First of all, I would like to thank the members of the jury, whose presindent is T. Cren, for their great involvement in my defense. The discussion had great value both for the construction of future projects from this manuscript and from a personal point of view. The jury's words have a motivating effect on new challenges and allow me to end these three years with the feeling of a job well done.

Concerning the last four years, I am grateful for the opportunity and the welcome of Laurent and Roman, my thesis supervisors, to do this doctoral work, who have trusted me since my Master 2 internship. This trust has allowed me to work very comfortably, overcoming the bad moments and enjoying the good ones, which have been many. The team formed by Laurent, Roman and myself has drawn moments far beyond research. Conversations on diverse topics such as economics and politics enriched by a debate with a plurinational vision joining France, Germany and Spain. Sport has been an important centre of discussion, discovering squash, and combining the warm afternoons of measurement with the emotion of the arrivals at the great hills of the Tour de France. Roman's impressive technical quality has made me fall in love with the bricolage. Of a creative but rigorous bricolage in the choice of solutions, always hitting the nail on the head. Everything that Roman has taught me in this aspect is a learning for life. Of course, there is no doubt that Laurent, with his infinite knowledge and experience, added to an unrivalled scientific and human value, has been fundamental to the success of this thesis work. His involvement and closeness in all areas of the project (I am sure he would have entered the clean room if necessary), both in the experiments and in the management of the same, has given me wings to remain active and confident, for which I will always be grateful. At the same time, I must highlight the figure of Alain Fontaine and the Fondation Nanoscience, who, in addition to trusting me by financing my doctoral work, has accompanied and encouraged me during these three years. This work would not have been possible without the professional collaboration, where I emphasize the high human level of the laboratory for this purpose.

iii

Abstract

In this doctoral work, we have developed a new scanning single electron transistor (SET) microscope that works at very low temperatures (T = 50 mK) and high magnetic fields (B = 18 T). A SET consists of a small metallic island connected to source and drain electrodes through two tunnel junctions. In the Coulomb blockade regime at low temperature regime (T ≪ 5 K), an external electric field tunes the current circulating through the SET. In addition, small electric field variations lead to large SET current changes that makes the device a highly sensitive charge detector, able to detect charges smaller than 0.01 e. Thus, when the SET scans above a surface, it maps the electrostatic properties of the sample. However, the implementation of a scanning SET microscope is extremely challenging since it combines scanning probe microscopy, low temperatures and sensitive nanoscopic devices. For this reason, only a few groups have succeeded its realization. Our technological choices to build the microscope improve certain aspects with respect to the already existing instruments. The breakthrough is that we fabricate the SET probe using standard lithographic techniques on commercial silicon wafers. For that reason, batch fabrication of SET probes is possible. Furthermore, by a combination of dicing and etching techniques, the SET is engineered extremely close to the edge of the Si chip (< 1 µm). In this way, the SET can be approached to a few nanometer from the sample surface by means of a atomic force distance control. Additionally, an on-probe gate electrode fabricated close to the island can be used to tune the operating point of the SET. A novelty of our instrument is that with this on-probe gate and a feedback loop we have been able to map directly the local electric field. We demonstrate this new feedback scanning method by imaging an interdigitated array of nanometer scale electrodes. Moreover, the SET is an ideal tool for the study of the localization of electronic states. In the future, our scanning SET will be used for the study of two-dimensional electron systems in the quantum Hall regime, topological insulators and the metal insulator transition.

Resumé

Dans le cadre de ce travail de doctorat, nous avons mis au point un nouveau microscope à balayage à transistor à électron unique (SET) qui fonctionne à très basse température (T = 50 mK) et à champs magnétiques intenses (18 T). Un SET se compose d'un petit îlot métallique relié aux électrodes de source et de drain par deux jonctions tunnel. En régime de blocage de Coulomb à basse température (T ≪ 5 K), un champ électrique externe règle le courant circulant dans le SET. De plus, de petites variations du champ électrique entraînent de grandes variations du courant SET, ce qui fait de l'appareil un détecteur de charge très sensible, capable de détecter des charges inférieures à 0,01e. Ainsi, lorsque le SET scanne au-dessus d'une surface, il cartographie les propriétés électrostatiques de l'échantillon. Cependant, la mise en oeuvre d'un microscope à balayage SET est extrêmement difficile car il combine la microscopie à sonde à balayage, les basses températures et les dispositifs nanoscopiques très sensibles. Pour cette raison, seuls quelques groupes ont réussi sa réalisation. Nos choix technologiques pour construire le microscope améliorent certains aspects par rapport aux instruments déjà existants. La percée est que nous fabriquons la sonde SET en utilisant des techniques lithographiques standard sur des plaquettes commerciales de silicium. C'est pourquoi il est possible de fabriquer des sondes SET par lots. De plus, grâce à une combinaison de techniques de découpage et de gravure, le SET est conçu très près du bord du substrat de Si (< 1 µm). De cette façon, le SET peut être approché à quelques nanomètres de la surface de l'échantillon au moyen d'un contrôle de distance de force atomique. De plus, une électrode de grille sur sonde fabriquée à proximité de l'îlot peut être utilisée pour régler le point de fonctionnement du SET. Une nouveauté de notre instrument est qu'avec cet électrode de grille et une boucle de rétroaction, nous avons cartographié directement le champ électrique local. Nous démontrons cette nouvelle méthode de balayage par rétroaction en imaginant un réseau interdigité d'électrodes à l'échelle nanométrique. De plus, le SET est un outil idéal pour l'étude de la localisation d'états électroniques. À l'avenir, notre microscope sera utilisé pour l'étude des systèmes d'électrons bidimensionnels en régime de l'effet Hall quantique, des isolants topologiques et de la transition métal-isolant. v vi the collaboration of Pole electronique. In this work I also thank the participation of Pole informatique, Pole Optique et microscopies, S. Pairis, Liquefacteur, Pole gestión financiere, Otmane and Angelique, magasin, biblioteque, securité with Jusseppe, and administration with special mention for Sabine Gadal and Florence Pois, for their help in these 5 years. I also thank Florence Fernandez for her dedication to the laboratory and her confidence in me. Many other names appear as keys during these years. H. Sellier, who welcomed me in M1 and made me discover scanning probe microscopy. David Barral, who has made the corridor a very pleasant place. P. Mallet and his almost familiar treatment. F. Balestro and O. Buisson, who have helped me in everything required, N. Roch and B. Sacepé, and their talent and youth. And David Ferrand, responsible in capital letters for the M2. And other names like A. Claudel, Henri... The presence of such committed people as J. Delahaye, P. Molho and L. Marty is also to be welcomed. In addition, my time with Néel has been very pleasant thanks to the involvement, respect and closeness that the laboratory direction, Klaus, Laurence and T. Fournier, and Etienne, have dedicated to me. I am also especially grateful to the AS Cea Basket, and CAE-SUG and their managers for offering us so many possibilities of escape, and in particular with the ski and its manager William.

