the WL effect, the phase coherence length l φ , can be measured by applying a perpendicular magnetic field to the sample. However, the presence of WL (perturbative correction to the conductivity) together with Coulomb blockade (nonperturbative treatment needed) is puzzling, and this question will also be addressed in this paper. The coefficient l φ is also correlated to α and shows a second universal behaviour (Sect. 3).

(3) Finally, the behavior of the conductivity at very low temperatures (typically below 1 K) shows an abrupt transition to a strong-localization like effect where the conductivity is driven by an activation process. The activation process allows another well-defined parameter to be measured which is the activation energy E a . It is shown that this transition, observed as a function of temperature, is destroyed by applying a critical bias voltage V c [START_REF] Fuhrer | [END_REF], and the Coulomb blockade regime is recovered at higher voltage. The parameters E a and V c are also correlated to the coefficients α (Sect. 4).

After presenting the sample fabrication in Section 2, the three kinds of measurements described above are reported and the corresponding empirical laws are dis-cussed. The challenge is to find a common picture able to describe the three types of the observed behaviors. The simplest picture of Coulomb blockade is given for a system composed by a small Coulomb island separated by two capacitive junctions. The conduction electrons have to overcome the Coulomb energy due to the electric charging of the island. Beyond this, however, some more general approaches have been developed in order to describe Coulomb blockade through tunnel barriers in disordered environment. Indeed, generalization of the concept has been developed in the context of single electron devices [START_REF] Grabert | Single Charge Tunneling[END_REF], extended junctions [START_REF] Pierre | [END_REF], disordered systems [44], and multiwall CNTs [45]. In this picture, the active part of the device is a tunnel junction connected to the rest of a circuit. During the tunneling process, the electron, electrically screened in the first contact, should recover a screening cloud after tunneling, and this process costs an amount of energy defined by the impedance of the rest of the circuit. In a more realistic picture (according to previous observations [3][4][5]), due to charge inhomogeneities within the wire or tube, the quasi 1D system is broken into conducting islands, or quantum dots, separated by poorly conducting regions, or potential energy barriers. In this case, WL effect occurs inside the quantum dots, and Coulomb blockade is generated at the interfaces between the islands and averaged over many junctions. The consequences of the chain of quantum dots, in terms of transport properties, are described formally in the last section (cf. Sect. 6) in relation with the observed effects.

Sample fabrication

The samples are obtained by the template synthesis method in which the growth of the wires is performed inside the pores of a membrane. The diameter of the wires is calibrated by the diameter of the pores (typically around 50 nm), and the length is fixed by the thickness of the nanoporous membrane (typically adjusted at 1 μm). A first electric contact is obtained on the bottom through the nucleation process in the electrolytic bath, and the second contact is obtained on the top of the membrane with the help of a second patterned metallic layer.

We will discuss three types of samples: Tellurium nanowires(TeNW) grown and contacted by electrodeposition (see Fig. 1), silicon nanowires (Si NW) and multiwall carbon nanotubes (MWCNT) grown with CVD on electrodeposited catalysts (see Fig. 2). All samples are obtained with bottom-up growth inside a nanoporous insulating membrane. The MWCNT were obtained by chemical vapour deposition (thermal CVD) of acetylene at 640 • C, after the electrodeposition of Ni or Co catalyst inside the nanopores of an alumina membrane (a precise description of the process is given in [17,22]). The Si NW are also obtained with thermal CVD of Silane (SiH4 ) in a 5% hydrogen atmosphere (a description can be found in [37]).

The first set of samples is composed of single contacted tellurium nanowires -TeNW -obtained by electrodeposition in nanoporous polycarbonate or alumina membranes of diameter d = 40 nm and d = 150 nm and a length of 6 μm. Figure 1b shows a zoom of the wire after dissolution of the membrane. The wires are polycrystalline (Fig. 1b) with small disorientations between grains (Fig. 1c). No other defects have been observed. With the electrodeposition technique [17], a single nanowire can be contacted in situ with a feedback loop on the intermem-

Coulomb blockade regime

The dynamic conductance measurements G(I ) = dI /dV | I were performed with a lock-in detection bridge LR7OO (using an AC current from 0.3 nA of amplitude for most samples to 10 nA for low resistance samples), together with a DC current I provided by a current source. The curves G(V ) plotted as a function of bias voltage are obbrane electric potential, after depositing a thin enough Au layer on the top. If the electrodeposition potential is main-tained after calculating the voltage V = 0 (G(i))

-1 di.

tained, dendritic growth are observed on the surface of the membrane (Fig. 1d). This is the typical form of the top contact. Both contacts are oxide-free, due to the electrochemical reduction at the Te interfaces during the in-situ electrodeposition.

For the second set of samples, the contact is performed in a second step. At the end of the CVD growth, MWCNT or Si NW are emerging from the pores at the top of the membrane, over a length of a couple of 100 nm. The density of CNT or NW emerging out of the top of the membrane is very low and not uniform. This is due to the pres-The static conductance G DC (V ) as a function of voltage is also measured directly with a nanovoltmeter in order to identify possible frequency dependent responses. The profiles G(V, T ) are measured with temperatures ranging from T = 300 K to T = 4.2 K. The results, plotted in Figure 3, show the typical profiles of the G(V ) curves at different temperatures, in a standard semi-log plot Figure 3a. The profile shows two remarkable asymptotic regimes: the nearly zero bias regime eV × kT (k is the Boltzmann constant, and e the charge of the electron), where the conductance is a power law of the temperature (Fig. 3b): ence of an oxide layer at the interface between the pore and the non-oxidized Al. In the electrolytic bath (during the lim

eV <kT G(T ) = G T T α (1)
step before CVD growth), the catalyst nucleates from the Al layer on the bottom of the wire, after the application of the potential. The catalyst is consequently deposited in and the high bias regime eV » kT , where the conductance is also a power law, but a power law of the bias voltage:

the wires that have no, or negligibly small, oxide layer. According to SEM observations, some islands of a couple of lim

eV >kT G(V ) = G V V α . ( 2
)
nanowires are emerging out of the membrane, from part to part, separated by a few microns. It is consequently easy to contact the emerging part of the CNT or NW with a layer of Au, or any other metal (Cu, Co, Ni and Al), through a mask, by evaporation or sputtering. Among all wires or tubes that can be contacted in parallel during the process, we observe (see Sect. 4) that only a couples of wires with good ohmic contacts effectively contribute to the signal.

