Rémi Emonet
email: remi.emonet@univ-st-etienne.fr

E Oberzaucher
email: elisabeth.oberzaucher@univie.ac.at

J.-M Odobez
email: odobez@idiap.ch

What to Show? Automatic Stream Selection Among Multiple Sensors

Keywords: stream selection, camera network, probabilistic models, temporal topic models

The installation of surveillance networks has been growing exponentially in the last decade. In practice, videos from large surveillance networks are almost never watched, and it is frequent to see surveillance video wall monitors showing empty scenes. There is thus a need to design methods to continuously select streams to be shown to human operators. This paper addresses this issue and make three main contributions: it introduces and investigates, for the first time in the literature, the live stream selection task; based on the theory of social attention, it formalizes a way of obtaining some ground truth for the task and hence a way of evaluating stream selection algorithms; and finally, it proposes a two-step approach to solve this task and compares different approaches for interestingness rating using our framework. Experiments conducted on 9 cameras from a metro station and 5 hours of data randomly selected over one week show that, while complex unsupervised activity modeling algorithms achieve good performance, simpler approaches based on amount of motion perform almost as well for this type of indoor setting.

Introduction and Previous Work

There is an ever increasing number of sensors deployed in the real world, being it in large scale sensor networks or closed-circuit television (CCTV) networks. Exploitation of these cameras fulfill several needs: safety, when the physical integrity of people in a given environment need to be ensured (e.g., when people go in or out a metro train, or by detecting events that could lead to accidents in traffic infrastructures); security and protection of equipment through the detection of anti-social behaviors and vandalism; and efficiency, e.g., through the identification of typical flux and detection of congestions that can be communicated appropriately to users. However, most of the time, surveillance network videos are never watched. For instance, in our case, 28 monitors are used in the control room to supervise more than 800 cameras. The probability to watch the right streams at the right time is therefore very limited. Moreover, the operator vigilance also appears to be a core criterion that strongly influences CCTV effectiveness, especially when a constant human monitoring is required. As vigilance studies confirm, operators who spend hours screen gazing at static scenes tend to become bored and less efficient, and are then likely to miss low-frequency events, such as a person falling or jumping over a turnstile.

Thus, along with top-down user-based protocols for visualizing and browsing the video networks, and the detection of predefined event, there is a need for bottom-up automatic and content-based selection systems that provide surveillance operators with the most salient and informative data streams within an adaptive real-time process. To address this issue, unusual event detection [START_REF] Zhong | Detecting unusual activity in video[END_REF][START_REF] Boiman | Detecting irregularities in images and in video[END_REF][START_REF] Hua | Abnormal activity recognition based on hdphmm models[END_REF] and abnormality rating algorithms [START_REF] Zelniker | Global abnormal behaviour detection using a network of cctv cameras[END_REF][START_REF] Varadarajan | Topic models for scene analysis and abnormality detection[END_REF][START_REF] Emonet | Multi-camera Open Space Human Activity Discovery for Anomaly Detection[END_REF] have been proposed in the past. Although these algorithms are obviously related to this paper, none of them has addressed the stream selection task as we propose in this paper.

Commercial systems tends to use the amount of motion and person tracking in order to select cameras with a high level of activity, in this paper we try to investigate how relevant this choice is. Recent related work also includes work that both model the content of the videos and monitor the operator with a camera such as in [START_REF] Atrey | Effective multimedia surveillance using a humancentric approach[END_REF]. We consider a setup where observing the operator is not an option and where only the content of the videos is used to select the stream to display.

In this article, we explore the stream selection issue. First, in Section 2, we formalize the task. Then, in Section 3, we propose an approach for collecting ground truth for the task, based on human estimation of interestingness done by 100 persons on videos coming from 9 cameras and spread over one week. Section 4 describes the approaches used to solve this task together with the performance measure we propose to evaluate these approaches. The experimental results are detailed in Section 5. Finally, our conclusions and future work is presented in Section 6.

Defining the Stream Selection Task

The ultimate goal of stream selection algorithms is to allow the automatic pre-selection of some audio or video streams to present to a human operator for further screening. More formally, given a set of N streams, the system must continuously select the M ones that are the most meaningful to show to an operator. Defining what relevance criterion has to be used is an open problem and in the following sections, we use human attention as a cue. Also, even if not considered in our evaluations, the selection must be consistent in time, for example it should avoid screen flickering and avoid showing continuously the same views (unless it is sure these are very relevant).

Figure 1 illustrates the context and the task. In Figure 1a), an operator looking at a video wall is depicted. In Figure 1b) the stream selection task is shown, with the N input streams and the M streams selected by the system and to be shown to the operator.

Social Attention and Ground Truth

The ground truth dataset eventually used for the evaluation of the stream selection task is based on an experiment on social attention, described in Section 3.1. We follow the idea that human attention can act as a guide for the selection of interesting sensors.

Button clicking experiments

We carried out an experiment where we simultaneously showed multiple movies to our subjects, and they had to press a button corresponding to the screen that showed something interesting. For this purpose, we created a set of 2-minute long movie chunks. We considered 9 cameras from a camera network deployed in a public transportation system. We randomly picked 150 starting times across a week period, and for each starting time, we created 2-minute chunks for all 9 camera views starting at these times. These 150×9 movies are the basis for the attention study.

An environment called the "Virtual Operator" have been used for data collection. Figure 2 shows a picture of the virtual operator setup. The virtual operator simultaneously shows four synchronized video streams on a 2 by 2 grid of screens. Each screen shows a shot from a different surveillance camera. Every two minutes, a new set of movies is randomly brought up to the screens.

Whenever the subject sees something interesting on one of the screens, he or she has to press a button indicating the respective monitor. The participants had to fill out some questionnaires prior to and after the experiment. In total 52 male and 51 female subjects (mean age 23.78) participated in the study. Each 2 minute chunk was watched between 14 and 18 times.

First, we see that attention is clearly not randomly distributed over the movies. The random probability of two subjects hitting the same button at the same time (same frame, at 5 frames per second) is 0.0016. By analyzing the distribution of the button clicks, we observe that clicks tend to cluster around some instants and views of interest.

Overall, we can retained the following observations: a) the attention is not randomly distributed, b) there are certain scene properties that suggest attention, and c) there is some degree of random clicks. This experiment suggests that, once cleaned, the button clicks can be a good indicator of the video stream interestingness. We thus use these interestingness clicks for the creation of a ground truth, making the assumption that the interestingness clicks are related to which stream should be selected in the task of stream selection.

Derived ground truth

From the button clicks, we derive a "annotation database" that we will use for the evaluation of stream selection algorithms. Overall, the annotation database can be seen as a list of annotated instants where people marked a single view. Each instant has a timestamp, the ordered list (set) of the views shown and the index of the view that was marked by the subject. More formally we can define the annotation database as:

A = {A i } i=1..n = {(t i ,V i , u i )} i=1..n
where n is the number of annotations, A i is the i th annotation in the list, t i is its timestamp, V i is the list of shown views and u i is the index of the view marked as interesting (V i (u i ) thus being the selected view). In the raw button click dataset presented before, there are 9 possible views (i.e., each element of V i can take 9 different values) and n is almost 10000 and realized by around 100 subjects. The timestamp is a pointer to any instant (a frame) within one of the 150 slots (each, long of 2 minutes) that were extracted over one full week. Each annotation provide information about the selected view versus the 3 others but does not provide information between these 3 other views.

There is a actually a mismatch between the button clicks and the goal of the stream selection system: the complete system should be able to continuously select which stream(s) to show, while the button clicks are sparse. For the evaluation measure presented in Section 4.1, we thus make the assumption that only the instant that have been clicked by some users are meaningful for the evaluation. Said the other way around: on the periods where there are no clicks, the selection algorithms can produce any results without being penalized.

Robust ground truth

We observed some user clicks are random. Also, as some subjects might be quicker to respond than others, it is problematic to instantaneously link the button clicks to the respective scene in the video. Apart from considering the raw button clicks we also considered a smoothed version of the "annotations".

We run a Gaussian smoothing function over the click results, generating a continuous index of interestingness (IoI). The IoI is taking into account multiple button presses within a small time span accounting for the temporal uncertainty of the raw button clicks. A threshold on the IoI value is used to decide whether an annotation should be kept. The threshold can be tuned to control the robustness of the resulting dataset. The provided results are obtained with around 10000 annotation for the raw case and around 2000 for the smoothed one.

Experimental setup

The goal of a stream selection system is to select the M among N streams to display to an operator. Almost equivalently, we can consider that the system task is to rank the streams by interestingness and the first ones are displayed. We thus suppose that the output of an algorithm algo at any time instant t eval and for a set of views V eval is a ranking of the camera views (or their indices). For instance, algo(t eval ,V eval ) = (3, 1, 2, 4) means that the third streams is the most interesting to show, then the first stream, then second and fourth. For performance measurement, we make the assumption that the instants where stream selection matters are only the ones which have been clicked by the human subjects.

Performance measure

At each marked instants, the ideal selection algorithm would rank first the stream that was marked by the human subject. A less effective algorithm would rank this stream second, and so on, the worse in our case being to rank it 4 (as there are 4 views shown at any time). To analyze the performance of an algorithm, we can thus look at the distribution of the rank produced by the algorithm for the clicked instant. More precisely, we compute the proportion of times the algorithm ranked the mark stream first, second, third and fourth: ∀r ∈ 1..4

(1)

p algo (r) = 1 n ×Card{i|algo(t i ,V i )(r) = u i } (2) = 1 n × num. clicked u i ranked r by algo (3)
This distribution (p algo (r)) in itself is a complete description of the performance of the algorithm. From this distribution, we could extract more synthetic numerical performance measures by giving different weights to the different ranking errors. For example, if we are interested in the percentage of time the algorithm performs in perfect accordance with the annotations, we can just take the value of p algo (1). Average precision is algo a common measure and corresponds to ∑ 4 r=1 1 r p algo (r).