Without a doubt, a signature of the Institut Néel is the collaborative and youthful character of its corridors, which have added dozens of names to the list of people to whom I would like to thank for thousands of moments, and if I forget one, don't take it into account. I start with the good atmosphere in the office thanks to Estelle, a real discovery as a person, Guilliam, Michael, and the old colleagues who have helped me so much, Michal, Alexis and Quambo. Another pillar in my thesis has been the people of the Z building, Rémi with whom I have shared my time in Neel since les Houches 2015, and I especially thank him for the strength he has transmitted to me in the last months of writing, and Javier, who has given that necessary touch of humor and Spanish abstraction to the life of the labo. Always ready to help me with my rookie programs, Python's great man, and a great friend for the future, not forgetting his wife Ana. I add the good coffees and football matches with Luca, Seb, Jovian. I also remember Arkanksha who also lived in that office, as well as Gonzalo and our incipient friendship. And more coffees and bar time with Gom, Hugo, Kimon, Vadim, Maria S., Thanasis, Roberto, Francesco, Stephano. I also remember my classmates, Lucas, Olivier, Mathias, Pauline and Damir, Cyril, Khaled. And so many others like Jose, Victor, Corentin, Louis, David, Marco, Everton, Martin, Jun-Lian, Michelangelo, vii Kate, Massimiliano, Lotto, Jesus, Fito, Nitika, Gabriel... And my new Doctoral colleagues, Jean, Roxane and Thibault and Marion.

Y como hogar, tengo que agradecer a Grenoble y a todos aquellos los que compartieron sus experiencias conmigo a lo largo de estos casi 6 años. Tomás, mi gran amigo Tomás, "y los amigos me ayudan" llevado al extremo. He llorado como un niño cuando nuestros caminos se han separado, no por tristeza sino por sentirme abrumado de todo lo que has dejado en mí, ya que a veces una sonrisa y un abrazo sincero no llenan todo lo que sientes. Todavía tenemos aventuras por vivir y kiches por compartir, y copas del Pro por ganar, en Australia, en Argentina y donde sea que el mundo nos lleve. Gracias. Y a Raquel también, Gracias. Con mayúscula. Una conocida convertida en vecina, y. . . en una gran amiga para siempre. Vanni y Marila, o Marila y Vanni. Os agradezco tantos buenos ratos, apoyo, compromiso, respeto y más buenos ratos. Tanto en vuestra versión conjunta de powder rangers, como por separado. Sois un ejemplo para mí, y esta tesis solo ha sido posible por teneros cerca. . . Ana y Dani. Mis fichajes estrellas. Ana, te agradezco que estuvieras perdida aquel día de octubre. Así, pude encontrar un tesoro. Tu amistad, soporte y la confianza que me has regalado estos años de duro trabajo, han sido mi cuerda para llegar hasta el final. Esta tesis es muy tuya. No me preocupa el futuro porque sé que os tendré muy cerca a ambos. Shridevi, the warrior, discreet, humble, committed... the enormous mark you have left me, even in deep Spain, has driven me to complete these three years successfully. My best gift is to have shared moments with you and to have learned so much from you, and I hope to continue to do so. Thank you Shri. No me olvido de Alvaro, no cabe duda de que ha sido mis muletas estos años. Quien nos lo hubiera dicho haya por las barbacoas de Torre. . . Eres "real" y me has mantenido "real" en mis rayadas. Este trabajo no hubiera acabado igual sin tu amistad dentro y fuera del labo. Gracias bro. Agradezco haber compartido con Jaione nuestro proceso de evolución, desde aquella jornada de presentación Erasmus en 2013. . . completando el mismo recorrido. A Xenia, su ejemplo y experiencia, con un espíritu aventurero envidiable y una persona íntegra y fiable de las que se encuentran pocas. Gràcies. A Isa, ese karma de buena persona tras una personalidad sin límites, siempre dispuesta a ayudar, colaborar y siempre presente y agradecida. A Euge, y Nico y su paz y buenas vibras. A Maddi, sus conversaciones de sabia, su ejemplo de lucha y su independencia compaginada con el respeto y el buen rollo. Y la fiesta con el gran Asi. Eskerrik asko. A Robert, mi guía, el hombre que controla el viii tiempo para hacer todo lo que se propone, una inspiración y un orgullo compartir estos años. También a Kike y sus breves, pero intensos momentos. Un ejemplo de aguante donde los haya, que se mereció mucho más. A Andrea y Arianna. My favorite playmaker, with his vozka style, and the person that is surely the most intelligent and hardworking scientist in the city. The anxiety could not with us, and now the future is ours, and Oxford will give good account of it. Ketty, que se ha convertido en amiga de forma inesperada,a base de canastas y conversaciones de viaje, cada vez mas sinceras. Ricardo e Inés. Bien venidos al club grenoblés en buena hora y que me han mostrado que una amistad se construye tan rápido como buena persona sea alguien. To my namesake Giorgios Pavlov, I thank him for his enormous commitment and dedication to friendship, and to the Alpe du Grand Serre, always ready to gather despite having 3 free minutes a day. I thank Diana, the fastest person I have ever met in generating complicity and friendship. A Ana y Marc, con los que he compartido menos tiempo del que me gustaría, y que ahora defienden mis orígenes madrileños, esperando que vuelva. A Javi, el viajero y la alegría de vivir. A Bob, compañeros de las canchas y amigo del día a día. Y a María, la última en llegar, la reflexión hecha persona, la integridad y entrega hecha amiga. Muchas horas de viajes hemos hecho y muchas nos esperan.

También me gustaría dedicar un agradecimiento a aquellas personas que me han llevado hasta aquí y me han seguido acompañando. Por supuesto, mi promoción de la Universidad Autónoma de Madrid, y mis compañeros de viaje, Javi y Marcos, David, Eze, Mario, Jose, Imanol. . . Mis compañeros de Erasmus en la Joseph Fourier, y sus comidas en el cuarto, mi trieja Vane e Inés, y el mexican Fer, María, Carlos, Alvaro. . . y a mi confidente Jaime. También Marta y María, y nuestras no visitas hebdomadarias. Guiño, guiño. Por supuesto, no me olvido de mis vecinos, Karel, Dan, Fraser, Ellie y esa gran persona que es Erik. Palabras de agradecimiento para mi familia Scout Alamos, que me han hecho sentir como en casa cada verano y me han ayudado a limpiar mi espíritu cada mes de julio a base de sonrisas y lágrimas de alegría. Mis pueblenses, Rachel, Miguel, Pablo, Ares. Agradezco también a mis amistades del mundo, que siempre encuentran tiempo para mí, Clara, Patri, Lara, Angela, Alba, Cristina, Ali. . . y a mis amistades de siempre, el núcleo duro, Pablito, Salme, Juli, Vacas, Fede, Danito, Potter, Darío, Yago y Maya, siempre ahí desde preescolar.