What is significant here is that the power coefficient α is the same in both regimes. The scaling law is valid for all intermediate regimes (Fig. 3c): introducing the scaling function f , all points collapse on a unique curve the function f , is typical and systematically measured in such samples: this is a robust structure that is obtained whatever the type of the wire (carbon nanotubes, Te and Si nanowires, and metallic wires), the nature and crys- single tunnel barrier. In this model, by assuming a well defined tunnel junction, contacted to the rest of the circuit (i.e. mainly the nanowire or nanotube), the following expression is obtained [START_REF] Grabert | Single Charge Tunneling[END_REF]44,45]:

tallinity of the contacts (Co, Ni, Cu, Au, Te...) [22], the length (300 to 6000 nm), the diameter of the wires (from

G T ≈ G V (k/e) α = e -γα παk α (4)
5 nm to 200 nm), and the nature of the supporting tem-R Γ(2 + α) eV 0 plate (polycarbonate and alumina) [23]. Note that a deviation from the scaling law can be observed for the Si nanowires. The origin of this deviation deserves further investigations, and may be related to the crystalline properties of the Si nanowires [46,47].

The histogram of the coefficient α obtained for all samples measured is presented in Figure 5. The fluctuations for fixed experimental parameters (e.g. CNT contacted with Ni, or CNT contacted with Co), are of the same order of magnitude as that of the total fluctuations for all materials (the individual distributions can hardly be distinguished). Accordingly, the parameter α is not directly defined by the intrinsic parameters, like the band structure or the density of states. This observation confirms the assumption that α is a measure of disorder. The values found for α are in agreement with that found in the literature for CNT measured with four terminal contacts [4,16,[START_REF] Forro | Carbon Nanotubes[END_REF][START_REF] Yao | Carbon Nanotubes[END_REF]22,23,[48][49][50][51][52][53][54][55].

It is possible to go further in this analysis by correlating the main parameters of this study with the coefficient α. The parameter G T (see Eq. ( 1)) is the conductance extrapolated (or measured) at T = 1 K at zero bias. The result is presented in Figure 6. All points approximately align in a log-log plot: beyond the singularity of each samples, a universal behaviour is obeyed by all samples as a function of the scaling parameter α.

The law followed by the experimental points plotted as a function of α can be fitted with a formula calculated in the framework of the Coulomb Blockade theories of a where γ = 0.577. . . is the Euler constant and Γ the Gamma function. The formula contains two adjustable parameters that are the resistance of the tunnel barrier R, and the energy eV 0 of the junction is the Coulomb energy (e.g. in the case of ultra-small tunnel junctions: eV 0 = e 2 /2C where C is the capacitance of the tunnel barrier) or the thermal energy eV 0 = kT . This formula is obtained identically for single electron tunneling for ultra small junctions [START_REF] Grabert | Single Charge Tunneling[END_REF], for extended junctions [START_REF] Pierre | [END_REF], or in the context of non perturbative dissipative tunneling (environmental Coulomb blockade) [44,45]. The fit gives a "universal" resistance of the tunnel junctions of the order of R = 4 kΩ, and a "universal" capacitance of the order of C = 10 -18 F.

However, this oversimplified picture of a single junction contacted to the rest of a circuit can hardly be accepted in our experimental study. In the case of a fixed material, e.g. CNTs, we would even expect the opposite situation: a fixed circuit impedance and a distribution of the parameters of the tunnel junction due to different contacts because we changed the nature and the crystallinity of the contacts (from the single crystalline Co, Cu, or Au, to the quasi-amorphous Ni) studied in reference [22]. The chemical disorder of the junction is also varying dramatically between electrodeposited Te, in which the contact is performed inside the electrolytic bath under a reducing potential, so is consequently free of oxides, and the CNT or Si nanowires, for which the contact is performed in a second step after the CVD growth, with sputtering, evaporation, or electrodeposition of metal. A universal tunnel junction (with R = 4 kΩ and C = 10 -18 F) for all these cases is difficult to understand. Another picture, developed in detail in the last section, can be proposed in terms of a chain of quantum dots (like that observed in Ref. [5]). In this model, the resistance of the junction is not an adjustable parameter but is contained in the variable α. The averaged capacitance C of the junctions between the quantum dots is still an adjustable parameter.

The following formula is obtained from simple assumptions developed in the last section of the paper: specificity of the Si nanowires will be reported elsewhere). WL appears in a disordered metallic system, into which electrons undergo Brownian motion due to elastic collisions with defects within a loop of size smaller than the phase coherence length l φ . Weak localisation comes from quantum phase interference effects of electron backscattering [38,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]. Because the interference terms cancel when applying a sufficiently strong magnetic field (which breaks the time reversal symmetry), WL manifests itself as a positive magnetoconductance.

Our samples have magnetoconductive behaviour shown in Figure 7a The corresponding fit is presented in Figure 6 for two values of the capacitance C , which delimited the best fits (7) obtained for all samples. Once C is fixed, the only adjustable parameter is the number N of tunnel junctions. In this framework, the experimental universal law observed is interpreted as the behavior of arrays of 2 to 7 tunnel barriers of average capacitance between 3 to 8 aF (depending on the sample).

G W L = 1 e 2 ( 1 π Ln 2 W 2 + 3l 4 \ -1/2
where l H = (n/(eH )) 1/2 is the magnetic length (H the magnetic field), W diameter of the wire, e the charge of an electron and n the Planck constant. The adjustable parameters are G(∞) and l φ . At each temperature, the phase coherence length l φ can be extracted (cf. Fig. 7b). The function l φ (T ) follows the Altshuler-Aronov-Khmelnitskii [38,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF] formula:

4 Magnetoresistance √ 2 φ 1/3 (8)
The conductance is also measured as a function of a perpendicular magnetic field H applied by a superconducting coil. The field is ramped between ±1.3 T. MWCNT and TeNW systematically feature a positive magnetoconductance (MC) which (in magnetic field and temperature) is in agreement with the formulas of quasi 1D weak localization [23,25,26] for diameters below 200 nm. The Si nanowires show a smaller WL effect, or no WL at all (the e 2 kT where L is the length of the sample, k the Boltzmann constant, D the diffusion coefficient, and G D the conductivity without WL correction (i.e. the Drude conductivity).

Hence, we can associate with each sample a couple of transport parameters (α, l φ (T = 4.2 K)). By representing these couple of parameters for all the samples studied we observed a strong correlation between α and l φ : the scaling exponent α is decreasing as a function of the phase coherence length. This curve can be reproduced at different temperatures, and also presents a universal behavior. The correlation at T = 4.2 K is shown in Figure 8. The data fits a power law of power coefficient: where the adjustable parameter c depends on the temperature. A derivation of this empirical law is reported in Section 6, for the model of the chain of quantum dots of typical size below l φ . Unfortunately, the adjustable parameter c does not give direct physical information on the system due to numerical constants difficult to express precisely (see the discussion of Eq. ( 15) in Sect. 6).