Modular architecture for selection

The stream selection algorithms are responsible for providing a ranking or a selection of the cameras to show to the operator. Most algorithms are providing an interestingness rating at each instant for a camera.

If we want to be able to integrate information across cameras and to mix different algorithms in a system (some can be more effective on some cameras), we have to consider a kind of normalization of the measures across cameras. The overall stream ranking system follows the architecture shown in the Figure 3:

• each video stream is processed independently by the interestingness/abnormality rating algorithm;

• a possible normalization step is performed in order to provide comparable abnormality measures across views and algorithms. Normalization parameters are typically learnt without supervision using statistics on the abnormality measure on some (unlabeled) learning videos.

• the normalized abnormality scores are used as the inputs of a ranking algorithm whose goal is to produce the final ranking of the camera views. The simplest ranking algorithm (that we have used in the reported experiments) just ranks the views according to their normalized abnormality score. Temporal integration (not done here), which ensures stability in the views shown to the operator, would also take place in the final ranking algorithm.

In the experiments, we consider a single stream ranking algorithm: it simply sorts the streams based on their normalized interestingness measure. For the stream interestingness rating algorithms and for the normalization algorithms, we considered different algorithms detailed in the following subsections.

Considered interestingness rating algorithms

We consider multiple algorithms: two simple baseline algorithms, a method from the state of the art [START_REF] Emonet | Multi-camera Open Space Human Activity Discovery for Anomaly Detection[END_REF] with variations, and a mixture of methods.

Random -we just simulate a random interestingness score at any instant. This pseudo-algorithm is used only to illustrate the random factor inherent to the task. This random baseline can also be seen as a random ranking of the 4 views of interest. it is equivalent to ranking all the views in a uniform random manner and thus having 25% of probability for any of the 4 ranks.

Motion -we use the amount of motion as an interestingness measure given that studies on human attention have shown correlation between motion and attention. The abnormality score for each view is simply the amount of detected motion (proportion of image pixels where a motion could be robustly detected) using a Lucas Kanade method. With no normalization, this score will tend to display the camera with absolutely more motion. With some per-view normalization, the score will favor cameras with more motion than the view average. PLSM -we use the Probabilistic Latent Sequential Motifs (PLSM) that has been used in [START_REF] Emonet | Multi-camera Open Space Human Activity Discovery for Anomaly Detection[END_REF] to extract an abnormality measure from video.

The principle of PLSM is to learn, in an unsupervised way, the temporal motifs that are recurrent in a temporal document. Applied to a video with localized motion features, the algorithm learns the typical activities, which are often typical trajectories of objects within the scene. The authors of [START_REF] Emonet | Multi-camera Open Space Human Activity Discovery for Anomaly Detection[END_REF] suppose that the recurrent motifs represent the usual, normal activities and use this supposition to derive an abnormality score. We use the abnormality score as an interestingness measure. PLSM has been run, on each view, on 2 hours of video data with the following parameters: 75 topic at low-level, 20 motifs of 15 seconds each. PLSM-LL -taking the exact same approach as PLSM, we also experimented with using the negative log-likelihood of the observations after the fitting of the PLSM model. The same parameters as for the PLSM measure have been used. MIX -we use a combination of methods, where half of the camera views are rated with "Motion" while the other half are rated with "PLSM".

Considered Normalization Methods

The interestingness rating produced by different algorithms and/or for different camera views might vary notably in their scale. To be able to merge the views in a meaningful manner, we need to re-scale them to a common range of values. Many ways of normalizing the scores can be imagined and implemented.

In our case we only consider automatic ways of re-normalizing the abnormality scores. All the normalization parameters are set using a learning datasets of 2 hours (without any supervision or annotation). The abnormality score is first extracted on this data (at each time instant) and then the resulting abnormality time series is analyzed to produce the normalization parameters. We considered two kind of normalization methods. nonorm (∅) -we just simply don't normalize the ab-normality measure. Given the discrepancy between different rating algorithms, this method is limited to the cases where only a single type of interestingness algorithm is used. Still, it makes sense to try this normalization (actually non-normalization). For example, the "motion" measure without normalization will actually favor the selection of the view with the absolute most motion. From the experiments, it can also be meaningful to not normalize the interestingness measure obtained from PLSM as it is also related to motion. percent99 (p99) -we explored a normalization based on the 99 th percentile. Using the learning set, we linearly re-normalize the abnormality using a scaling factor p 99 such that the observed interestingness score falls below a fixed threshold (e.g., 1) 99% of the time. The scaling factor is then used to proportionally rescale the abnormality, i.e, norm.score = score-minscore p 99 -minscore . This normalization can be seen as a robust equivalent of re-normalizing using the maximum observed value. percent95 (p95) -as with "percent99", we experimented with a normalization based on other percentiles, only 95% being shown (see results). We also experimented with other percentiles: for high percentiles, results are comparable to the 99 th and 95 th one, while smaller percentiles lead to lower results.

Results

We provide results using the two versions of the button clicking annotations. First with the raw annotations (about 10000 clicks), then with the Gaussiansmoothing approach (about 2000 clicks).

Results using the raw ground truth

We first evaluate the algorithms introduced previously on the complete ground truth. As mentioned in Section 3.3, this ground truth is actually noisy as the human subject do not agree exactly on the interesting streams. We evaluate the different algorithms together with the different normalization methods. Our algorithms are compared against a theoretical random algorithm that would rank the views randomly (no normalization is meaningful).

As for example in Figure 4, the result figures show the rank distributions for various interestingness algorithm, grouped by normalization method. In a given graph, there is a line per considered method. For a given method, the colored bars show the proportion of clicked views ranked 1 st , 2 nd , 3 rd and , 4 th by the algorithms. The red bar corresponds to p algo (1) and so on. More weight on the first (leftmost) ranks is better. We see for example that random (which is repeated on each graph) has a probability of 0.25 for each of the ranks. Methods that will be above 0.25 for the first rank, can be considered as better than random.

The normalization based on percentiles below 95% give systematically lower performance than the one with 95%. To improve readability, we thus omit these in the shown graphs. The results show that the consider algorithms, both motion based and PLSM based, perform better than the random guess. Despite the fact that the raw annotations have been found relatively noisy, the gain over random algorithms is important.

The results also show that the simple motionbased criterion is almost as good as the more elaborated PLSM algorithm. The main observation is that in this metro setting, human social attention, encoded in the form of our annotations, is very well linked to the amount of motion present in the video. Both kind of abnormality measures derived from PLSM seem also to be following the behavior of "Motion". However, in other contexts like traffic videos, where motion is present in the form or more regular patterns, the capacity of algorithms such as PLSM at filtering out normal activities would be beneficial to detect anomalies as compared to relying on motion only.

In this setup, we observe that normalization has a small negative on the results. The interpretation can be that human attention is directed toward the absolute amount of motion and not a motion amount relative to the normal amount. Actually, noise in the annotations is also part of the reason as shown in the following section.

Results on filtered dataset using the Gaussian smoother

We also evaluate the performance of the different algorithms on the ground truth after smoothing by a Gaussian kernel, as explained in Section 3.3. We varied the threshold on the smoothed index of interest so as to keep around 2000 evaluation points. Figure 5 shows results with the corresponding results. Globally, the methods exhibit better accuracy on the cleaned ground truth. We also observe a more marked effect of method mixing: results are notably lower than with each individual methods. When mixing methods, normalization also helps. However, as observed in the raw case, normalization degrades the results for individual methods. This again means that the attention is directed to absolute motion.

Overall, the results with the smooth annotations consolidate and complete what has been observed with the raw annotations:

• non-normalized version are preferable for both the motion-based measure and the PLSM-based one,

• when mixing methods of different scales, as expected, results degrades and the normalization helps,

• motion-based and PLSM-based selection both significantly outperform the random guess algorithm especially on cleaned ground truth,

• on the raw ground truth, the motion-based selection works best,

• motion-based methods and PLSM-based methods perform comparably,

• we reach good ranking accuracy of up to 0.6 meaning that 60% of the times the stream clicked by the human subject is ranked first among four streams; 85% of the times, the clicked stream is among the two firsts in the rank provided by the algorithm.

Conclusions and future work

We have introduced the stream selection task and proposed an evaluation protocol for it. The proposed evaluation is based on a social attention experiment where human subjects were shown four video streams at once and were asked to mark a video at any time they spotted something interesting in it. We tested various stream selection algorithms to rate the interestingness of the streams at any instant. The video instants marked by the subjects are used to evaluate how well the marked video has been ranked by a selection algorithm.

Our evaluations have shown that, considering all the raw annotations produced by the human subjects, all test approaches were better than random but mostly equivalent to a simple criterion based on the amount of motion. By filtering the annotations (removing the isolated ones) we obtained a more trustful Figure 4: Ranking results on the raw ground truth (9980 evaluation instants) coming directly from the button clicking experiment. For each method (row), the proportion of each rank position is given for the considered annotation points. More weight on the first (leftmost) ranks is better. set of annotations. On such more robust evaluation dataset, it has been shown that mixing interestingness measures of different scale actually poses problem and that the proposed normalization method helps in this case.