Bien plus que des remerciements à Marie Laure, ma compagne d'aventure ix infatigable grenobloise, avec moi depuis le tout début. J'ai beaucoup appris de toi et avec toi. T'avoir tous les jours, ton calme, ton sourire et ton regard m'ont gardée sereine, concentrée et très vivante, générant de bons moments, et toujours là dans les moments les plus compliqués. Je remercie également sa famille, ses parents et ses oncles pour leur accueil, faisant des Hautes Alpes ma deuxième maison. Y, por último, aunque, en primer lugar, debo agradecer muy especialmente a mi familia, mucho más que un hogar, que ha sido la base sólida que me ha mantenido a flote durante toda mi etapa en Grenoble. Mi madre, mi padre y mi hermana Paula, han sido parte fundamental de este trabajo doctoral, mostrándome su confianza, promoviendo mi optimismo y disipando mis momentos de soledad. Considero a Lila también como parte de esa familia que tantos años ha cuidado de mí y que no le podré pagar nunca. Siendo una familia tan numerosa, quiero también agradecer el apoyo del resto de sus miembros, tíos y primos, y mil gracias a Pilar por venir a disfrutar de la defensa. Mención especial para mis abuelos, José, Conchita, Natalia y Antonio, que han construido tanto desde tan poco, emisarios de honor del sacrificio y el esfuerzo. Antes de vosotros, todo esto era campo. Gracias.

xi Chapter 1

Imaging Electric fields at the nanoscale 1.1 Introduction and motivation

The existence of inhomogeneous electronic systems has been known for quite some time. Charge density waves for example have been discovered in organic quasi-one dimensional systems [START_REF] Heger | Combined X-ray and neutron diffraction study of K 2 [Pt(CN) 4 ] X 0.3 .3H 2 O with X = Br, Cl(KCP) between 31 K and room temperature[END_REF] as well as in the inorganic layered material NbSe 3 [START_REF] Monceau | Electronic Properties of Inorganic Quasi-One-Dimensional Compounds[END_REF]. A direct imaging of a charge density wave has not been possible until local probe microscopies have become widely available [START_REF] Carpinelli | Direct observation of a surface charge density wave[END_REF]. Materials that develop inhomogeneous electronic states tend to do so on relatively short length scales, which requires a microscopy with nanometer and not micron scale resolution. Scanning Tunneling Microscopy (STM) achieves this to a high degree as it can image the local density of states with subnanometer resolution. But STM requires the injection of a tunnelling current in the sample probed, which is only possible in metallic states. While a broad range of inhomogeneous systems are indeed metallic, a number of other systems are not. For example, Mott insulators and strongly correlated oxides can in appropriate circumstances develop various types of droplet phases [START_REF] Shenoy | Electronic Phase Separation in Correlated Oxides: The Phenomenon, Its Present Status and Future Prospects[END_REF]. Another issue is that a local density of states is not always the observable of interest: In a dopant layer inserted in a semiconductor, it is the local electric field, or the local electro-chemical potential, which is the relevant physical quantity.

Techniques to image electric fields are also relatively recent. One of the first instrument used to image an electric field pattern inside a material is the holographic electron microscope [START_REF] Tonomura | Electron-holographic interference microscopy[END_REF]. An electron beam travels through the thin material of interest and subsequently interferes with a reference beam coming from the same coherent source. Along the path of an electron an additional phase, which depends on the electric field, is accumulated. The interference pattern reflects the map of this additional phase, and with some analysis, an electric field map can be inferred. Simpler and more direct methods derived from the atomic force microscope have been conceived. The Electric Force Microscope (EFM) [START_REF] Terris | Contact electrification using force microscopy[END_REF] and the Kelvin probe force microscope (KPM) [START_REF] Nonnenmacher | Kelvin probe force microscopy[END_REF] are similar in spirit, but measure different quantities. In an EFM, a charge q at the end of a cantilever experiences the local electric field at that point and transfers a force q|E| to the cantilever, up to some geometrical factor. As in the Atomic Forces Microscope (AFM), the measured force produces a map of the electric field magnitude. In a KPM setup, the metallic tip at the end of the cantilever is used as one side of a capacitor, the other side being the sample. The tip is biased at a potential V DC and the sample at a potential V CDP known as the contact difference potential, usually the difference between the sample Fermi level and the vacuum level. By modulating the tip position, the difference V DC -V CDP is measured [START_REF] Girard | Electrostatic force microscopy: principles and some applications to semiconductors[END_REF]. Since V CDP is closely related to the sample workfunction, it is commonly used today to extract workfunctions in a variety of setups.

In spite of this variety of tools available, studies of electric field mapping at solid state surfaces are rare. In conducting materials the scanning gate microscopy [START_REF] Topinka | Imaging coherent electron flow from a quantum point contact[END_REF] has been used to map the current distribution in point contacts, quantum Hall edge states and more generally in one and two dimensional conductors. While the interpretation of the image obtained is not always straightforward, it is a valuable tool for the investigation of low dimensional conductors.

In 1997, a new scanning probe technique [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF] consisting of a single electron transistor (SET) placed at the end of a glass fiber was developed in Harvard University. In this work, single charge sites in a GaAs heterostructure were resolved. The success of this technique came in large part from the extreme sensitivity of single-electron transistors [START_REF] Kastner | The single-electron transistor[END_REF] to local electric fields, which has been the object of an entire field of studies. The instrument also functioned in high magnetic field and exhibited remarkably detailed information on the sample density of states, the local electron compressibility of the 2D electron gas, all with a few hundred nanometer resolution. For instance, this SET microscope could image the electron/hole charge distribution in graphene [START_REF] Martin | Observation of electron-hole puddles in graphene using a scanning single-electron transistor[END_REF]. In spite of its success, subsequent attempts to develop an instrument for multipurpose charges and electric field imaging have been limited and mostly unsuccessful, mainly for technical reasons. Only one 1.2. The state of the art 3 scanning SET instrument built recently in Weizmann Institute of Science, [START_REF] Honig | Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3[END_REF] has a level of performance comparable to Yacoby's machine at Harvard. In this doctoral work, a new multipurpose scanning SET microscope has been built using somewhat different technological choices than in these two pioneering works. It is effectively the desired multi-purpose instrument which can be used to image the charge and electric-field distribution at the surface of insulating, metallic or semiconducting materials with charge sensitivities of ∼ 0.01 e or better. This allows to detect potential differences above a sample as small as ∼ 2 -5 mV at submicrometers distances and with a lateral spatial resolution of 300 nm. It can be operated in different modes and access several electric properties of the sample of interest. Its limitations come from the SET characteristics, namely the charging energy and the capacitance, and therefore the temperature range where the instrument functions. A metallic SET being nearly insensitive to magnetic fields, the instrument can also work in high magnetic fields.