It is worth pointing out that, in contrast with usual systems exhibiting WL (e.g. in metallic samples), the conductivity without WL (at saturating magnetic field) is not the usual Drude conductivity but the Coulomb blockade We studied the transport behavior of 6 samples at very low temperature (cf. Tab. 1): 4 MWCNT (samples A-D) of 1 μm long with a diameter D = 30 nm, one TeNW (sample E) of 6 μm long with a diameter of 150 nm, and one TeNW (sample F) of 6 μm long with a diameter around 40 nm. For all samples, at zero bias, we observed an abrupt transition in the electronic transport regime at a critical temperature T a (Fig. 10). The transition occurs from the power law regime previously described to an activated regime at T < T a (see inset in Fig. 10). More precisely, the samples A,B,C,D,F showed a transition from the power law regime described in the previous section. The observation G ∝ T α to an activated law G ∝ exp ( E a , with E a desof the WL and the description with equations ( 7) and ( 8) is consequently not trivial. As an illustration, Figure 9 shows that the two effects, WL and Coulomb blockade, are not independent. The conductance G(V, T , H ), with the typical zero-bias anomaly profile, is plotted at different magnetic fields H : it can be observed that bias voltage kT ignating the activation energy. The parameter E a is also a characteristic of each sample (cf. Figs. 10a-10e). Sample E (whose diameter is 5 times bigger than the other samples) shows a behavior which cannot be fitted to a simple activation law, but has rather a profile of the form E a destroys the weak localization correction. The curves are G ∝ exp kT (see Fig. 11). We will come back to plotted for the two temperatures, 4.2 K and 8 K, in order to show that the destruction is not produced indirectly by the Joule heating (otherwise, the profiles at 8 K would scale to those at 4.2 K at lower bias).

Low temperature transition

The low temperature measurements are made in a dilution refrigerator down to 50 mK. Resistance is measured by traditional two contact lock-in detection, with AC excitation between 10 and 100 μV. A DC bias is applied in the reason for this difference below. The results are summarized in Table 1.

The behavior of the conductance as a function of the bias voltage at low temperature is shown in Figure 12 in a semi-log plot for samples A to F. At T > T a , the conductance follows the scaling law reported in the previous sections. For T ≤ T a , we systematically observed a critical voltage V C below which the samples change from the scaling law regime to an exponential dependence:

G ∝ exp(C (T )V ). ( 10
)
where C(T) is a function of the temperature. G(Ω -1 ) T(K) 22 -ln(G) The transition can be very sharp, with a jump of at least three orders of magnitude of the resistance (cf. Figs. 12b and12d). The plateau observed for the conductance G = 10 -9 Ω -1 is an artifact due to the limitation of the measurement set-up. It is important to note that the transition is not that observed for metal -insulator transition [56], because it occurs from a Coulomb blockade regime to the insulator state, as shown in the scaling plot Figure 13. Whatever the temperature, the Coulomb blockade regime, with the correct scaling coefficient α is recovered below T a , under a high enough bias voltage. In other terms, the scaling law is maintained whatever the temperature, for high enough scaling parameter eV /kT .

Once again, in the activation regime, a strong correlation appears when we measure a couples of parameters (α, E a ) for all samples studied (Fig. 13): E a follows an increasing function of α (Fig. 14). The same trend is observed for E C = eV C plotted as a function of α (not shown). An interpretation of this third universal law is provided by the model of a chain of quantum dots of high conductance developed in the last part of this article (see Sect. 6). In the framework of this model, it appears that there exists a critical temperature T a at which the conductance goes to zero. The activation energy is related to the temperature T a through the activation law: critical energy E a = k B T a ln(τ /τ 0 ) where τ is the typical measurement time and τ 0 is the typical electronic relaxation time). Each tunneling junction (of resistance R) enters into an activated behavior [57,58], and gives to the sample conductance an averaged activation process. In order to fit the data, the following expression is used (see discussion in Sect. 6, Eqs. ( 25) and ( 27)):

In this section we develop a theoretical model, which has been used to fit the experimental data. We consider a homogeneous one-dimensional chain of N 1 macroscopically identical quantum dots (see Fig. 15).

Let us first briefly discuss the main features of this model and its relation to the experiment. First of all, the experimentally observed suppression of the conductance with decreasing temperature and bias voltage is attributed to Coulomb blockade in the tunnel junctions. This assumption allows us to reproduce the main features observed in the experiment. Indeed, at sufficiently high temperatures and at low resistance of a tunnel junction, R × 2πn/e 2 , the regime of weak Coulomb blockade is realized, in which the model predicts logarithmic dependence of the conductance on temperature. The latter is similar to the power law dependence observed in the experiment. At the same time at temperatures lower than certain effective charging energies the strong exponential suppression of the conductance is expected, which is again in line with experimental observations. Next, both in theory and in experiment the non-linear differential conductance shows similar behavior as a function of bias voltage. Finally, in order to describe the magnetic field dependence of the

E a ≈ N 1 exp α N 1 . ( 11
) α conductance we include the weak localization correction.

We now turn to the quantitative analysis of the proposed model. The quantum dots are coupled to each other The best fit is obtained for an average number of 5 junctions (dotted line of Fig. 14).

The transition observed from the power law regime to the activated one (cf. Fig. 13) could be explained by the transition to a strong tunneling regime of the most resistive junctions. In this framework, the voltage behavior of and to the leads by N identical tunnel junctions with resistance R. We also introduce a capacitance C between neighbouring quantum dots. Provided R is sufficiently low, namely R × 2πn/e 2 , the differential conductance of this system is given by the sum of three contributions CB the resistance below T a can also be understood. When a DC bias is applied, the barrier level of each junction is

dI 1 = dV NR + G WL + dI dV . ( 12
)
decreased. Once a critical value V C is reached, the sample comes back to the scaling law regime. Another estimation of the number of barriers existing in our sample can be obtained by the ratio of the critical energy E C to the activation energy E a which, depending on the sample, takes the value E C /E a ≈ 59. These values are in good agreement with those deduced from the universal law of Figure 6.