Overall, the results are encouraging and more interestingness rating algorithms need to be tested and evaluated in this framework. More advanced normalization methods needs also to be explored: we can expect to find a normalization procedure that allows to mix various algorithms without degrading their individual accuracy. We plan to explore a normalization based on robustly aligning every percentile of all interestingness measures. A possible per-view bias (e.g., a particular view being almost never interesting) could also be integrated in the normalization process if we accept some slight human supervision. Collecting more ground truth, especially with real operators which have been knowing the camera setup for months, and making it available for research purpose is also in our future work direction.

Figure 2 :

 2 Figure 2: The setup for the Virtual Operator experiment on social attention. The subject is exposed to 4 synchronized videos. Whenever something interesting is happening on one of the screen, the user have to press the button corresponding to the screen. Every two minutes, the videos are changed; and b) an illustration of the window-based and Gaussian-smoothing-based filtering for button clicks.

  Figure 1: a) An operator in front of a video wall. b) Stream selection task: at any time instant, select the M streams, among N, that are the most meaningful to show to a human operator.

Figure 3 :

 3 Figure3: Architecture for the stream selection. Each stream is processed independently: a first algorithm is extracting an interestingness measure and this measure is then normalized. The normalized measures are used by a simple algorithm that the streams based on their measures.

Figure 5 :

 5 Figure 5: Ranking results on the smoothed ground truth obtained by removing annotations with unsufficient supporting evidence (total, 2003 evaluation instants).

Acknowledgement

This work was funded by the European Union FP7 project VANAHEIM (248907) under the 7th framework program.


C Jourdana
email: clement.jourdana@imag.fr

P Pietra
email: pietra@imati.cnr.it

A hybrid classical-quantum transport model for the simulation of Carbon Nanotube transistors

Keywords: Quantum-classical coupling, Schrödinger equation, drift-diffusion, interface conditions, effective mass, confined nanostructures, carbon-nanotube FETs AMS Subject Classification: 65M60, 65Z05, 78A35, 82D37, 82D80

In this paper, we propose a hybrid classical-quantum approach to study the electron transport in strongly confined nanostructures. The device domain is made of an active zone (where quantum effects are strong) sandwiched between two electron reservoirs (where the transport is considered highly collisional). A one dimensional effective mass Schrödinger system is coupled with a drift-diffusion model, both taking into account the peculiarities due to the strong confinement and to the two dimensional transversal crystal structure. Interface conditions are built preserving the continuity of the total current. Self-consistent computations are performed coupling the hybrid transport equations with the resolution of a Poisson equation in the whole three dimensional domain. To illustrate this hybrid strategy, we present simulations of a gate-all-around single-walled Carbon Nanotube Field-Effect Transistor.

Introduction

The extreme miniaturization reached in nanoelectronics brings the necessity of using new models to describe accurately the electron transport. Obviously, in the recent semiconductor devices, quantum effects play an important role due to the extremely small dimensions. Nevertheless, some of these quantum effects generally take place in a localized region (for instance, around the double barrier in resonant tunneling diodes or in the active zone in short channel transistors). Since quantum transport simulations are complex and above all computationally expensive, it can be interesting to follow a geometrical hybrid strategy: use a quantum model in regions where quantum effects are strong and couple it to a model governed by classical mechanics in the rest of the device domain. A hybrid strategy is also motivated by the fact that collisions of charged particles are not easily included into quantum models but they are relevant for the functioning of devices. In Field-Effect Transistors FETs, an active zone is sandwiched between two largely doped regions (Source and Drain) considered as electron reservoirs, and there the transport is expected to be in a highly collisional regime.

In [START_REF] Abdallah | A hybrid kinetic-quantum model for stationary electron transport[END_REF], a coupled kinetic-quantum model has been introduced. A Boltzmann equation is used to define the density in the classical zones, instead a Schrödinger equation is chosen to describe the density in the quantum domain. So, the transport in the quantum region is considered ballistic, whereas the classical regions can be highly collisional with an appropriate collision operator. At interfaces, reflection-transmission coefficients are defined to give the boundary conditions of the Boltzmann equation. Inversely, the distribution function is used to construct the quantum density. In that paper, the author proves that the reflection-transmission conditions preserve the current.

Next, in [START_REF] Degond | A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations[END_REF], the Boltzmann equation and the reflection-transmission conditions are replaced by a drift-diffusion equation with appropriate interface conditions. These connection conditions are derived from those of [START_REF] Abdallah | A hybrid kinetic-quantum model for stationary electron transport[END_REF] through a diffusive approximation and a boundary layer analysis. In [START_REF] Baro | A 1d coupled Schrödinger driftdiffusion model including collisions[END_REF], the strategy to couple the drift-diffusion Schrödinger system is quite different since the coupling is direct and authors get an analytic expression of the connection conditions by writing the exact continuity of the current at the interfaces. Differently, in [START_REF] Degond | A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations[END_REF], due to the diffusion approximation, the continuity of the classical and the quantum current is only preserved up to an order α, where α is the small parameter of the diffusion approximation. We also mention [START_REF] Ayyadi | Semiconductor simulations using a coupled quantum driftdiffusion Schrödinger-Poisson model[END_REF] where a hybrid strategy is studied with a quantum drift-diffusion equation. We point out that all these hybrid approaches are different from a "dimensional hybrid coupling" (see [START_REF] Ben Abdallah | Diffusive transport of partially quantized particles: existence, uniqueness and long-time behaviour[END_REF] e.g.) where electrons are described by a quantum model in the confined direction and a classical drift-diffusion equation along the transport direction.

In this paper, we extend the method of [START_REF] Baro | A 1d coupled Schrödinger driftdiffusion model including collisions[END_REF] to the framework of strongly confined nanostructure (like nanowires or nanotubes) for which both the quantum transport and the classical collisional transport need new formulations. Indeed, in these structures, the dimension of the transversal cross section is so thin that the transport of charged particles is restricted to the one dimensional longitudinal direction. When the cross-section diameter is below 3 nm, the strong confinement affects the energy band structure and bulk material quantities cannot be used in the simulations (see [START_REF] Gnani | Band effects on the transport characteristics of ultrascaled snw-fets[END_REF], e.g., and references therein). In particular, the assumption of infinite periodic structure in the wire cross-section, which allows to derive the usual effective mass theorem, is not reasonable anymore. Using an envelope function decomposition, a new effective mass approximation, describing the ballistic transport of electrons in ultra-scaled confined nanostructures, has been obtained in [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF]. The model consists of a sequence of one dimensional device dependent Schrödinger equations, one for each energy band, in which quantities retaining the effects of the confinement and of the transversal crystal structure are inserted. On another hand, a drift-diffusion model which describes the confined diffusive transport, taking into account the interactions of charged particles with phonons, has been derived and analyzed in [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF]. It consists of a single macroscopic equation in which atomistic quantities are integrated. The novelty of the present paper is to spatially couple the effective mass model [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF] with the nanowire drift-diffusion model [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF], preserving the current continuity, as it was announced in the proceeding [START_REF] Ben Abdallah | A hybrid classical-quantum approach for ultra-scaled confined nanostructures : modeling and simulation[END_REF].

Here, the coupling approach requires to take into account different bands, preserving the continuity of the total current. Moreover, while in [START_REF] Degond | A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations[END_REF][START_REF] Baro | A 1d coupled Schrödinger driftdiffusion model including collisions[END_REF] numerical simulations are performed for a resonant tunneling diode, we consider in this work a strongly confined nanostructure, where self-consistent computations include the resolution, in the whole three dimensional device domain, of a Poisson equation describing a slowly varying macroscopic potential. For an illustration, we have chosen a gate-all-around Carbon Nanotube Field-Effect Transistor (CNTFET).

Carbon Nanotube (CNTs) are rolled-up sheets of graphene. A pair of indices (n, m), called the chiral vector, indicates how the sheet of graphene is wrapped. The CNT is called zig-zag if m = 0, armchair if n = m, and chiral otherwise. The chirality is crucial in regards to the electronic properties of CNTs. Indeed, if n -m is a multiple of 3, CNTs are metals. Otherwise, they are semiconductors (see [START_REF] Saito | Physical Properties of Carbon Nanotubes[END_REF][START_REF] Wildöer | Electronic structure of atomically resolved carbon nanotubes[END_REF] e.g.). Semiconducting CNTs have emerged as promising candidates to build the future FETs because of their superior electrical characteristics over usual FETs (see [START_REF] Martel | Single-and multi-wall carbon nanotube field-effect transistors[END_REF][START_REF] Pourfath | Numerical study of quantum transport in carbon nanotube transistors[END_REF][START_REF] Marchi | Investigating the performance limits of silicon-nanowire and carbon-nanotube FETs[END_REF][START_REF] Gnani | Quantum-mechanical analysis of the electrostatics in silicon-nanowire and carbon-nanotube FETs[END_REF][START_REF] Koswatta | Dependence of DC characteristics of CNT MOSFETs on bandstructure models[END_REF] and references therein for details). The modeling of such devices is very important in order to predict their behavior, to access their performance limits and to design new configurations. That is why we propose in this paper a hybrid approach that allows for computationally efficient numerical simulations, that can be used in a device design framework.

The paper is organized as follows. In Section 2, we present the zig-zag (10,0) singlewalled carbon nanotube used all along this article to illustrate the hybrid approach. We also define the relevant physical quantities that retain the atomistic description of the strongly confined cross-section, recalling the model derived in [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF] . In Section 3, we describe the geometrical coupling, detailing the equations in the classical regions (introduced in [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF]), those in the quantum zone (obtained in [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF]) and explaining the derivation of interface conditions. We also present the self-consistent formulation which consists in coupling the 1D transport equations with a 3D Poisson equation. Section 4 describes the iterative algorithm and the numerical issues. Finally, the numerical experiments are collected in Section 5. After a description of the gate-all-around CNTFET, we compare the hybrid approach with the full drift-diffusion model [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF] and the full quantum one [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF]. In particular, we discuss the impact of the electron mobility and the influence of interface positions.