The state of the art

We now briefly present how an SET's can measure electric fields. We next review the fabrication techniques of the scanning SET probes used in previous works and the results obtained so far. This is intended to motivate this doctoral work and to outline the possibilities offered scanning SET microscopy. Some of the potential applications will be presented in the final part of the chapter.

The SET as an electrometer

An SET is a device where electrons tunnel from a metal electrode (source) to a metallic island and subsequently to a second electrode (drain) through two thin insulating barriers (tunnel junctions) between the electrodes and the island (see Fig. 1.1 (A)). Depending of the potential difference V SD = V S -V D applied to the source and drain electrodes, a small DC current (I SET ∼ nA) flows. In this process, the island is charged and discharged. As in a field effect transistor, the current can be modulated by an external voltage V G applied on a gate electrode, capacitively coupled to the island. This voltage changes the device transconductance by shifting the charging energies of the island. The dependence of I SET (V g ) with the gate voltages is sketched on

SET probe fabrication

Due to the advances in nanofabrication and characterization techniques, devices consisting of tunnel and Josephson junctions (SETs, SQUIDs...) have been wide spread and accessible to many groups. Typically, a tunnel junction is formed by an oxide layer over metal, over which a second metallic layer is deposited. This is usually done with an electron beam physical vapor deposition (e-gun evaporator) (see for instance [START_REF] Sreeharsha | Principles of physical vapor deposition of thin films[END_REF]). This technology allows to deposit metallic films of only some nm thickness on a different surfaces. However, adapting such procedure to make a scanning probe is not straightforward. First the device should be engineered close to a sharp tip. The device should approach the surface of interest surface to a known height. This was first achieved by Yoo et al in 1997 [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF] in Harvard University. They were able to fabricate an SET using thin film metal depositions at the end of a sharp glass fiber. Some years later, a different approach using a carbon nanotube based SET was implemented into a new scanning SET microscope at Weizmann Institute of Science [START_REF] Honig | Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3[END_REF]. We briefly present the fabrication methods of the SET probes of these two instruments. We also describe the fabrication of a third SET probe developed at Max Planck Institute for which some preliminary results have been reported. Two other attempts for scanning SET measurements were not successful, but their fabrication approach has some bearing on the choices made for this instrument.

The scanning SET microscope of Harvard University

The fabrication process of the SET probe used in Yacoby's group [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF], is shown in Fig. 1.2 (A). Their scheme was adapted from ref [START_REF] Finkler | Self-aligned nanoscale SQUID on a tip[END_REF] where a similar process was used to fabricate a SQUID probe at the apex of the glass fiber. Their SET was made by depositing two Aluminium films (of 10 to 20 nm thickness) inside an e-gun evaporation chamber on the sides of a glass fiber in two separate steps (1) and [START_REF] Monceau | Electronic Properties of Inorganic Quasi-One-Dimensional Compounds[END_REF]. They are the source and drain leads (grey regions in Fig. 1.2 (A)). In a third step, the structure is oxidized in situ. The thin (∼ 1 nm) Al 2 O 3 insulting layer acts as the tunnel barrier. A last metal evaporation (3) towards the fiber apex creates the SET island which overlaps with the previous metallic films and forms the two tunnel junctions. The island size is the fiber diameter which was 100 nm at the apex. In the SEM image in Fig 1 .2 (B) from ref [START_REF] Finkler | Self-aligned nanoscale SQUID on a tip[END_REF], the Al regions on the flanks of the glass fiber (source and drain) appear in bright contrast separated by the bare quartz fiber which looks darker in the image. A schematic image of the SET fabricated on the Chapter 1. Imaging Electric fields at the nanoscale Si 3 N 4 sharp cantilever protruding from a Si substrate as seen in the SEM images in Fig. 1.5 (C) magnified in (D). Both methods showed the possibility of a batch fabrication of SET probes. Also, in both cases the cantilever were glued on a tuning fork (see Fig. 1.5 (C)) which, as explained in the chapter 3, allows to control reliably the tip-to-sample distance. In spite of the promising fabrication developments, no scanning SET measurement have been reported by any of them.

Properties mapped with a scanning SET

In a scanning SET the quantity that one can directly measure is the SET current I SET flowing from source to drain. While scanning, the SET current oscillates as a function of the electrostatic interaction between the SET and a sample at a certain height above the sample's surface, in a similar way as represented in Fig. 1.1 (B) for the gate voltage. In analogy with a transistor, a simplified view is that the sample acts as a local gate for the SET in each point of the scanned region. With this concept, from the SET current measurement, once can reconstruct maps of the electrostatic surface potential above a certain sample and other properties derived from this quantity. In the following sections, the main properties measured and results that have been obtained using a scanning SET are presented.

Surface potential

Measurements of electrostatic potential at surfaces with a scanning SET were first demonstrated in ref [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF]. They studied single electron sites arising from the photoactivity in a semiconductor GaAs/Al x Ga 1-x As heterostructure. In order to do so, an image of the surface potential is obtained before and after illumination of the sample with a light emitting diode. Subtracting the images, the difference can be related to the new charge arrangement after the light-matter interaction. The images (B), (C) and (D) in Fig. 1.6 are the result of subtracting maps of the calculated surface potential made before (A) and after a brief, low intensity illumination at three different wavelengths 850 nm, 770 nm, and 690 nm respectively. The typical scan height was z = 25 nm. The light and dark spots seen in the image after the subtraction can be associated with individual photo-ionized (light spots) and neutralized (dark spots) sites coming from a positive (resp. negative) charges. The spots, probably donor atoms or traps, have an apparent size of ∼ 100 nm which coincides with the spatial resolution of the instrument, the SET island size (Fig. 1

.2 (C)). The

Compressibility of electronic states

The SET can also be used as a sensor for physical properties related to the electrostatic field and the surface potential. Particularly, a number of the works done with a scanning SET focussed on the compressibility of the electronic states in a 2D electron systems (2DES). The electronic compressibility is defined as [START_REF] Eisenstein | Compressibility of the two-dimensional electron gas: Measurements of the zero-field exchange energy and fractional quantum Hall gap[END_REF]