The correlations observed between E a and α can be fitted to equation (11) quite well, and give a mean value of the number of junctions around 5, which is in agreement with the previous values. Sample E is different, because of its 3 times bigger diameter (150 nm). As the diameter of the wire increases, the equivalent behavior goes from a 1-dimensional

The first and leading term in this expression, 1/N R, comes from the classical addition of junction resistances. The second term, G WL , is the weak localization correction to the conductance, while the last term, dI CB /dV , is the interaction correction to the current originating from Coulomb blockade effects. Both G W L and dI CB /dV are proportional to e 2 R/2πn × 1 and formally supposed to be small compared to the leading classical contribution.

Let us first consider the weak localization correction to the conductance G WL . Provided the dephasing length l φ exceeds the size of a single quantum dot and the magnetic field is zero, the weak localization correction reads [59] 2 chain of tunnel junctions to a 3-dimensional one. Then, G WL = e l φ , (13) many chains in parallel appear, and electrons have many paths to go through the wire. Therefore, the final resistance results from adding up several activation expo-πnN d where l φ is the dephasing length and d is the size of a quantum dot. Note that if we introduce the length of the sample L = N d, equation ( 13) coincides with the standard result derived for diffusive metals (see e.g. Ref. [START_REF] Altshuler | Electron Electron Interactions in Disordered Systems[END_REF]). Since the temperatures in the experiment, T ;:: 1 2 K, are relatively high and exceed the level spacing in a single this case and for τD » RC the correction dI CB /dV takes the form [60] quantum dot δ, we can use the expression for l φ derived dI CB e 2 N 1 = Re Ψ 1 + n by Altshuler, Aronov, Khmelnitski [38] dV πn N 2 2πk B T RC

DG D Ln 2 1/3 + ieV + n + ieV l φ = a e 2 k B T . ( 14
) 2πk B T 2πk B T RC n ieV 2πk B T
Here a is a numerical prefactor. In order to adapt this formula to the array of quantum dots we use the expression

Ψ 1 + 2πk B ieV + T RC 2πk B T
ieV for the diffusion constant (see e.g. [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]59]) defined by the Drude conductivity

σ ≡ d/(W R) = 2e 2 D/(W d δ) (W is Ψ 1 + 2πk T 2πk B T
the diameter). Besides that, we rewrite the product G D L in the form

G D L = L/N R = d/R. Thus we obtain Ψ 1 + ieV × 2πk B T l ( 16
)
2 δ l φ = ad 1/3 . ( 15
)
where Ψ (x) = Γ t (x) is the digamma function. In the limit 2e 4 R 2 k B T Next we turn to the interaction correction dI CB /dV . The dwell time is defined by the expression τ D = e 2 R/δ. In the Likewise for 2πk B T ;S eV ;S n/RC we find

1 G(V ) = 1 + N RG WL + N 1 e 2 R NR eV RC l N πn × ln n . ( 18
)
As we have discussed earlier, equations ( 17) and ( 18) are the result of the lowest order expansion in the dimensionless parameter e 2 R/πn × 1. In this limit and at sufficiently high temperatures and voltages (k B T, eV ;:: E a , where Ea will be defined below, see Eq. ( 25)), we may formally replace equations ( 17) and ( 18) by the following ones:

e 2 2 α -(1+γ )α π αk B T G(T ) ≈ πnN α e e 2 E C παeV α , (19)
G(V ) ≈ πnN α 2E C . ( 20
)
Here we defined the charging energy E C = e 2 /2C and the exponent Although the weak localization correction

G WL ∝ T -1/3 (N 1)e 2 R α = πN n (1 NR|G WL |) . ( 21
)
depends on temperature, this dependence is weak and in the wide range of temperatures α approximately remains constant. This result justifies the analysis of the data presented in Figure 6. At the same time α is sensitive to magnetic field through the same weak localization cor-case the exponent in E a is mostly determined by the junctions with lower resistances, while α is mostly sensitive to high resistances. Therefore the activation energy turns out to be suppressed, E a ;S e -1/α . Alternatively, additional suppression of the activation energy may be explained if we assume that the tunnel junctions have few channels with significant transmission probability T n ~ 1. According to Nazarov [62] in this case one finds the activation energy in the form (27) with c > 1.

T ≈ C ≈ ≈
Although equations ( 19), ( 20), ( 23), ( 25) are formally justified in the limit α × 1, they seem to describe experimental data quite well even for α > 1. Since the rection G WL . In the metallic limit we have NR|G WL | × 1, N » 1 and the exponent α acquires a very simple form range α ~ do not c 1 is most difficult for theoretical studies, we pletely understand why it happens. We can

e 2 R α . ( 22

) πn

Let us now express the dephasing length l φ (15) in terms of α. The level spacing in a quantum dot depends on its size as δ ~ 1/N 0 d 3 , where N 0 is the density of states at the Fermi level. Therefore, b om speculate that if the junctions in the array have different resistances one can approximately split the system into several parts consisting of one highly resistive junction and several low resistive ones. The low resistive junctions form an Ohmic environment for the poorly conducting one. The conductance of every part is then described by the theory of Coulomb blockade in the presence of electromagnetic environment [START_REF] Ingold | Single Charge Tunneling[END_REF], which predicts power law de-

α l φ = , ( 23
) pendence G(T ) ~ T . The resistance of the whole sample α 2/3 (N 0 k B T ) 1/3
is simply the sum of resistances of the wire sections, and α where b is another numerical prefactor. Here we used the therefore the total conductance ∝T as well. More acsimplified expression for α (22). This result is in agreement with the empirical law equation ( 9) observed in Figure 8. Finally, we estimate the temperature T a at which the crossover to the activation behavior should occur. We define T a as the solution of equation G T (T a ) = 0, where G T is given by equation [START_REF] Yao | Carbon Nanotubes[END_REF]. T a sets the scale of temperatures below which the Coulomb blockade correction to the conductance becomes comparable to the linear conductance 1/N R + G WL . In this way we obtain curate approach to this problem should probably rely on the renormalization group equations formulated in reference [63].

Conclusion

Transport measurements of an ensemble of semiconductor nanowires and carbon nanotubes have been performed. The samples are obtained by electrodeposition and CVD E C a π 2 αk B e 1+γ -1/α . (24) in nanoporous membranes. Three different kinds of samples have been measured and compared: carbon nan-The activation energy E a should be approximately equal to k B T a (assuming that the prefactor ln(τ /τ 0 ) = 1: see last section), i.e. we can put otubes, Te nanowires, and silicon nanowires. Qualitatively, the transport parameters of all samples follow the same profile as a function of bias voltage, temperature, and magnetic field, whatever the material and the contacts.