Preliminaries

We are interested in simulations of the electron transport in a semiconducting carbon nanotube. More precisely, we consider a gate-all-around CNTFET. To fix ideas, we describe here the device that will be simulated in the results presented in Section 5. It is however clear that the model and the numerical algorithm can be applied to general strongly confined nanostructures (such as silicon nanowires or different CNTs), with different gate geometries. We consider a (10,0) zig-zag single-walled CNT (see Fig. 1 for a representation of atom positions), surrounded by a layer of dielectric acting as an insulator of 1.4 nm thickness. The carbon atoms are placed on a circle of 0.78 nm diameter. In [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF], a novel quantum effective mass model has been derived by performing an asymptotic process which consists in using an envelope function decomposition to obtain a new effective mass approximation (see also [START_REF] Barletti | Quantum Transport in Crystals: Effective Mass Theorem and K•P Hamiltonians[END_REF] for a similar approach for 3D periodic crystals). We recall it briefly here. Let us consider an infinite wire defined in a physical domain R × ω ǫ , where ǫ is the typical spacing between lattice sites. As starting point, the transport is described by a scaled Schrödinger equation in R × ω ǫ containing a potential W L generated by the crystal lattice, fast oscillating in the scale defined by the crystal spacing, and a slowly varying external potential

V i ∂ t ψ ǫ = - 2 2me ∆ψ ǫ + 1 ǫ 2 W L x ǫ , z ǫ ψ ǫ + V x, z ǫ ψ ǫ (x, z) ∈ R × ω ǫ , ψ ǫ = 0 for z ∈ ∂ω ǫ . (2.1)
Here, is the reduced Planck constant and m e is the electron mass. Since the 2D crosssection ω ǫ comprises few ions, W L is considered periodic only in the longitudinal x-direction (transport direction) and the variable z of the transverse section can be considered as fast variable, rescaled as z ′ = z ǫ . Denoting by ω the scaled cross-section, we consider the following Bloch-type problem (with a quasimomentum equal to 0) in the 3D cell

U = (-1/2, 1/2) × ω          - 2 2me ∆χ n + W L χ n = E n χ n , χ n (y, z ′ ) = 0 on ∂ω, χ n 1-periodic in y, U |χ n | 2 dydz ′ = 1. (2.2)
Here y denotes the transport variable in the cell. The peculiarity of the strongly confined structure is reflected in the choice of the unit cell problem (2.2) of Bloch type. We point out that this unit cell U comprises the entire cross-section of the nanostructure (for instance, in the case of the (10,0) carbon nanotube structure, it contains all the red atoms highlighted in Fig. 1 by the solid lines that connect them). Thus, the eigenvectors depend on the device under consideration, for instance on the device geometry, on the number of atoms, on the chirality, and so on. Moreover, the homogeneous Dirichlet condition imposes confinement in the transverse directions, while periodicity is considered only in the transport direction. Consequently, the eigenvectors are 3D quantities but the Brillouin zone and the associated energy bands are one dimensional.

The asymptotic process [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF], by using the functions defined in (2.2) as basis for the envelope function decomposition, allows to average out not only the lattice potential, but also the lateral dimension. The quantum electron transport is then modeled by an infinite set of Schrödinger equations, one for each band, where relevant averaged quantities, based on the Bloch functions, are incorporated. In particular, in the non-degenerate case (i.e. assuming that all the eigenvalues of the problem (2.2) are simple, as it will be considered all along this paper), the equations are decoupled and have the form

ı ∂ t h n (t, x) = - 2 2m * n ∂ xx h n (t, x) + V nn (x)h n (t, x), x ∈ R.
(2.

3)

The n -th band effective mass m * n is defined by

m e m * n = 1 - 2 2 m e n ′ =n P nn ′ P n ′ n E n -E n ′ , where P nn ′ = U ∂ y χ n ′ (y, z ′ )χ n (y, z ′ ) dydz ′ . (2.4)
Also, the effective potential is given by

V nn (x) = ω V (x, z ′ )g nn (z ′ ) dz ′ , with g nn (z ′ ) = 1/2 -1/2 |χ n (y, z ′ )| 2 dy. (2.5)
For the degenerate case, we simply recall that the equations corresponding to the same eigenvalue are coupled through the potential term, while the kinetic part is diagonal. We refer to [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF] for details.

To complete the description of the nanostructure under consideration, in Fig. 2 we present the energy bands. They are calculated solving in the 3D cell U the eigenvalue problem for the fibered Hamiltonian for different wavevector k and with W L as the pseudopotential given in [START_REF] Mayer | Band structure and transport properties of carbon nanotubes using a local pseudopotential and a transfer-matrix technique[END_REF]. Piecewise linear finite elements on prisms with an appropriate non-uniform mesh have been used (see Fig. 6 for an illustration of the 2D cross-section mesh). Fig. 2 is in good accordance with the results presented in Fig. 4 in [START_REF] Mayer | Band structure and transport properties of carbon nanotubes using a local pseudopotential and a transfer-matrix technique[END_REF] for the same choice of pseudopotential. Notice that similar results can be provided by different methods (for instance using tight-binding models [START_REF] Charlier | First-principles study of the electronic properties of simple hexagonal graphite[END_REF][START_REF] Saito | Electronic structure of chiral graphene tubules[END_REF]). We point out that the model described in [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF] does not rely only on the knowledge of the energy bands. This is only the first step in the construction of the effective mass model. The important and novel issue is the link between the 1D transport equations and the 3D confined nanostructure, incorporating the atomistic information of the 2D cross-section by means of the functions g nn 's. In actual computations, we only solve problem (2.2), which corresponds to H L (0), in order to obtain the eigenvalues E n , the effective masses m * n and the functions g nn 's. In Figs. 3 and4, we present the g nn 's corresponding to the three first conduction bands for the (10,0) carbon nanotube structure, showing that the g nn 's retain information about the confinement and the cross-section structure. These same averaged quantities enter in the diffusive model derived in [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF] and used in the following of the paper where we propose a hybrid strategy coupling the fully quantum model [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF] with the diffusive one [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF].

H L (k) = - 2 2m e ∆ -i 2 m e k∂ y + 2 k 2 2m e + W L (2.6)

Electron transport with a hybrid strategy

In this paper, we consider the CNTFET described above. In the transport direction we consider a channel region sandwiched between largely doped Source and Drain regions. It is defined by the bounded domain Ω = (x L , x R ) × ω ǫ , where (x L , x R ) is the transport domain and ω ǫ is the two dimensional strongly confined cross-section. The novelty is that we present a hybrid strategy to couple, spatially in the transport direction, the Schrödinger system (derived and tested numerically on a "toy" problem in [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF]) with the drift-diffusion model (introduced and analyzed in [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF]). So, we assume that the device domain in the transport direction x is divided into a quantum zone

Q = (x I 1 , x I 2 ), with x L < x I 1 < x I 2 < x R and a classical zone C = (x L , x R )\Q.
The different regions of the domain (x L , x R ) are illustrated in Fig. 5.

We assume at this point that the electrostatic potential V is given (and consequently so is the n -th band effective potential V nn ). We write first the transport equations in the classical regions, as well as the equations in the quantum region. Afterward, we describe the interface conditions which preserve the continuity of the total current between the classical and the quantum domains. Finally, we present the self-consistent Poisson equation.

The classical region

The classical transport is described by the diffusive model for confined nanostructures which has been derived and analyzed in [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF]. Here we consider that stationary nanowiredrift-diffusion model on the disconnected domain

C d dx J C (x) = 0, (3.1)
where the classical current is written in terms of the quasi-Fermi energy ϕ as

J C (x) = -µ(x) ∂ x ϕ(x) F V s (x) -ϕ(x) . (3.2)
Here µ is the electron mobility coefficient which, in the particular case of a constant scattering kernel in the collision operator that gives rise to this diffusive model, is defined by

µ(x) = m e µ +∞ n=1 e -En+Vnn(x) /(k B T ) m * n +∞ m=1 e -Em+Vmm(x) /(k B T ) , (3.3)
where the constant µ = qτ me , with τ denoting the relaxation time, can be interpreted as the low field mobility constant. We notice that the mobility coefficient µ(x) exhibits some spatial variations due to the variations of V nn . An analogy can be seen with the electric field dependent mobility (see i.e. [START_REF] Pietra | Modeling and simulation of the diffusive transport in a nanoscale double-gate mosfet[END_REF][START_REF] Markowich | Semiconductor equations[END_REF]). Moreover, V s is the effective potential defined by

V s (x) = -k B T ln Z(x) with Z(x) = +∞ n=1 e -En+Vnn(x) /(k B T ) , (3.4)
where k B is the Boltzmann constant and T is the lattice temperature. Finally, we recall that the 1D density is defined by

N s = F (V s -ϕ), (3.5)
which, in the case of the Boltzmann statistics, is defined by

F (s) = n i e -s/(k B T ) , (3.6)
where n i is the 1D intrinsic density. We point out that the choice to use the quasi-Fermi energy formulation is crucial, motivated by the fact that we want to find an analytic expression of the interface conditions, as we will explain in the following. Dirichlet conditions are given at the boundary, imposing quasi-neutrality between the density N s and the background ion density N D . More precisely, denoting by x b the abscissa at the boundary (b = L or b = R), we take