κ = 1 n 2 dn dµ (1.1)
where n is the average density of carriers of the 2DES and µ the chemical potential. The compressibility, in simple words, measures the capacity of the electronic system to screen electric fields. A measurement of the local compressibility can be done with the SET using a back-gate electrode placed below the 2DES. The back-gate V BG tunes the Fermi level of the 2DES and thus the average carrier density n. The electric field created by the back-gate will be screened to different degrees depending whether the 2DES is compressible or incompressible. If the electrons have delocalized wavefunctions in certain regions (like in a metal), they respond to the external electric field coming from the back-gate effectively screenning its effects. This is a compressible behavior. On the contrary, if the electrons are localized (like in an insulator), they are "frozen" and the electric field can penetrate through the 2DES with little or no screening. This is an incompressible response. The magnitude of the penetration field (referred as transparency) is the quantity detected by the SET. As a result, the SET above the 2DES feels the local variations in the surface electrostatic potential φ which depends on the screening by the 2DEG 1 . When a strong magnetic field is applied perpendicular to the 2DES (and at low temperatures), integer and fractional quantum Hall phases appear which are specific to different 2DES. In this phases, the filling of Landau levels is characterized by ν is the filling factor. Macroscopic transport measurements done with a Hall-bar mesa configuration show a quantization of the Hall conductance σ = ν e 2 h for filled Landau levels (ν is an integer). A plateau in the Hall conductance appears for a range of magnetic fields as illustrated in Fig 1 .8 (extracted from [START_REF] Weis | Metrology and microscopic picture of the integer quantum Hall effect[END_REF]). In this regime, a current flows without dissipation along the edges of the sample along edge channels (grey 1 More technically this is done by modulating the back gate ∆V BG ∝ ∆n and measuring the ∆φ ∝ ∆µ with a lock-in amplifier [START_REF] Yacoby | Electrical imaging of the quantum Hall state[END_REF]. This operation allows to obtain dµ dn which is proportional to the inverse of the compressibility.

GaAs/AlGaAs heterostructure, and their evolution with the magnetic field and the carrier density was detected with an SET in refs [START_REF] Wei | Single-electron transistor as an electrometer measuring chemical potential variations[END_REF][START_REF] Weis | Probing the depletion region of a two-dimensional electron system in high magnetic fields[END_REF]. These experiments were not performed with a scanning SET but instead, with a fixed SET fabricated directly on top of the heterostructure. This work was the first indirect measurement of the electronic compressibility using an SET and is the precursor to other scanning SET compressibility experiments.

In this context, Yacoby's group in ref [START_REF] Yacoby | Electrical imaging of the quantum Hall state[END_REF] implemented a sophisticated set-up using a lock-in detection of the SET signal for a direct measurement of the compressibility od 2DES using a scanning SET. In such way, incompressible stripes associated with edge states were spatially resolved in a GaAs/AlGaAs heterostructure. This gave a qualitative picture of the current flow in the quantum Hall state. In their work, the SET was also used to measure simultaneously the Hall voltage profile between two edges of the 2DES. They observed a drop in the potential profile along incompressible strips where the edges decouples from the bulk. These measurements clarified the current distribution in the quantum Hall regime. Similar Hall voltage observations were observed later using an electrostatic force microscope [START_REF] Weitz | Hall-potential investigations under quantum Hall conditions using scanning force microscopy[END_REF][START_REF] Ahlswede | Hall potential profiles in the quantum Hall regime measured by a scanning force microscope[END_REF][START_REF] Ahlswede | Hall potential distribution in the quantum Hall regime in the vicinity of a potential probe contact[END_REF][START_REF] Panos | Current distribution and Hall potential landscape towards breakdown of the quantum Hall effect: a scanning force microscopy investigation[END_REF].

With the scanning SET, individual localized states in a GaAs/AlGaAs heterostructure were observed in ref [START_REF] Zhitenev | Imaging of localized electronic states in the quantum Hall regime[END_REF] where the effect of disorder was investigated. Some years later, in ref [START_REF] Ilani | The microscopic nature of localizazion in the quantum {Hall} effect[END_REF] the many regimes and phase transitions in the quantum Hall regime were shown incompatible with a single particle vision of the phenomena [START_REF] Huckestein | Scaling theory of the integer quantum Hall effect[END_REF]. This is also the case in graphene, where the localization in quantum Hall regime was also studied with an scanning SET in ref [START_REF] Martin | The nature of localization in graphene under quantum Hall conditions[END_REF]. Such localization seems to arise from the competition between the disorder potential and the repulsive Coulomb interactions.

The compressibility of electronic states is intrinsically related to the thermodynamic density of states dn dµ [START_REF] Martin | Observation of electron-hole puddles in graphene using a scanning single-electron transistor[END_REF]. With the scanning SET, it is possible to study the evolution of the compressibility as a function of the average carrier density n and the magnetic field B and build Landau fan diagrams at different sample positions above the 2DES.

Such diagrams give a direct signature of the state changes across the sample, and can be compared with the averaged results obtained from transport measurements. Studies with an SET were reported in refs [START_REF] Martin | Observation of electron-hole puddles in graphene using a scanning single-electron transistor[END_REF][START_REF] Martin | The nature of localization in graphene under quantum Hall conditions[END_REF] in monolayer graphene flakes, and in refs [START_REF] Martin | Local compressibility measurements of correlated states in suspended bilayer graphene[END_REF][START_REF] Feldman | Unconventional sequence of fractional quantum hall states in suspended graphene[END_REF][START_REF] Benjamin E Feldman | Fractional quantum hall phase transitions and four-flux states in graphene[END_REF][START_REF] Kou | Bilayer graphene. Electron-hole asymmetric integer and fractional quantum Hall effect in bilayer graphene[END_REF], in suspended high mobility graphene. In this last sample, fractional quantum Hall states were also observed (see ref [START_REF] Venkatachalam | Single Electron Probes of Fractional Quantum Hall States[END_REF] for more information). Two examples of Landau fans measured with an SET are shown in Fig. 1.9 (A) for a monolayer graphene Chapter 1. Imaging Electric fields at the nanoscale studies presented in ref [START_REF] Honig | Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3[END_REF] similar to the compressibility of 2DES have revealed the movement of the domains with the backgate voltages. This is an extrinsic piezoelectricity. This behavior may be explained by the formation of an anomalous tetragonal striped domains in the material which was also revealed in ref [START_REF] Honig | Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3[END_REF].

Some potential applications for the scanning

SET microscope

Scanning SET can, measure not only a charge or an electric field distribution but also the local compressibility of an electronic system in the vicinity of its surface. The sensitivity is very high, but the present resolution (of the order of a few 100 nm) may not always be sufficient to resolve the electronic structures which have been measured recently using STM techniques. On the other hand, the local compressibility is often a key quantity in a number of electronic problems, as will be illustrated in this section, where some of the outstanding problems in condensed matter physics are reviewed in the context of scanning SET.

Metal insulator transitions

In condensed matter physics, two mechanisms are usually invoked for driving a metallic state to an insulator.