E = k T = EC e a B a π 2 α 1+γ -1/α . ( 25
)
The Coulomb blockade regime dominates in the temperature range between about 50 K down to about 1 K.

Likewise we find the critical voltage

The Coulomb blockade regime is characterized by a typical scaling law of the temperature and bias voltage, from

V 2E C e - παe 1/α . ( 26
)
which a scaling coefficient α is deduced. The parameter α describes the disorder, and is a fingerprint of each sample.

In the Coulomb blockade regime, a weak localization like The activation energy (25) agrees with the estimates of the renormalized charging energy of a tunnel junction obtained in references [61][62][63]. Our experimental data rather suggest e -c/α behaviour is also observed in the magnetoresistance profile. These measurements allow the phase coherence length to be measured as a function of α. Beyond, we observe that the weak localization is destroyed at high voltage: an interplay between both effects is evidenced. Finally, below E a ~ , (27) α 1 K (down to 60 mK), a transition to activation conduction regime is observed at low enough bias potential. The where c ≈ 4, i.e. the measured activation energy is lower than that predicted by equation (25). Two different mechanisms may account for this discrepancy. First, the junctions in the array may have different resistances. In this activation energy has been measured as a function of the coefficient α.

We have observed that the three following parameters: conductance, phase coherence length, and activation energy are universal functions of the coefficient α (in the sense that it is valid for almost all samples measured).

In order to explain the universal laws, a theoretical model of a chain of quantum dots has been developed. This physical picture is supported by recent imaging experiments based on scanning probe techniques showing that nanowires or nanotubes break into a one-dimensional chain of quantum dots [3][4][5] at low temperatures.

Due to charge inhomogeneities, the wire is composed by high conducting regions separated by poorly conducting ones. Schematically, the low conducting regions are associated with tunnel barriers, that separate high conducting regions, quasi-ballistic at this scale, i.e. quantum dots.

The problem is then to understand the consequences of such a picture in terms of transport properties measured in the configuration of two point contacts. The conclusion is that the universal laws associated with the three different regimes are indeed rather well described within this model. The parameter α is defined in relation to the scaling law of the conductance, the phase coherence length, and the activation energy. However, the generalization of the theory to large exponents α is still unclear. Some arguments are proposed in order to relate the model of chain quantum dots to the environmental Coulomb blockade. Another point that remains to be clarified concerns the observed interplay between Coulomb blockade and weak localization for large α.

However, this study shows that a change of paradigm should be taken into account while considering the transport properties in carbon nanotubes or semiconductor nanowires. In the context of room temperature nanoelectronic devices, the tubes or wires are usually characterized with the parameters related to bulk semiconductor physics like band structure and mobility of charge carriers. At low temperature, the process of "breaking the wire into a chain of quantum dots" takes place, and the same object changes to single electron device, for which the relevant parameters are related to the disorder through the barriers that separates the quantum dots, and the quasiballistic properties inside the quantum dots. The relevant parameters are then reduced to the scaling coefficient α, the phase coherence length of the charge carriers, and the activation energy.

Fig. 1 .

 1 Fig. 1. (a) Sketch of a Te nanowire inside the polycarbonate nanoporous membrane. (b) Transmission microscopy of the wire. (c) High resolution transmission microscopy. (d) View (SEM) of the top of the membrane with the typical dendritic surface growth above the wires.

Fig. 2 .

 2 Fig. 2. (a) Sketch of a CNT or Si nanowire inside the alumina nanoporous membrane. Lateral view of the top of the membrane by transmission microscopy of (b) CNT, (c) Si nanowire emerging from membrane.

Fig. 3 .

 3 Fig. 3. (Color online) Typical transport properties of the samples studied (here, of a MWCNT): (a) conductance as a function of bias voltage at different temperatures, (b) zero bias conductance as a function of the temperature, (c) scaling plot GT -α (eV/kT).

Fig. 4 .

 4 Fig. 4. (Color online) Examples of scaling laws GT -α = f (eV /kT ) observed for: (a) a MWCNT of diameter d = 30 nm, (b) a TeNW of diameter d = 40 nm, and (c) a Si NW of diameter d = 30 nm.

Fig. 5 .

 5 Fig. 5. (Color online) Histogram of coefficients α obtained for TeNW and CNT samples measured. The parameter α is not directly defined by the intrinsic parameters of the samples.

Fig. 6 .

 6 Fig. 6. (Color online) Prefactor of the conductance G T (α) for about 100 different samples: CNT and TeNW with different contacts (Co and Ni). Lines: fit to equation (5) with 2 and 7 junctions.

Fig. 7 .

 7 Fig. 7. (Color online) (a) Magnetoconductance of a Te nanowire at several temperatures. Dotted lines: fit to equation (7) with parameter l φ . (b) Phase coherence length l φ (T ) as a function of the temperature. Grey line: fit to equation (8).

  at several temperatures. The variation due to the magnetic field is measured by the quantity e ∞) G(H ), where G(∞) is the saturation G T ≈ πnN α value of the conductance, i.e. at a field for which the weak localization correction is destroyed. The doted lines of Fig-G V ≈ πnN α 2E C . (6) ure 7a represent the fit to the following quasi 1D WL expression[38,[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]:with the charging energy E C = e 2 /2C , and N referring to the number of tunnel barriers (cf. Sect. 6).

  φ

Fig. 8 .

 8 Fig. 8. (Color online) Correlation between the phase coherence length l φ at 4.2 K and α for the 20 samples studied. The dotted line is the Fit defined by equation (9).

Fig. 9 .

 9 Fig. 9. (Color online) Zoom of the conductance as a function of bias voltage for different magnetic fields (0 T to 1.2 T), at 4.2 K and 8 K.parallel to probe the bias voltage dependence of the resistance. The response of the samples is measured by the lock-in after having been amplified by a 10 8 current amα(l φ ) = cl -3/2 (9) plifier, which allowed us to probe resistance up to the GΩ range. where the adjustable parameter c depends on the temperature. A derivation of this empirical law is reported in Section 6, for the model of the chain of quantum dots of typical size below l φ . Unfortunately, the adjustable parameter c does not give direct physical information on the system due to numerical constants difficult to express precisely (see the discussion of Eq. (15) in Sect. 6).It is worth pointing out that, in contrast with usual systems exhibiting WL (e.g. in metallic samples), the conductivity without WL (at saturating magnetic field) is not the usual Drude conductivity but the Coulomb blockade

10 -

 10

Fig. 10 .