N s (x b ) = ωǫ N D (x b , z)dz, (3.7)
where N D is the prescribed 3D doping density. For the Boltzmann statistics (3.6), the boundary condition (3.7) leads to

ϕ(x b ) = k B T ln ωǫ N D (x b , z)dz n i + V s (x b ). (3.8)
Finally, the two disconnected classical zones are connected by means of the conditions

J C (x I 1 ) = J C (x I 2 ) = h(ϕ(x I 1 )) -h(ϕ(x I 2 )), (3.9)
which relate the quasi-Fermi energy ϕ at the two interface points. If h is a real-valued and monotonously increasing function, then system (3.1)-(3.2) with boundary conditions (3.8)-(3.9) is well posed (see [START_REF] Baro | A 1d coupled Schrödinger driftdiffusion model including collisions[END_REF] and references therein). We will determine the function h in the following subsections, by means of the quantum expression of the current. Here we notice that without applied voltage, ϕ(x I 1 ) = ϕ(x I 2 ) and thus J C = 0. We also see that ϕ(x I 1 ) > ϕ(x I 2 ) gives a positive current and inversely. Finally, in preparation for self-consistent computations, from N s we need to define a charge density for each band. We do so, recalling that the starting point of [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF] to obtain the nanowire-drift-diffusion model (3.1),(3.2) is a sequence of decoupled Boltzmann equations, one for each band, and that the approximate solution of the n -th band kinetic equation, up to order 1 in the scaled mean free path parameter, is given by the distribution function

g n (x, k) = N s (x) 2πm * n k B T Z(x) e -( 2 k 2 2m * n +En+Vnn(x))/(k B T ) . (3.10)
Using (3.5), (3.6) and (3.4), the previous formula takes the form

g n (x, k) = n i 2πm * n k B T e - 2 k 2 2m * n +En+Vnn(x)-ϕ(x) /(k B T ) . (3.11)
Proceeding as in [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF], we integrate (3.11) over k and we obtain

N n C (x) = F V s (x) -ϕ(x) e -En+Vnn(x) /(k B T ) Z(x) . (3.12)
We notice that the sum over all the bands gives the density N s .

The quantum region

The quantum transport in Q is given by the sequence of Schrödinger equations (3.13).

The system is considered as an open system in (x I 1 , x I 2 ), with Transparent Boundary Conditions supplementing the equations (see [START_REF] Lent | The quantum transmitting boundary method[END_REF], [START_REF] Ben Abdallah | On a one-dimensional Schrödinger-Poisson scattering model[END_REF], e.g.). To fix ideas, we detail the case V nn (x I 1 ) ≥ V nn (x I 2 ). For each band and for each wave vector k, we consider the following stationary Schrödinger equation

- 2 2m * n ∂ xx ψ k n (x) + V nn (x)ψ k n (x) = E n,k ψ k n (x), in Q, (3.13)
where

E n,k =      E + n,k = 2 k 2 2m * n + V nn (x I 1 ) if k > 0, E - n,k = 2 k 2 2m * n + V nn (x I 2 ) if k < 0.
Defining the coefficients

p ± n (k) = 2 k 2 ∓ 2m * n (V nn (x I 2 ) -V nn (x I 1 )), (3.14)
the TBCs are written for k > 0 as

∂ x ψ k n (x I 1 ) + ikψ k n (x I 1 ) = 2ik and ∂ x ψ k n (x I 2 ) = ip + n (k)ψ k n (x I 2 ), (3.15)
and for k < 0, we have

∂ x ψ k n (x I 2 ) + ikψ k n (x I 2 ) = 2ik and ∂ x ψ k n (x I 1 ) = -ip - n (k)ψ k n (x I 1 ). (3.16)
The reflection and transmission amplitudes r n (k) and t n (k) of the wave functions are determined by

r n (k) = 1 2 ψ k n (x I 1 ) + i 2k ∂ x ψ k n (x I 1 ) and t n (k) = ψ k n (x I 2 ) for k > 0, r n (k) = 1 2 ψ k n (x I 2 ) + i 2k ∂ x ψ k n (x I 2 ) and t n (k) = ψ k n (x I 1 ) for k < 0.
Finally, we define the reflection coefficients as R n (k) = |r n (k)| 2 and the transmission coefficients T n (k), corresponding to the proportion of incident electrons which are transmitted, as

T n (k) =      p + n (k) k |t n (k)| 2 if k > 0, - Re p - n (k) k |t n (k)| 2 if k < 0.
(3.17)

The transmission coefficients have the following properties

T n (k) + R n (k) = 1 for all k ∈ R, (3.18)
T n (k) = T n -p + n (k)
for all k > 0, (3.19)

T n (k) = T n p - n (k) for all k < -p + n (0), (3.20)
T n (k) = 0 for all -p + n (0) < k < 0, (3.21)
Next, the 1D density carried by the n-th band N n 1D in the Q region is given by superimposing the densities of states injected from the reservoirs, that is

N n Q (x) = R φ n (k)|ψ k n (x)| 2 dk, x ∈ Q, (3.22)
where φ n (k) is a given statistical function which characterizes the electron injection from reservoirs and whose choice will be discussed in the next subsection. Finally, the current is defined by

J Q (x) = +∞ n=1 J n Q (x), (3.23)
where the n th band current is given by

J n Q (x) = q m * n R φ n (k)I ψ k n (x)∂ x ψ k n (x) dk, x ∈ Q. (3.24)
It can be easily seen that the current does not depend on x. Furthermore, using the TBCs of the Schrödinger equation (3.15) and (3.16), as well as the expression of the transmission coefficients (3.17) with the properties (3.18)-(3.21), J n Q can be expressed in the following form

J n Q = q m * n +∞ 0 kT n (k) φ n (k) -φ n (- p + n (k) ) dk. (3.25)

The interface conditions

So far, we still did not make an explicit choice for the monotone function h in (3.9) and for the statistical function in (3.22) and in (3.25). Our choice will lead to the definition of h in terms of the quantum current J Q .

As statistical function in (3.22) and in (3.25) we choose the distribution function related to the diffusive model at the interface, (assuming, for the moment, that the quasi-Fermi energy ϕ is known). More precisely, we take

φ n (k) =      g n (x I 1 , k) if k > 0, g n (x I 2 , k) if k < 0,
with g n defined in (3.11). Introducing f n as

f n (s) = n i 2πk B T m * n e -s/(k B T ) , (3.26)
and observing that E - n,-p + n (k)/ = E + n,k , we can write (3.25) as

J n Q = q m * n +∞ 0 kT n (k)f n (E n + E + n,k ) e ϕ(x I 1 )/(k B T ) -e ϕ(x I 2 )/(k B T ) dk. (3.27)
Notice that the term containing the quasi-Fermi energy variables enters (3.27) as a multiplication factor, and this is crucial to write the interface conditions in an explicit way. With the definition

h(ϕ) = +∞ n=1 q m * n +∞ 0 kT n (k)f n E n + E + n,k -ϕ dk. (3.28)
we obtain immediately from (3.23) and (3.27) that

J Q = h(ϕ(x I 1 )) -h(ϕ(x I 2 )).
We point out that the function h(s) is monotone increasing since f n (s) is monotone decreasing. The function h can be expressed explicitly, by means of

h(ϕ) = Θ -1 e ϕ/(k B T ) , (3.29)
where Θ is a positive number defined by

Θ -1 = +∞ n=1 q m * n +∞ 0 kT n (k)f n (E n + E + n,k )dk. (3.30)
Consequently, imposing J C = J Q , we obtain the final interface conditions

J C (x I 1 ) = J C (x I 2 ) := J C , (3.31)
and e ϕ(x I 1 )/(k B T ) -e ϕ(x I 2 )/(k B T ) = ΘJ C .

(3.32)

Remark 3.1. The interface conditions (3.31) and (3.32) have been constructed preserving the continuity between the classical and the quantum current, which is the relevant physical property. It is however clear that continuity of the charge density is not assured.

Self-consistent formulation

In order to define the self-consistent electrostatic potential, a 3D charge density is required. First of all, the 1D charge density of the hybrid model in the n -th band N n 1D is defined by

N n 1D (x) =    N n Q (x) for x ∈ Q, N n C (x) for x ∈ C,
where N n Q and N n C are given in (3.22) and (3.12), respectively. Then, as in [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF] and in [START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF], the transformation from the one dimensional transport direction to the entire nanostructure is done by means of the (properly scaled) quantities g nn 's (2.5). It leads to

ρ(x, z) = n∈N N n 1D (x)g ǫ nn (z) (x, z) ∈ (x L , x R ) × ω ǫ , (3.33)
where g ǫ nn (z) = 1 ǫ 2 g nn ( z ǫ ). Finally, the transport equations are coupled with the following Poisson equation for the electrostatic potential V

P -∇ ǫ r (z)∇V P (x, z) = q ǫ 0 N D (x, z) -ρ(x, z) , (x, z) ∈ (x L , x R ) × ω ǫ , (3.34)
where q is the elementary charge, ǫ 0 the permittivity in vacuum, ǫ r the relative permittivity and N D the prescribed doping density. Equation (3.34) is then supplemented by boundary conditions at Source and Drain contacts, at the gate contact and at the insulating boundaries, for which we refer to [START_REF] Ben Abdallah | An effective mass model for the simulation of ultra-scaled confined devices[END_REF], Section 5.5. Then, V in (2.5) is given by V = -qV P .