• For the Anderson transition, it is the disorder which drives a metal to a poor metal and eventually to an insulator [START_REF] Anderson | Absence of Diffusion in Certain Random Lattices[END_REF][START_REF] Abrahams | Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions[END_REF]. The mechanism is the random diffusion on impurities and defects, which leads to destructive interference between all the possible electronic paths. The nature of the diffusive paths of the charge carriers depends strongly on the spatial dimension: interference occurs on closed paths, which are relatively infrequent in three dimensions while they are ubiquitous in two and one dimension. This is the reason why Anderson localisation is so important for bidimensional electronic systems (heterostructures, graphene, topological insulators). For disordered systems, the spatial extend of wavefunctions is characterized by a localization length ξ. In one and two dimensions, ξ is controlled by the mean free path ℓ e

1.3. Some potential applications for the scanning SET microscope

19 ξ 1D = c 1 ℓ e (1.2) ξ 2D = ℓ e (exp c 2 k F ℓ e ) (1.3)
where c1 = 2w λ F and c 2 = π w 2λ F . Here w measures the width (resp. thickness) of the 1D (resp. 2D) structure and λ F is the electron wavelength. c1 and c2 are usually of order of unity. In 1D, the localization length is of the order of the mean-free path ℓ e , which is of the order of the impurity distance, a microscopic length scale. In 2D, the exponential factor contains the ratio ℓ e /λ F , usually significantly larger than unity, and the localization length in 2D can often exceed the 10-100 nm range, not too far from the current resolution of an SET microscope.

In the immediate vicinity of the Anderson transition and on the insulating side, charges are localized on the scale of ξ and Coulomb interaction starts to play a role. This affects transport properties and critical exponents without changing fundamentally the nature of this localization driven mechanism. Studies of the Anderson transition using real space and imaging techniques are relatively recent and mostly confined to STM imaging on the metallic side of the transition [START_REF] Richardella | Visualizing critical correlations near the metal-insulator transition in Ga(1-x)Mn(x)As[END_REF]. To this date, most of the experimental literature on the Anderson transition comes from transport measurements and the localization length is inferred from scaling which is in fact quite reliable.

For an Anderson transition, the electronic compressibility is a continuous function, since the density of state remains finite on the insulating side: states are localized on a relatively large length scale, where their polarizability remains large. This is in sharp contrast with the Mott-Transition, driven by Coulomb interactions:

• The other common mechanism which drives a metal to an insulating state is Coulomb interaction [START_REF] Imada | Metalinsulator transitions[END_REF]. This has been widely studied theoretically, by Mott, Hubbard and many more and dominates the physics of oxides. When an electronic band is half filled, there is an electron on each site. Adding an electron is possible, but requires paying a Coulomb repulsion U on each site: this opens a (Hubbard) gap. For this reason, the state is usually insulating unless the kinetic factor measured by the hopping amplitude t is large. There are two ways the system can become metallic: one departs from half filling by increasing the electron Inhomogeneous electronic phases often occur in the vicinity or at a metalinsulator transition, either for intrinsic reason (charge density waves or stripe phases), or because of slight inhomogeneities in the stochiometry and or disorder. For example, the latter was observed in the graphene experiments of Yacoby et al. in ref [START_REF] Martin | Observation of electron-hole puddles in graphene using a scanning single-electron transistor[END_REF]. The spatial scale of these larger structures are much easier to resolve and are within reach of the SET microscope.

Two dimensional electronic systems and bilayers

Coulomb blockade devices are unaffected by large magnetic fields: the SET microscope works well in high magnetic fields and can image all 2D gases or layered electronic systems. 2DEG form naturally series of conducting and insulating phases as a function of magnetic field because the quantization of electrons in a magnetic field organize their spectra in one or more series of Landau levels. In electronic transport, this manifests itself as a succession of quantum Hall plateaus where the longitudinal conductivity vanishes. This zero conductivity signals an insulating state, but the presence of a quantized Hall resistivity also indicates that some form of transport takes place: it is through an integer number of edge states (one dimensional conducting channels) related to the Landau level index. There is in fact a topological description of the quantum Hall effect which relates directly the Hall conductivity to the presence of edge channels [START_REF] Thouless | Quantized Hall Conductance in a Two-Dimensional Periodic Potential[END_REF]. These conducting channels are conducting "filaments" on equipotential lines. Their width depends on the magnetic length and the shape of the local potential and ranges between 5 to 50 nm depending on the system studied and the magnetic field.

In two dimensions, disorder plays a critical role and is a central element in quantum Hall physics: Landau levels broaden and the spatial extend of localized states depends on the energy difference with respect to the center of the Laudau levels. For smooth disordered potentials, localized states in the quantum Hall regime take the form of closed filament loops around the equipotentials of the disordered potential (inner edges) and lie within the 2D sample. When the filaments connect opposite edges, bulk conductance becomes finite and a transition between quantum Hall plateaus occurs.

STM microscopes have been used to image localized states in graphene, as the two dimensional electron gas which coincides with the surface, is directly accessible to the tunneling tip. Unfortunately, there are a number of systems where a cap layer (GaAs/GaAlAs) or an oxide layer exists (topological insulators, polar oxides), and the 2D system lies below an insulating layer. The filaments corresponding to localized states are charges that accumulate between insulating and incompressible regions and produce a detectable electric field which can be accessed using an SET microscope. The localization length is large and well within the resolution of the instrument as shown in the experiments of Yacoby's group.

Today, the physics of 2D electron systems, including graphene are relatively well understood. However, there are a number of bi-layer systems or systems involving several surfaces which are researched everywhere. We use to illustrate the physical problems with 3D topological insulators [START_REF] Brüne | Quantum Hall Effect from the Topological Surface States of Strained Bulk HgTe[END_REF]. In these materials, a conducting layer forms on the entire surface at the interface with vacuum or a "standard" non-topological material. For a parallelepiped and a magnetic field oriented perpendicularly to two faces, top and bottom faces have Quantum Hall States which are coupled together by the vertical faces. This pins the chemical potential and, depending on the coupling strength, can favor the formation of bilayer states. In some materials, a coupling can also take place through the bulk bands which can hybridise weakly with the surface states. The quantum Hall states of topological insulators have resemblance with graphene and coupled graphene layers, which are also intensively studied. This is because the dispersion relation of the surface states form a Dirac cone as it is the case in graphene. There are notable difference also as spin and orbital degrees of freedom are locked in topological insulators because of the strong spin-orbit coupling.

There is much interesting physics related to bilayer 2D electron gases some of which have been studied in quantum wells [START_REF] Eisenstein | Exciton Condensation in Bilayer Quantum Hall Systems[END_REF]. For example, 2D coupled quantum wells exhibit a Josephson effect related to the formation of an excitonic liquid where electron states in one layer are coupled to hole states in the other layer. The nature of the physics changes with the strength of the coupling as well as its nature. Different phenomena are expected to arise in different 2D systems. The SET microscope appears to be ideally suited to make real space images of single and bilayer 2D electron systems. The presence of a cap or an oxide layer at the surface is not a problem, provided it is thin enough and that charges do not accumulate on these insulating surfaces.