 10 Fig. 10. (Color online) Conductance as a function of temperature at zero bias voltage for six samples: (a-d) 4 MWCNT (samples A,B,C and D), (e): TeNW of 150 nm diameter (sample E), and TeNW of 40 nm diameter (sample F). Dotted line: power law of the Coulomb blockade regime. Inset: Arrhenius plot at law temperatures (T ≤ T a ).

Fig. 11 .

 11 Fig. 11. (Color online) Arrhenius-like Plot of the conductance of sample E (bigger diameter: 150 nm) at zero DC bias and below T a . Dotted line: fit with the law G ∝ exp E a

Fig. 12 .

 12 Fig. 12. (Color online) Conductance versus DC bias voltage for each of the sample A to F at different temperature below T a , in a semi-log scale. We observe for each sample a strong increase of the resistance bellow a critical voltage V C .

ΓFig. 13 .

 13 Fig. 13. (Color online) Scaling plot of the function GT -α as a function of the ratio eV /kT (same data as Fig. 12). Dotted line: scaling law.

Fig. 14 .

 14 (Color online) Activation energy E a (α) as a function of the scaling coefficient α for 6 samples. Dotted line: fit with equation(11) (see text).

Fig. 15 .

 15 Fig. 15. Model of the chain of quantum dots coupled by tunnel junctions.

Table 1 .

 1 Sample description: name, material, length L (μm), diameter d (nm), scaling exponent α, activation energy E a , critical energy E C = eV C , and transition temperature T a .

	A	MWCNT	1	30	2.5	0.81	7.25	7-8
	B	MWCNT	1	30	2	0.82	5	5-6
	C	MWCNT	1	30	0.85	0.08	0.48	0.2-0.3
	D	MWCNT	1	30	1.2	0.45	3	4-5
	E	TeNW	6	150	0.17	0.04	0.18	6-7
	F	TeNW	6	40	1.1	0.22	1	4-5

Name

Material

L (μm) d (nm) α E a (meV) E c (meV) T a (K)


Aurelien Latouche
email: aurelien.latouche@cnam.fr

Arthur Allignol

Jan Beyersmann

Myriam Labopin

J P Fine

A competing risks analysis should report results on all cause-specific hazards and cumulative incidence functions

Keywords: Competing risks, Cumulative Incidence, endpoints, bone marrow transplant

published or not. The documents may come

A competing risks analysis should report results on all

cause-specific hazards and cumulative incidence functions.

Introduction

In hematopoietic stem cell transplantation (HSCT), the competing risks endpoints are usually relapse and treatment-related mortality (TRM). For each patient, we record an observation time that is the minimum of a failure time, and a censoring time and a status indicator that capture the failure type information.

The status indicator is generally coded as 0 if the observation is censored, 1 if the observed cause of failure is the event of interest, and 2, 3, . . . if the observed cause of failure arises from the competing events.

The quantities of clinical and statistical interests are the cause-specific hazard and the cumulative incidence function. The former refers to the instantaneous rate of occurrence of a given event amongst the patients still event-free, whereas the latter is the probability of occurrence of a given event by time, t. In other words, the cumulative incidence denotes the expected proportion of patients with a certain event over the course of time. It has been well documented that the analysis of the cause-specific hazard of a particular event does not suffice for estimation of that event's corresponding cumulative incidence function. As a result, the Kaplan-Meier estimator, which naively disregards censoring from competing event, is an inappropriate method for estimating the cumulative incidence in the presence of competing events such as death in remission [1,2].

A critical point is that the effect of a covariate on the cause-specific hazard for a particular cause can be different from its effect on the cumulative incidence of the corresponding cause [START_REF] Gray | A Class K-Sample Tests for comparing the Cumulative Incidence of a Competing Risk[END_REF]. Thus, investigators are usually advised to choose carefully that quantity which has the most relevant clinical interpretation and importance from a biomedical perspective. However, as both the cause-specific hazard and cumulative incidence provide particular insights, this advice might be too restrictive in practice and lead to difficulties in interpretation and study planning.

Competing risks endpoints are commonly analyzed by using proportional cause-specific hazard models and/or proportional subdistribution hazards models. The first approach requires that each cause-specific hazard follows a Cox model [START_REF] Andersen | Counting Process Models for Life History Data: A Review (C/R: P141-158)[END_REF][START_REF] Keiding | Multi-state models and outcome prediction in bone marrow transplantation[END_REF][START_REF] Klein | Statistical methods for the analysis and presentation of the results of bone marrow transplants. Part 2: Regression modeling[END_REF]. The second approach, also known as the Fine-Gray model, is also a Cox model, but for the subdistribution hazard [START_REF] Gray | A Class K-Sample Tests for comparing the Cumulative Incidence of a Competing Risk[END_REF] attached to the cumulative incidence [START_REF] Fine | A proportional Hazards Model for Subdistribution of a Competing Risk[END_REF]. In the sequel, we reserve the term Cox model for Cox regression on the cause-specific hazard (CSH), and we use Fine-Gray regression for proportional hazards modeling of the subdistribution hazard (SH).

Often only one set of analyses is reported, either cause-specific hazard or cumulative incidence. When both sets of analyses are reported, the interpretation rarely connects the results. The aim of this paper is to offer practical guidance on how to synthesize findings across causes. Crucial points are:

1. An important issue when simultaneously conducting all of the analyses mentioned above is that under one of the popular model choices, the other model will be misspecified. In other words, if a Cox model is postulated for the event of interest and the competing event, the proportionality assumption of a Fine-Gray model for the event of interest may not hold and vice versa [START_REF] Latouche | Misspecified regression model for the subdistribution hazard of a competing risk[END_REF][START_REF] Beyersmann | Misspecified regression model for the subdistribution hazard of a competing risk[END_REF]. We advocate a practical approach to addressing such misspecification by including covariate time interaction terms, as originally suggested in [START_REF] Fine | A proportional Hazards Model for Subdistribution of a Competing Risk[END_REF]. This singular setting, often encountered in day-to-day data analysis, requires some guidance for use and interpretation of these models. Such issues have not been addressed in previous overview papers [2,[START_REF] Logan | Regression models for hazard rates versus cumulative incidence probabilities in hematopoietic cell transplantation data[END_REF][START_REF] Scrucca | Competing risk analysis using R: an easy guide for clinicians[END_REF][START_REF] Scrucca | Regression modeling of competing risk using R: an in depth guide for clinicians[END_REF]. 2. The presentation of the results may also lead to confusion in interpretation. Often, the analyses of cause-specific hazard and cumulative incidence are reported side-by-side, simply as hazard ratio or HR, without clearly distinguishing between the two approaches [START_REF] Aoudjhane | Comparative outcome of reduced intensity and myeloablative conditioning regimen in HLA identical sibling allogeneic haematopoietic stem cell transplantation for patients older than 50 years of age with acute myeloblastic leukaemia: a retrospective survey from the Acute Leukemia Working Party (ALWP) of the European group for Blood and Marrow Transplantation (EBMT)[END_REF][START_REF] Armand | Allogeneic transplantation with reduced-intensity conditioning for Hodgkin and non-Hodgkin lymphoma: importance of histology for outcome[END_REF][START_REF] Tomblyn | Similar and promising outcomes in lymphoma patients treated with myeloablative or nonmyeloablative conditioning and allogeneic hematopoietic cell transplantation[END_REF]. This underlines a common misconception that the two models are essentially the same and capture the same information. We thus suggest a terminology for each model as recently advocated in the analysis of epidemiological data by [START_REF] Lau | Competing risk regression models for epidemiologic data[END_REF]. 3. Although originally proposed for the summary analysis of a single event of interest, it is now not unusual that Fine-Gray models are fitted for each of the competing events [START_REF] Herr | HLA-identical sibling allogeneic peripheral blood stem cell transplantation with reduced intensity conditioning compared to autologous peripheral blood stem cell transplantation for elderly patients with de novo acute myeloid leukemia[END_REF][START_REF] Putter | Tutorial in biostatistics: competing risks and multi-state models[END_REF][START_REF] Scrucca | Regression modeling of competing risk using R: an in depth guide for clinicians[END_REF]. However, Fine-Gray models cannot generally hold simultaneously for all causes if one postulates such models at all time points. Importantly, it is possible that such models may hold over restricted time ranges, which has practical implications for studies with limited longitudinal follow-up. That is, the models may hold simultaneously up to the longest follow-up time. For situations where non-proportionality is evidence, covariate time interaction terms may be employed to improve model fit.

While we offer guidance on how to deal with these issues, we also note that there appears to be no final consensus on how to analyse competing risks endpoints.

In this paper, we argue that a complete understanding of the effect of prognostic factor on competing risk endpoints requires modelling both cause-specific hazards and cumulative incidences side-by-side.

Worked example: MAC versus RIC

For ease of presentation, the formal definition of the hazard functions and the proportional hazards regression models for the cause-specific hazard and cumulative incidence function are deferred to the supplementary material.

We consider data from the EBMT acute leukemia working party comparing outcomes from a reduced intensity conditioning regimen (RIC) in the HLA identical hematopoietic stem cell transplantation (HSCT) with those after myeloablative conditioning regimen (MAC) in patients with acute myeloblastic leukaemia (AML) over 50 years of age. With this aim, outcomes of 315 RIC HSCT recipients were compared with those of 407 MAC HSCT recipients [START_REF] Aoudjhane | Comparative outcome of reduced intensity and myeloablative conditioning regimen in HLA identical sibling allogeneic haematopoietic stem cell transplantation for patients older than 50 years of age with acute myeloblastic leukaemia: a retrospective survey from the Acute Leukemia Working Party (ALWP) of the European group for Blood and Marrow Transplantation (EBMT)[END_REF]. We begin by focusing on prognostic factors for Relapse. The competing endpoints are thus Relapse (n=182) and TRM (n=164), with 376 patients censored over 2 years of follow-up . The main prognostic factor in these analyses is the status of the disease at the time of transplant. The majority of patients achieve remission before transplantation but some diseases are resistant to chemotherapy so patients are transplanted in the refractory or relapse phase of the disease (referred to as advanced status). The covariates of clinical interest considered in this example are the conditioning regimen (MAC (56%) vs RIC (44%), as well as the disease status at transplantation (Other (72%) vs. advanced status (28%)). A proportional hazards regression model for both the CSHs and the SHs will be employed for the univariate and multivariate analyses, i.e. including regimen and status at transplant. The estimated effects of regimen and disease status are summarized in Table (1) and Table (2), using the notation Cause-Specific Hazard Ratio (CSHR) and Subdistribution Hazard Ratio (SHR).

Cumulative cause-specific hazards for both endpoints as well as the cumulative incidences are displayed in Figures (1) and (2) respectively. The reason for displaying these curves is to facilitate the interpretation of the effect of a covariate on the cause-specific hazard or the cumulative incidence.

Goodness of Fit

Both the Cox model and the Fine-Gray model rely on the key assumption of proportionality of CSH or SH. It is assumed that the CHSR (or SHR) does not depend on time. The alternative to the proportional hazards (PH) assumption is to allow the hazard ratio to vary over time. This can be checked by various methods, as described by [START_REF] Klein | Statistical methods for the analysis and presentation of the results of bone marrow transplants. Part 2: Regression modeling[END_REF][START_REF] Klein | Survival Analysis: Techniques for Censored and Truncated Data[END_REF][START_REF] Bradburn | Survival analysis Part III: multivariate data analysis -choosing a model and assessing its adequacy and fit[END_REF].

Using Schoenfeld residuals, the PH assumption of the CSH of relapse was met for treatment (p=0.39) and for status at transplantation (p=0.17). On the contrary, the PH assumption of the CSH of TRM was met neither for the treatment (p=0.02) nor for status at transplantation (p=0.01). However, the regression parameter estimates can be interpreted as the average effect on the rate of TRM. In depth study of the time-dependent effect with CSH can be found in [START_REF] Scheike | Analyzing Competing Risk Data Using the R timereg Package[END_REF]. The cox.zph function was employed with multivariate models including treatment and status at transplantation.

Checking the proportional assumption for a Fine-Gray model is seldom done but can be conducted using the log minus log of subdistribution hazard or using Schoenfeld residuals tailored for the subdistribution hazards, as described in [START_REF] Fine | A proportional Hazards Model for Subdistribution of a Competing Risk[END_REF].

If the PH assumption is not met for a given covariate, stratification on this factor is advisable if the effect on this variable on the competing endpoint is not of prime interest. This has been recently extended to the Fine-Gray model [START_REF] Zhou | Competing risks regression for stratified data[END_REF]. In addition to stratification, one may include time by covariates interaction terms, which enable the testing of covariate effects in the context of regression modelling, unlike stratification.

The PH assumption for the Fine-Gray model was investigated by testing for time by covariate interaction in a multivariate analysis. The PH assumption for the SH of relapse was met for both covariates (e.g. no significant time by treatment interaction, nor time by status at transplantation interaction). On the contrary, the PH assumption for the SH of TRM was not met for the status at transplantation (borderline significant p=0.05).