Algorithm approach

We describe here the numerical issues related to the solution of the coupled 1D transport equations with the 3D Poisson equation, once the atomistic quantities are computed on the unit cell, by means of the solution of the eigenvalue problem (2.2). The 3D mesh for the Poisson equation consists of prisms, with a 2D cross-section triangular mesh as in Fig. 6 and with 151 equally spaced discretization points in the longitudinal direction (discretization step: 0.2 nm). We point out that the 2D cross-section mesh is the same as the one used for the solution of the eigenvalue problem (obviously the 3D domain is different), so that the functions g nn (z ′ ) are incorporated in the Poisson problem without resorting to any interpolation. Finally, due to the highly oscillatory behavior of the wave functions with high energy, a fine mesh size is required for the 1D transport computations. We take 751 equally spaced discretization points (discretization step: 0.04 nm).

Figure 6: Mesh of the circular cross-section : entire mesh (left) and zoom of the carbon domain (right).

For a given gate voltage V G , we first consider the whole system for zero applied Drain-Source voltage V DS and then we increment V DS step by step. Finally, this procedure is repeated for different gate voltage V G .

Because of the highly nonlinear coupling between the density and the potential equations, the inner iteration procedure is done by an iterative method of Gummel type [START_REF] Gummel | A self-consistent iterative scheme for one-dimensional steady state transistor calculations[END_REF]. We detail here the different steps of one iteration of this Gummel type algorithm.

1. Let V old P be a given approximate electrostatic potential in the finite element space of piecewise linear polynomials on the prismatic mesh described above.

2. We define the potential energy V = -qV old P and compute the quantities V nn (2.5), V s (3.4) as well as µ (3.3).

3. For each band n and each wave vector k, we solve on Q the Schrödinger equations (3.13), with the TBCs (3.15). A single equation for k > 0 is first rewritten, by means of the transformation

ψ k n (x) = 2ikΨ k n (x)/[∂ x Ψ k n (x I 1 ) + 2ikΨ k n (x I 1 )
], as an initial value problem. Then, it is discretized with a Crank-Nicolson scheme, which is a conservative scheme and avoids numerical dissipation for large k's. The negative wave vector case is treated analogously. Then, the transmission coefficients T n (3.17) are obtained and, subsequently, the number Θ (3.30) is computed. ) is discretized by means of a mixed finite element scheme in hybrid form [START_REF] Arnold | Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates[END_REF], which, after the static condensation process, takes the form

-D j 0 ϕ j-1 + D j 0 + D j+1 0 ϕ j -D j+1 0 ϕ j+1 = 0, (4.1)
where ϕ j denotes the value of the approximate quasi-Fermi energy ϕ h at the discretization point x j . The coefficients D 0 are given by

D j 0 = D j 0 (ϕ h ) = -n i µ j + µ j-1 2 e ϕ j +ϕ j-1 2k B T /I(V s /k B T ), (4.2)
where µ j are the values of the given (within this iteration step) functions µ(x) at x j , and I(u) =

x j+1

x j e u(x) dx, which here can be computed exactly, because the (given) V s is piecewise linear.

The (globally constant) current is then given by

J j = D j 0 * ϕ j -ϕ j-1 . (4.
3)

The non-linear equation (4.1) is solved by means of a Newton algorithm. We obtain the quasi-Fermi energy ϕ. We recall that mixed finite elements schemes for the drift-diffusion in the quasi-Fermi energy formulation have been used in [START_REF] Hecht | Mixed finite element simulation of heterojunction structures including a boundary layer model for the quasi-fermi levels[END_REF][START_REF] Hecht | Semiconductor device modelling for heterojunctions structures with mixed finite elements[END_REF].

5. We compute the classical density on each band N n C (3.12), and, using ϕ(x I 1 ), ϕ(x I 2 ) in (3.22), the quantum density on each band N n Q . Finally, we compute the three dimensional density ρ (3.33). 6. We solve the 3D Poisson equation using piecewise linear finite elements on the prismatic mesh described above, modified according to the Gummel iteration algorithm, as in [START_REF] Ph | Finite element approximation of electrostatic potential in one-dimensional multilayer structures with quantized electronic charge[END_REF], that is

-∇ ǫ r (z)∇V new P ) = q ǫ 0 N D - q ǫ 0 ρ[V old P ] 1 + q k B T (V new P -V old P ) , in (x L , x R ) × ω ǫ . (4.4)
7. We repeat the last five steps until the quantity V old P -V new P L ∞ becomes sufficiently small. Once the convergence is reached, we increment the applied Drain-Source bias V DS and start a new iteration. For the device presented in Section 5, the convergence algorithm is not very sensitive to the size of the V DS increment, since the Current-Voltage characteristics do not present a stiff behavior. We numerically found that a uniform increment step equal to 0.02 V is a reasonable choice. Moreover, as usual for MOS devices, the number of Gummel iterations increases for large gate voltages V G (see Table 1), but convergence is still reached with the same V DS increment.

Numerical results

Numerical simulations are carried out for a Carbon Nanotube Field-Effect Transistor CNT-FET. A section along the transport direction (x-axis) is presented in Fig. 7. It contains a (10,0) zig-zag single-walled CNT (see Fig. 1) surrounded by a layer of dielectric SiO 2 (ǫ r,ox = 3.9) of 1.4 nm thickness (represented in red) acting as an insulator. The relative permittivity of a CNT is a device dependent quantity, that here has been taken as ǫ r,c = 13. The transport direction is composed of a 10 nm active zone, with a doping concentration N - D = 10 21 m -3 , sandwiched between a 10 nm Source region and a 10 nm Drain region, with large doping (N + D = 10 27 m -3 ). Finally, a Gate is imposed all-around the transversal structure to modulate the number of free electrons. In the sequel, two different cases will be illustrated. In the first case (non-overlap case), the gate is positioned in correspondence to the active zone (black G in Fig. 7), whereas in the overlap case, the gate is 20 nm long and it acts also on a part of Source and Drain regions (black+grey G).

In the present simulations, the hybrid interfaces are located at x I 1 = 10 nm and x I 2 = 20 nm, which correspond to the doping discontinuities. In Subsection 5.4, we will discuss how the interface position affects the device behavior. The transport-Poisson problem is only solved for the 9 first conduction bands. This choice is widely sufficient as we will see in Subsection 5.5. Among these 9 modes, there are degenerate eigenvalues. Nevertheless, the off-diagonal potentials V nm (analogous of (2.5)) are virtually zero and it is reasonable to neglect the coupling of these bands. It amounts to assume that all the eigenvalues are simple.

The electron mobility

First, we would like to make some comments about the electron mobility constant µ (3.3). For strongly confined structures, this is a device dependent physical parameter and a well established value is not found in the literature. Therefore, we discuss how different choices affect our simulations with the hybrid strategy. In Fig. 8, the output current-voltage characteristics (for the fixed gate voltage V G = -0.1 V) are presented for different mobility constants µ. The right picture shows the current value for a fixed applied voltage V DS = 0.2 V (which corresponds to a saturation regime). This behavior is peculiar of the hybrid approach. While solving the transport problem with the drift-diffusion model in the entire longitudinal domain provides a current which is proportional to the mobility constant, in the hybrid case we observe that for mobilities larger than 10 -3 m 2 V -1 s -1 the current is virtually not modified. This means that the quantum model in the channel region is predominant in the computation of the electron transport. On the contrary, for smaller mobilities, the hybrid model is able to capture the limited transport behavior due to the large number of collisions.

In order to choose the value for µ for our simulations, we compare the current-voltage characteristics for the three possible models (full drift-diffusion model, full quantum model and hybrid approach). It turns out that µ = 0.5 × 10 -4 m 2 V -1 s -1 is a reasonable choice. Indeed, with this choice (see Fig. 9), the drift-diffusion current is much smaller than the two others and the quantum current is higher than the hybrid one. This means on one hand that ballistic quantum effects are well considered in the active zone. On the other hand, electron-phonon collisions are taken into account in the reservoirs. Moreover, since the hybrid current is quantitatively closer to the quantum one, the quantum ballistic effects are here considered predominant with respect to the collisional effects, which is preferable in modeling electron transport in a strongly confined structure.

Hybrid results for the overlap and non-overlap cases

Now, we present the numerical results obtained with the hybrid approach. First, Fig. 10 shows the 3D density for V DS = 0 V and V G = -0.1 V in the overlap case. This figure allows to visualize the formation of channels between Source and Drain. Also, we observe the atomic circular cross-section structure of the carbon nanotube, due to the incorporation of g nn 's (presented in Figs. 34).

Figs.11-12 represent the self-consistent potential for V DS = 0 V in a 2D slice along the transport direction (x-axis in the pictures), cutting the cross-section in the middle. Fig. 11 corresponds to a gate voltage V G = -0.1 V, whereas Fig. 11 corresponds to V G = 0.1 V. We recover these values at top and bottom boundaries of the pictures (with the reverse sign since we plot energies expressed in eV). The overlap case is presented in the left figures and the non-overlap case in the right ones. In both cases, we observe a clear influence of the gate and of the doping. We detect also the cross-section structure of our device. Indeed, we recall that the external diameter of the CNT cross-section is 0.92 nm and so, at z 1 ≈ ±0.5 nm, the different profiles have relevant variations. Electron channels are apparent between Source and Drain. 1D profiles of the potential for V DS = 0 V are presented for different gate voltages in Fig. 13. These 1D curves are averaged quantities resulting from an integration of 3D quantities over the 2D wire section divided by the wire section surface. For low gate voltages, we observe that the potential in the channel has a value close to V G , confirming that the transport is mainly controlled by the gate. Increasing the gate voltage, the channel potential energy always decreases but it does not reach V G . In this case, the action of V G on the potential in the active zone is governed by complex phenomena. Finally, in Fig. 14, we represent the output characteristics for seven different gate voltages in the overlap case (similar behavior is obtained for the non-overlap case). The current in the CNTFET increases with the gate potential V G . We notice the two typical regimes: an ohmic regime for values of Drain-Source voltage smaller than a threshold voltage V th and a quasi-saturation regime for larger V DS . The value of V th increases with the gate potential. Indeed, for V G = 0 V, V th is about 0.2 V, instead for V G = 0.1 V, it is about 0.4 V. All these results are in accordance with those reported in the literature for similar devices (see [START_REF] Gnani | Quantum-mechanical analysis of the electrostatics in silicon-nanowire and carbon-nanotube FETs[END_REF][START_REF] Koswatta | Dependence of DC characteristics of CNT MOSFETs on bandstructure models[END_REF][START_REF] Pourfath | Numerical study of quantum transport in carbon nanotube transistors[END_REF] e.g.).