Imaging characterization

For most active devices used today in nanoelectronics, charge carriers are introduced by doping specific areas. The dopants and the carriers generate local electric fields which in turn modify locally the semiconductor bands. Much of this, can be simulated by solving transport equations with Poisson equations for the electrostatics. However, new materials are constantly being introduced, for example high-κ oxides used to quench tunneling effects between gates and conducting channels. Not everything can be simulated as the complexity increases. Characterization of devices has therefore become an absolute necessity, to ensure the proper functioning of devices before mass production. The ability to map electric fields is of great value for the semiconductor industry. At present, the scanning SET is not a sufficient general purpose instrument for routine checks for the semiconductor industry, because it requires low temperature. However, with a graphene based SETs [START_REF] Ihn | Graphene single-electron transistors[END_REF], it is possible to increase charging energies by more than a decade and in the long run, this instrumentation may become a commercially viable technology for the characterization of semiconducting devices.

SET microscopy

The instrument of Yacoby's group showed that scanning SET microscopy was not only possible but could be exploited in a number of physical problems. After Yacoby's group instrument, only the scanning SET microscope implemented in Weizmann's Institute has shown similar results. This is an indication of the level of complexity that requires the realization of a machine of this type. Various aspects of these instruments can be identified to understand the strong points and the complications that need to be faced in order to succeed in the construction of a scanning SET. About Yacoby's group instrument we have noted that:

• The absence of a local gate in their SET makes the control of the operating point of the SET difficult. In their experiments, the sample potential, controlled via a gate or a bias was used to pick an adequate operating point. This is possible for a conducting 2D electron gas, but is impractical in a number of other problems where conducing and insulating regions co-exist. There is also an issue of sensitivity. In most of his experiment, the SET was operated in a unique charge period of the single electron transistor. This means that the effect of the sample acting as the SET local gate is not very large, i.e. the sensitivity is relatively weak.

• The SET device is fabricated at the end of a glass fiber. Each SET probe requires a full cycle of fabrication, and the variability in the SET tunneling resistance could not guarantee a functional probe at the end of each fabrication cycle. This is a major drawback as SET are fragile devices: the very small tunnel junction can easily blow up when some stray charges are present somewhere in the circuit used to measure them.

• There is no scheme to control the SET-sample distance other than the local capacitance, which can only be measured if the sample is metallic. This makes the microscope prone to crashing accident, a real setback considering the complex fabrication scheme and their low yield.

• Despite the complexity of the evaporation method, they have managed an instrument with an SET based on a thin film metal evaporation scheme.

For the Weizmann's Institute instrument,

• The SET is fabricated using a handmade nanoassambly of the nanotube, which requires a technological development and expertise that is not commonly accessible. This approach is very challenging and only one probe can be fabricated at a time which reduces the yield.

• A nice aspect of these probes comes from the small diameter of the nanotubes: the charging energies of the final SET probe are much higher. Scanning measurements can be made at higher temperatures compared to Yacoby's instrument.

• Additionally, a local gate is included close to the nanotube improving the control of the operating point of the SET.

• The SET is fabricated very close to the edge of an AFM cantilever which provides the distance control. Being a very challenging technique, this safety is critical.

One appreciate the major challenges facing SET microscopy and the technical aspects which need be brought together in order to build a functioning instrument. Eventually both the sample and the SET probe should work in a cryogenic environment (readout of small signals, filtering...) and it is necessary to scan at small SET-to-sample distances while maintaining the probe characteristics to perform the experiments. The fabrication scheme should be kept simple and generate many probes easily. Our objective was therefore 1.5. Thesis layout to simplify the probe fabrication, insisting on the standard lithographic fabrication of SETs where a gate electrode and a batch fabrication are possible. of tips. Also the fabrication scheme is compatible with distance control my means of a the tuning fork.

Thesis layout

In the next chapter, the SET is reviewed theoretically, taking the point of view of electrometry: for this application, the gate dependence of the tunneling current is key and needs to be investigated with the bias dependence. Issues of sensitivity, feedback mechanisms are also best described in this context. The fabrication scheme to bring the SET at the very tip of a wafer is new and is essential for the SET instrument fabricated in this thesis. The considerations which led to this fabrication scheme are described in the second part of chapter 2.

The implementation of the fabricated SET into a microscope requires a scanning stage, a scheme to control the distance to the surface, a measurement circuit which shields the device from all radiation sources, a cryogenic environment with a source of large magnetic fields. All these elements are discussed in chapter 3.

Chapter 4 is devoted to the imaging of test devices which have both insulating and metallic regions in order to test the microscope abilities in most situations of interest. Using a new SET current feedback, we were able to image samples with a large charge density, far exceeding the one-electron period of the SET. Very close to a metallic surface, we observed on a number of samples, a quenching of the SET Coulomb blockade which cannot be explained with the standard electrostatic model of the SET described in chapter 2.

Finally, some of the perspectives are given for the application of this instrument to some of the problems described in the preceding section. Some of the obvious improvements to the instrument can be made to make the imaging of a variety of surfaces simple.

Chapter 2

The instrument

Introduction

A single electron transistor (SET) is based on the Coulomb blockade phenomenon (CB) [START_REF]Single Charge Tunneling[END_REF]. When CB is active, the number of charges on a small island are quantized in integer numbers n of the electron charge e and can only increase or decrease one at a time, usually through tunneling across one of the junctions in contact with the small island. The electrostatic energy of a charge state n or n + 1 can be tuned continuously by inducing a polarization charge with an electrostatic gate. When this polarization charge is equivalent to 1 2 of an electron charge, the electrostatic energy of the n and n + 1 charge state are equal. At this gate potential, it no longer cost any energy to add or subtract an electron to the island and a current can flow through the stochastic addition or subtraction of a charge. If a DC bias between the source and the drain is applied, the range of gate voltages where a current can flow through the device becomes larger. The phenomenon does not depend on the exact number of charge on the island: if a polarization charge equivalent to one full electron charge is induced on the island, the device characteristics are equivalent to that of an island with one electron missing. Hence the SET characteristics are periodic with gate voltage: it is not a linear device. This behaviour was first observed in [START_REF] Fulton | Observation of single-electron charging effects in small tunnel junctions[END_REF]. In many respects an SET can be compared to a superconducting interference device (SQUID) [START_REF] Likharev | Single -electron transistors: Electrostatic analogs of the dc squids[END_REF] which characteristic is periodic with the applied magnetic flux.