MAC VS. RIC

The RIC regimen is significantly associated with an increase in the rate of relapse (CSHR 1.56, 95% confidence interval [1.16; 2.10]) while it significantly decreases the rate of TRM (CSHR 0.50, 95% confidence interval [0.36; 0.71]).

These results are in line with those displayed by the cumulative incidence analyses. Indeed, the RIC regimen increases the probability of relapse (SHR 1.83, 95% confidence interval [1.36; 2.45]) while reducing that of TRM (SHR 0.50 [0.35; 0.68]).

The CSHRs for the two competing events are in opposite directions. This facilitates the interpretation of the impact of treatment on the cumulative incidence of relapse. Indeed, a higher rate of relapse for RIC patients associated with a reduced rate of TRM implies that we will observe more relapse in the RIC group at the end of the study. Moreover, as both the cause-specific and the cumulative incidence analyses are consistent with each other, we can interpret the regimen effect on the cumulative incidence of relapse as an actual effect (without being necessarily causal), and not as an indirect effect on the competing event.

Disease Status

Patients having advanced disease have a higher rate of relapse (CSHR 3.70 [2.77; 4.96]) and also a slightly elevated (not statistically significant) rate of TRM (CSHR 1.30 [0.92; 1.82]). Advanced disease status also displays a strong effect on the cumulative incidence of relapse (SHR 3.20 [2.40;4.27]) but a rather small non-significant effect on the cumulative incidence of TRM (SHR 1.07 [0.75; 1.51]).

Both CSHRs are greater than 1. Such unidirectional treatment effects on the CSHRs is the most difficult situation to interpret in terms of cumulative incidence functions [START_REF] Lau | Competing risk regression models for epidemiologic data[END_REF][START_REF] Allignol | Understanding competing risks: a simulation point of view[END_REF]. Because the effect of disease status on the CSHR of relapse is so strong (Figure (1)), it appears that the direct effect of disease status via the CSHR of relapse may be the primary reason for the large differences in the corresponding cumulative incidence functions.

Multivariate analyses

Results of the multivariate analyses i.e., including regimen and status at transplant, are displayed in Table (2) and are in agreement with those from Table (1), but with tighter confidence intervals.

Finally, we recall that the case study is not a randomized experiment so some confounders are present.

Discussion

The current proposal supplements previous works on the analysis and the reporting of competing endpoints [2, [START_REF] Logan | Regression models for hazard rates versus cumulative incidence probabilities in hematopoietic cell transplantation data[END_REF][START_REF] Scrucca | Competing risk analysis using R: an easy guide for clinicians[END_REF][START_REF] Scrucca | Regression modeling of competing risk using R: an in depth guide for clinicians[END_REF][START_REF] Tourneau | Reporting of time-to-event end points and tracking of failures in randomized trials of radiotherapy with or without any concomitant anticancer agent for locally advanced head and neck cancer[END_REF][START_REF] Lau | Competing risk regression models for epidemiologic data[END_REF]. When analysing the effects of prognostic factors on competing risk endpoints, we suggest using the Cox model and Fine-Gray model, presenting the results for all causes side by side. This was illustrated in a recent data set where insights from the different models were connected and reconciled in a unified interpretation.

From a practical point of view, if a particular endpoint is of clinical interest, we recommend:

• Using a distinct terminology for each model of the hazard ratio namely CSHR for Cox model and SHR for Fine-Gray model

• Reporting all the CSHRs

• Reporting the SHR for the event of interest and the SHR for the competing event

• Presenting the results in a unified interpretation, so as to connect and reconcile results from the two sets of models

• Explicitly checking the PH assumption for Cox and Fine-Gray models

• Providing plots of all cumulative incidences for categorical variables, in order to better understand whether the effect of such factors on the subdistribution hazard ratio of a particular endpoint is either direct (e.g, on the cause-specific hazard ratio of that endpoint), or indirect (e.g., on the cause-specific hazard ratio of competing endpoints), or both direct and indirect

If there is more than one competing event, and the research question focuses on one event of interest, the competing causes of failure may be aggregated together in a single endpoint to simplify the analysis. We have not detailed other nonproportional hazards regression models for the cause-specific hazard and the cumulative incidence function which have been proposed for analyzing competing risks data, owing to their lack of use in real applications. A comprehensive discussion of such models can be found in [START_REF] Logan | Regression models for hazard rates versus cumulative incidence probabilities in hematopoietic cell transplantation data[END_REF][START_REF] Scheike | Analyzing Competing Risk Data Using the R timereg Package[END_REF][START_REF] Klein | Modelling competing risks in cancer studies[END_REF][START_REF] Zhang | Modeling cumulative incidence function for competing risks data[END_REF]. Further details on including time-dependent covariates in proportional hazards models for competing risks data, which has been used in practice, may be found in [START_REF] Cortese | Competing risks and time-dependent covariates[END_REF].

Figure 1 :

 1 Figure 1: Cumulative hazards of relapse (top) and TRM (bottom) for the binary covariates treatment (left) and disease status (right)

Figure 2 :

 2 Figure 2: Cumulative incidences of relapse (top) and TRM (bottom) for the binary covariates treatment (left), and disease status (right)

Table 1 :

 1 Estimated cause specific hazard ratio and subdistribution hazard ratio for Relapse and TRM using univariate regression model. .16; 2.10] p < 10 -3 0.50 [0.36; 0.71] p < 10 -3 Status(other vs advanced) 3.70 [2.77; 4.96] p < 10 -3 1.30 [0.92; 1.82] p = 0.13 10 -3 0.50 [0.35; 0.68] p < 10 -3 Status (other vs advanced) 3.20 [2.40; 4.27] p < 10 -3 1.07 [0.75; 1.51] p = 0.71

		Cox: cause specific hazard ratio
	Covariates	Relapse	TRM
	MAC vs RIC	1.56 [1Fine-Gray: subdistribution hazard ratio
		Relapse	TRM
	MAC vs RIC	1.83 [1.36; 2.45] p <

Acknowledgments

The research reported here was supported by the EU funded CLINT project, (Facilitating International Prospective Clinical Trials in Stem Cell Transplantation, LSHB-CT-2007-037662). We wish to thank the Acute Leukemia Working Party for providing the data.

Supplementary material

Supplementary data and R code related to this article can be found online at [2] Kim HT. Cumulative incidence in competing risks data and competing risks regression analysis. Clin Cancer Res. 2007;13(2 Pt 1):559-565.