Comparison of the three approaches

In this part, we shall compare the three approaches: full drift-diffusion model (dashed green curves in Figs. [START_REF] Gummel | A self-consistent iterative scheme for one-dimensional steady state transistor calculations[END_REF][START_REF] Hecht | Mixed finite element simulation of heterojunction structures including a boundary layer model for the quasi-fermi levels[END_REF][START_REF] Hecht | Semiconductor device modelling for heterojunctions structures with mixed finite elements[END_REF][START_REF] Jourdana | Analysis of a diffusive effective mass model for nanowires[END_REF][START_REF] Koswatta | Dependence of DC characteristics of CNT MOSFETs on bandstructure models[END_REF], full quantum model (dashdotted red curves) and hybrid approach (solid blue curves). In Figs. 1516, 1D profiles of the potential are presented for V G = -0.1 V, for V DS = 0 V and for V DS = 0.2 V, respectively. The same representation is done for 1D profiles of the density in Figs.17-18. As previously, the overlap case is presented in the left figures and the non-overlap case in the right ones. The two cases exhibit similar behaviors. The differences between the three methods are mainly in the active zone and at the doping discontinuities. In these pictures, differences look rather small but this is mainly due to the scale of the plots. For a better comparison, it is interesting to plot average velocities, defined as v(x) = J/[q n N n 1D (x)]. In Fig. 19, they are calculated for V G = -0.1 V and V DS = 0.2 V. We observe three distinguishable curves, due not only to the different current values (see Fig. 9), but also to the different densities. Moreover, for the hybrid approach, Fig. 19 allows also to detect discontinuities of the density at the interfaces, which were not evident in Figs.17-18. We recall, that the interface conditions (3.31) and (3.32) enforce current continuity, but, as pointed out in Remark 3.1, density discontinuity may occur. Now, we want to compare the full quantum model with the hybrid one from a computational point of view. The full drift-diffusion model is clearly cheaper, but, as we already mentioned, it does not take into account the relevant quantum effects that are essential in confined nanostructures. The total execution time necessary for computing a currentvoltage characteristics curve at a fixed gate voltage (from V DS = 0 V up to V DS = 0.2 V) is approximatively 65% lower with the hybrid approach than with the quantum one. An obvious reason is that the quantum simulation, that requires the solution of a large number of Schrödinger equations at each iteration step, is performed on a smaller domain. Also, the hybrid method needs less Gummel iterations to reach convergence, as it is illustrated in Table 1. This is partly due to the boundary conditions since in the hybrid approach the quasi-neutral boundary conditions are imposed in a classical macroscopic manner. These numerical results aim at showing that our choice to use a hybrid approach including device dependent parameters allows for computationally efficient simulations.

V G = -0.1 V V G = 0.1 V Schrödinger Hybrid Schrödinger Hybrid Nb. Gummel V DS = 0 V 37 10 46 20 iterations V DS = 0.02 V 28 5 31 13
Table 1: Number of Gummel iterations for the quantum model and the hybrid approach.

Interface positions

We now study the influence of the interface positions on the current. We remind that in the previous simulations the interfaces were located at x I 1 = 10 nm and x I 2 = 20 nm (at doping discontinuities). In Fig. 20, we present the current value once the saturation regime is reached. This test is run for V G = -0.1 V in the overlap case. We point out that a similar dependence on the interface position occurs for different gate voltages and we do not report the pictures. In the left picture, x I 2 is fixed at 20 nm and we move x I 1 . Inversely, in the right picture, x I 1 is fixed at 10 nm and we move x I 2 . The dashdotted red line and the dashed green line correspond to the current value obtained, respectively, with the quantum and the drift-diffusion model in the entire longitudinal domain (without interfaces). In the two pictures, we observe a perceptible lowering of the current when one interface is placed well inside the active zone. It confirms that quantum ballistic effects play an essential role in the channel. However, moving the interfaces well inside the reservoirs, we observe two different behaviors. The current stays almost unchanged when the interface x I 2 goes to Drain, while the current increases up to the quantum value when the interface x I 1 reaches Source. This is not surprising since, once the saturation regime is established, the electrons flow almost unidirectionally from Source to Drain, and consequently in reservoirs it is the modeling of the electron transport (quantum ballistic vs classical collisional transport) in Source that is crucial to determine the current value.

Contribution of the different bands

Finally, we check that our initial choice of using only the 9 first conduction bands is relevant.

In Table 2, we present the current distribution between the different bands for four different cases. In all cases, we see that only the two first bands give a significant contribution to the total current. This is due to the fact that for our (10,0) zig-zag single-walled CNT, the energy levels E n increase quickly. Notice that in this table we can clearly recognize the multiplicity of the different E n 's.

V G = -0.1 V V G = 0.1 V Overlap
Non overlap Overlap Non overlap 1 st band 2.25×10 -2 2.36×10 -2 2.09×10 -1 2.12×10 -1 2 nd band 2.24×10 -2 2.36×10 -2 2.08×10 -1 2.11×10 -1 3 rd band 1.60×10 -12 1.69×10 -12 1.43×10 -11 1.48×10 -11 4 th band 1.59×10 -12 1.68×10 -12 1.42×10 -11 1.47×10 -11 Current 5 th band 3.23×10 -22 3.40×10 -22 3.17×10 -21 3.29×10 -21 (µA) 6 th band 3.21×10 -22 3.38×10 -22 3.15×10 -21 3.27×10 -21 7 th band 6.27×10 -33 6.62×10 -33 6.11×10 -32 6.36×10 -32 8 th band 6.25×10 -33 6.60×10 -33 6.13×10 -32 6.34×10

Conclusion

In this paper, we propose a hybrid approach in the framework of strongly confined nanostructures. We spatially couple an effective mass Schrödinger system with a nanowire drift-diffusion model, preserving the current continuity. In these one dimensional transport models, the quantum confinement is taken into account averaging the effects of the two dimensional crystal structure. Simulations of a single-walled Carbon Nanotube Field-Effect Transistor aim at testing the capability of the approach to describe the electron transport with strong confinement. The reduced computational effort of this approach makes it suited for a device design framework.

Figure 1 :

 1 Figure 1: 3D (left) and 2D (right) representation of atom positions in a (10,0) "zig-zag" carbon nanotube. The solid red part corresponds to the atoms in a unit cell.

Figure 2 :

 2 Figure 2: Energy bands obtained for a (10,0) carbon nanotube.

Figure 3 :

 3 Figure 3: g nn (z) for the first conduction band.

Figure 4 :

 4 Figure 4: g nn (z) for the second (left) and the third (right) conduction band.

Figure 5 :

 5 Figure 5: Schematic representation of hybrid model regions.

4 .

 4 We solve the stationary drift-diffusion equation (3.1) on C with the boundary conditions (3.8) and the two connection conditions (3.31) and (3.32). The non-linear equation (3.1

Figure 7 :

 7 Figure 7: Longitudinal section of the CNTFET.

Figure 8 :

 8 Figure 8: Mobility influence on the current-voltage characteristics (left) and the saturation current (right) with the hybrid approach for V G = -0.1 V.

Figure 9 :

 9 Figure 9: Comparison of the Current-Voltage characteristics between the three approaches for µ = 0.5 × 10 -4 m 2 V -1 s -1 and V G = -0.1 V.

Figure 10 :

 10 Figure 10: 3D density (m -3 ) for V DS = 0 V and V G = -0.1 V.

Figure 11 :

 11 Figure 11: Potential energy (eV) for V DS = 0 V and V G = -0.1 V : overlap (left) and non-overlap (right) case.

Figure 12 :

 12 Figure 12: Potential energy (eV) for V DS = 0 V and V G = 0.1 V : overlap (left) and non-overlap (right) case.

Figure 13 :

 13 Figure 13: Averaged potential energy (eV) for V DS = 0 V and for different V G : overlap (left) and non-overlap (right) case.

Figure 14 :

 14 Figure 14: Current-Voltage characteristics for different gate voltages V G .

Figure 15 :

 15 Figure 15: Comparison of the averaged potential energy (eV) for V DS = 0 V and for V G = -0.1 V : overlap (left) and non-overlap (right) case.

Figure 16 :

 16 Figure 16: Comparison of the averaged potential energy (eV) for V DS = 0.2 V and for V G = -0.1 V : overlap (left) and non-overlap (right) case.

Figure 17 :

 17 Figure 17: Comparison of the averaged density (m -3 ) for V DS = 0 V and for V G = -0.1 V : overlap (left) and non-overlap (right) case.

Figure 18 :

 18 Figure 18: Comparison of the averaged density (m -3 ) for V DS = 0.2 V and for V G = -0.1 V : overlap (left) and non-overlap (right) case.

Figure 19 :

 19 Figure 19: Comparison of the averaged velocity (m.s -1 ) for V G = -0.1 V and V DS = 0.2 V : overlap (left) and non-overlap (right) case.

Figure 20 :

 20 Figure 20: Current saturation obtained with the hybrid approach, moving the left interface position x 1 (left) and the right interface position x 2 (right) for V G = -0.1 V.

  -32 9 th band 4.14×10 -36 4.37×10 -36 4.04×10 -35 4.20×10 -35

	Total	4.49×10 -2	4.72×10 -2	4.18×10 -1	4.23×10 -1

Table 2 :

 2 Current distribution in the 9 first conduction bands.

Acknowledgements : Authors acknowledge partial support from the Galilée Project no 25992ND of the Hubert Curien program: Modèles numériques du transport collisionnel dans des dispositifs nano-électroniques.


M Núñez-Regueiro

D C Freitas

R Brusetti

J Marcus

Effect of pressure on the superconductivity of Rb 0.19 WO 3

Keywords: 74.62.Fj, 74.70.Dd Pressure, Superconductivity, Phase transitions, oxides

We have performed electrical resistivity measurements under pressures up to 20GPa between 1 and 300K on monocrystalline hexagonal Rb 0.19 WO 3 . For pressures lower than ~5GPa, we observe a decrease of the metallic-like resistivity at room temperature as well as a small decrease of T c . At this pressure, the resistivity starts to increase slowly up to 10GPa accompanied by a sharper decrease of T c .The resistivity curves above 10GPa denote an activated behaviour and a T c lower than 3K indicating that there is a phase transition that takes place gradually between 5 and 10GPa.

We interpret our measurements as the result of structural transformations under high pressure.

Introduction

The study of tungsten bronzes was given a strong momentum after the report of evidence for surface superconductivity with a superconducting transition temperature T c of the order of 100K [1,2,3,4]. Tungsten bronzes have been intensively studied in the past due to its highly soft perovskite structure that induces a strong interaction between carriers and soft phonons leading to polaron and bipolaron formation [5]. The bulk structure of WO 3 corresponds to a tetragonal lattice formed by WO 6 octahedra that can be stacked into a ABO 3 perovskite structure with the W occupying the B sites and with the A site being empty. Band structure calculations [6] have shown that the material is a semiconductor with a gap of around 1eV (though experimental optical gaps are higher [START_REF] Granqvist | Handbook of Inorganic Electrochromic Materals[END_REF], indirect gap = 2.5eV), with the valence bands originating mainly from O 2p orbitals and the conduction bands from Wd ones. Reduction of the compound results in a small oxygen non-stoichiometry and conducting samples. The stoichiometric material shows thermally activated hopping conductivity with activation energies between 0.15 and 0.27 eV, depending on the crystal quality [START_REF] Berak | [END_REF]9,10,11]. At temperatures below 130K there is a change to a regime with much smaller activation energies. At even lower temperatures the electrical conductivity becomes nearly temperature independent as expected by hopping small polaron conduction affected by disorder [12]. In non-stoichiometric samples bipolarons [5] are observed for carrier concentrations <3.7x1021 cm-1. On the other hand, sheet superconductivity at 3K in reduced WO 3 was also reported [13]. Though the superconducting structure is unknown [14], the behaviour of the electrical resistivity, obviously the result of the carriers at the origin of the superconducting transition, is abnormal with an activated region at low temperatures.

Carriers can be introduced also by alcaline ion intercalation. Structural changes occur together with the superconducting properties expected for such strong electron-phonon interaction materials. In particular the case of Rb intercalation generates hexagonal tungsten bronzes, with form 3-and 6-membered rings, leaving trigonal and hexagonal channels along the c axis. The radius of the hexagonal cavities is ~2Å. These materials are superconducting with T c 's that can reach 7K [15]. Their properties can be very complex and different reports have claimed some rather different normal transport properties [16]. Recently, doping with oxygen replacement by fluorine 17 has yielded the compound WO 2.6 F 0.4 with a T c =0.4K.

Experimental

The samples we studied have been obtained by acid etching stoichiometric single crystals to reduce their Rb content. The Rb 0.33 WO 3 crystals were grown electrolytically from a melt consisting of Rb 2 CO 3 and WO 3 -according to the method developed by Sienko and Morehouse [18]. The homogeneity of the samples and the final x value (0.195 ± .01) was checked by measuring the magnetic anomaly associated with the superconducting transition (T c = 5.5 K at ambiant pressure) and confirmed by a micro-probe analysis.

The electrical resistivity measurements were performed in a sintered diamond Bridgman anvil apparatus using a pyrophillite gasket and two steatite disks as the pressure medium. The Cu-Be device that locked the anvils can be cycled between 4.2K and 300K in a sealed dewar. The overall uncertainty in the quasi-hydrostatic pressure is estimated to be ±15%. The pressure spread across the sintered diamond anvils was previously determined on Pb-manometers to be of about 1.5-2GPa depending on the applied pressure, through the measurement of the superconducting transition temperature of Pb at low temperatures. The temperature was determined using calibrated Cernox thermometer with a maximum uncertainty (due mainly to temperature gradients across the Cu-Be clamp) of 0.5K. Four probes electrical resistivity d.c. measurements were made using a Keithley 2182 nanovoltmeter combined with a Keithley 238 current source and by using platinum wires to make contacts on the sample.

Results

We show on Fig. 1 the electrical resistance as a function of temperature for different pressures. We observe that for the first pressures there is an irreversibility of the curves : the up curves show an increase of the resistance that results in a bump. Behaviour similar to this has been reported by Stanley et al. [16] who attributed it to a phase transition, that can be a charge density wave (CDW), an ordering of the alkaline, or a coupled version of both phenomena, as in some organic compounds [19]. With increasing pressures, the irreversibilities become less visible and are totally inexistent in the curves obtained at 4 GPa.

As pressure is increased we note a change in the temperature dependence of the resistivity : it passes from a metallic behaviour to an activated, i.e. increasing with decreasing behaviour. It is clear that there are important changes in the sample, possibly a phase transition under pressure. A lattice distortion has been reported in Rb 0.31 WO 3 [START_REF] Houser | AIP Conference Proceedings[END_REF] for pressures higher than 2GPa : the octahedra that form the channel tilt around the c axis of the lattice. Considering that our pressure gradient is of about 2GPa (as determined by the width of the superconducting transitions of Pb manometers), the transition temperature reported for this distortion coincides with the disappearance of the irreversibilities. It may thus explain why we do not see the apparent ordering of the Rb atoms at 4GPa, as the tilted octahedra can impede the normal displacements of the Rb atoms along the channels. It is possible that as pressure is increased different or more pronounced distortions of the structure may appear in this soft lattice.

We show on Fig. 2 a detail of the superconducting transitions. The transition temperature decreases as a function of pressure as shown on Fig. 3. It is clear that there is a structural phase transition between 5 and 10GPa, as both the resistance at 10K and T c show a significant change around this pressure. It is interesting to note that both the T c and the resistance curves at the higher pressures look very much like those of superconducting [13] reduced WO 3 . The activated behaviour does not seem to indicate that the material has become semiconducting as, it that case we would expect an insulator at low temperatures, not a superconductor. We have verified that the superconducting transition is due to the whole of the sample by testing the critical currents at the foot of the superconducting transition. These were of the same order of magnitude for the sample at low pressures and the sample at high pressures (filamentary superconductivity would have yielded much smaller critical currents.

Discussion

The picture that evolves from our results seems to be the following. At low pressures, we observe irreversibilities in the electrical resistance with pressure that are most probably due to an hysteretic ordering of the Rb atoms. As the material undergoes the pressure-induced distortion [START_REF] Houser | AIP Conference Proceedings[END_REF],

the Rb atoms are pinned down by them, and can no longer move along the channels. As a consequence, the carriers that coupled to the Rb ordering are now also pinned down by the static Rb atoms, and can only conduct by activated hopping. We thus expect a 3D variable range hopping ( exp (T 0 T ) 1 4 ! " # $ ) or a polaronic (T ! exp T H T [ ] ) type conductance or eventually a plain semiconducting type ( exp ! H T [ ] ). There are clearly two régimes for the electrical resistance, first one above ~60K and a second one between this temperature and the superconducting transition. We

have not been able to fit the high temperature régimes for the different pressures with any of the mentioned dependences. In the low temperature régime, the fit with the largest temperature range (20K) corresponds to the plain semiconducting type ( exp ! H T [ ] )However, the gap values obtained are too small for an actual semiconductor, ~ one tenth of k B T. Thus, another interpretation is necessary for this dependence. Obviously it would be interesting to confirm the phase transition around 10GPa with crystallographic measurements under pressure, in order to determine the distortion involved.

Conclusions

Our pressure measurements show at low pressures an irreversibility of the electrical resistance, most probably due to an ordering of Rb atoms at low temperatures. At about 10GPa there is a transition to another phase with a lower T c =3K. The proposed new high pressure phase shows in the electrical resistance an activated behaviour, that is flattened at around 25K towards an apparent Arrhenius behaviour. Further theoretical and experimental studies are needed to understand the origin of the unusual pre-transitional behaviour in Rb 0.19 WO 3 .

Figures

Figure 1 Electrical

 1 Figure 1

Figure 2 Detail of the superconducting transition at low temperature Figure 3

 23 Figure 2

Acknowledgements D.C.F. gratefully acknowledges support from the Brazilian agencies CAPES and Cnpq. This work was partially supported by the project TetraFer ANR-09-BLAN-0211 of the Agence Nationale de la Recherche of France.