In the pioneering work of Yacoby and co-workers [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF], the only gate acting on the SET island was the sample itself. This means that the electric field coming from the sample induced the polarization charge modulating the measured SET current. Because of the periodic nature of the transfer function between gate and SET current, a linear relation between the sample electric field and the measured current occurs only if the resultant induced charge on the SET island is small compared to an electron charge e, the SET period. In Chapter 2. The instrument this regime and at an appropriate working point, the measured SET current is proportional to the sample electric field up to geometrical factors. However, there are many cases where the electric field produced by a solid state sample can induce a polarization charge larger than e on the island, i.e. exceeding the linear limit. A complete control of the SET working point requires the addition of a control gate, which can compensate for the sample electric field and allows locking the SET current to the best working point, in terms of SET sensitivity and linearity. The latter suggest a lithographically defined SET [START_REF] Dolan | Offset masks for lift-off photoprocessing[END_REF], where a good control and reproducibility of the tunnel junctions can be achieved on a routine basis. The manufacturing and the integration of small side gates very close to the SET is also a standard procedure in nano devices. This additional gate gives a full control of the device and the SET operating point can be maintained by compensating the sample electric field by this SET gate, de facto transforming the device in a linear probe through this feedback mechanism. The technology of small junction and SET's has been in the group for a number of years, but transforming a lithographed SET into a usable tip for near field microscopy involved some real challenges. To achieve a few nanometer resolution in SET-microscopy, the probe needs to be placed at comparable distances to the sample. To this end, a technology to fabricate an SET at the extreme corner of a silicon wafer has been developed. Combined with standard top down fabrication techniques, these SET probes can be fabricated in large batches where all their junctions are identical. Because of their small junction size and high impedance, the probes are very sensitive to electrostatic charge, and require a careful and controlled procedure when they are mounted in place on the microscope.

Coulomb blockade imposes some constraints on the SET operating conditions, i.e. at temperatures below the device charging energy, of the order of 3 -7 K in our probe design. This energy could be increased further by replacing the Aluminium island used for these probes by carbon nanotubes as it is done in [START_REF] Waissman | Realization of pristine and locally tunable onedimensional electron systems in carbon nanotubes[END_REF], with considerably smaller junctions and larger charging energies. However most of the physics related to inhomogeneous electronic phases, usually occurs at low temperature and the device is well suited to these solid state studies.

In this chapter, the SET physics will now be reviewed and the schemes to make the SET probes presented. The probe design and fabrication are the central technological steps enabling the SET microscopy.

Chapter 2. The instrument for the source C s and drain C d leads, the gate C g and any other external capacitive coupling to the island C env . For our devices C s ≃ C d ≫ C g , C env . At finite temperature, thermal fluctuations blur this electrostatic effect. If on the contrary the system is frozen out T ∼ 0 K, the charging energy becomes dominant. This regime can be approximately reached when

E ch ≫ k B T, (2.1)
Eq 2.3 makes requires very small capacitances to be able to observe the Coulomb blockade even at very small temperature. Nanotechnology allows to do such junctions on the nanometer scale [START_REF] Dolan | Offset masks for lift-off photoprocessing[END_REF]. Thinking of a flat capacitor model for the tunnel junction, it is possible to fabricate easily junction areas of some tens of nanometer A ∼ 50 × 50nm 2 , and controlled oxide thicknesses around a nanometer, d ∼ 1 nm. For Al-AlOx-Al junctions, ǫ r = 10 for the oxide layer. The total capacitance considering only the source and drain elements (we will see that for our devices

C s ≃ C d ≫ C g , C env ) is of the order of, C ∑ = 2 × ǫ 0 ǫ r A d ∼ 5 × 10 -16 F (2.2)
This gives an charging energy of e 2 2C ∑ k B ∼ 2 K. It is simple to cool a system below this temperature.

In this low temperature regime, the charge state of the island is well defined and lead to an electrostatic blockade of the current at small bias. If the bias is not large enough to overcome the charging energy, the electron can not charge the island and the tunneling probability and thus the current drops to zero. This phenomenon is known as Coulomb blockade and is the main characteristic of an SET.

In addition, the gate (C g ) or any external capacitance (C env ) coupled to the island change slightly the charge present on it. The latter modulates the electrostatic barrier which can even disappear. As can be imagined, such effect change the tunnel probability. Controlling the conductance with a gate electrode makes the system look similar to a field effect transistor: this is why the device is called a single electron transistor.

Using the gate electrode, we can actually tune the system from a Coulomb blockade to a conducting regime with a polarization charge on the island. Between these regimes, the current variation may be fast and abrupt which makes this device a very good candidate for a sensitive electrometer. Such current variations may also come from a local capacitive coupling between the SET and a sample C sample . By measuring the variations in current at different positions above a sample, the SET becomes a electrostatic field sensor. This is the main objective of this work.

In order to fully describe the transport properties of the SET, tunneling and electrostatic effects need to be included. An standard model describes the energetic conditions making tunneling favorable from an electrostatic point of view. This model, known as the orthodox theory, is presented in the following sections and will help us to see the different features and characteristics of the SET electron transport between source and drain.

This model can still be used if the metal electrodes and island become superconductors at low temperatures creating a superconducting SET (S-SET). This system also exhibits the same features and can be used as a charge detector.

Single electron current

As seen in the introduction, thermal fluctuations should be quenched to make the charging effects dominant. This requires low temperatures

E ch ≫ k B T. (2.3)
Under this condition, an electron that tunnels from source to island leaves behind a screening cloud which increases temporarily the electrostatic barrier. The build-up in energy is enough to reduce or even cancel the probability for another electron to enter the island until the first one has escaped through the drain junction. The current is driven by sequential electron tunneling from source to drain.

Quantum fluctuations impose a second condition: the tunnel resistance has to be much larger than the quantum of resistance:

R t ≫ h e 2 = 25.8 kΩ. (2.4)
This ensures that the time delay δt = eR t V sd of an electron crossing from source to drain is much larger than the characteristic time of a single tunneling event τ ∼ h eV sd . The latter can be derived from the uncertainty principle and stands for the quantum limit for which quantum fluctuations blur charge quantization. In other words, as explained in [START_REF] Ferry | Transport in nanostructures[END_REF], this condition ensures that an electron stays longer on the island than the time it takes to cross the junctions, i.e. the charge is well localized on the island during the transfer from source to drain. An estimation of a tunneling time (measured in [START_REF] Hagmann | Determination of barrier traversal time by modulation of the incident wave[END_REF]) is of the order of 10 × 10 -14 s while typical SET currents are of the order of nA, meaning a transfer time of 10 × 10 -9 s.

Tunneling rates

With the two conditions 2.3 and 2.4, one can solve the quantum problem of tunnelling through a double junction using what is known as Orthodox theory where the thin oxide layer between two metal contacts can be treated as a potential barrier acting as a small perturbation of the system [START_REF] Nazarov | Coulomb blockade of tunneling in isolated junctions[END_REF][START_REF] Ingold | Charge Tunneling Rates in Ultrasmall Junctions[END_REF]. Such quantum model was first used in [START_REF] Kulik | Kinetic phenomena and charge discreteness effects in granulated media[END_REF] and further completed by [START_REF] Averin | Coulomb blockade of single-electron tunneling, and coherent oscillations in small tunnel junctions[END_REF][START_REF] Glazman | Coulomb oscillations of the conductance in a laterally confined heterostructure[END_REF]. Quantum perturbation theory is the most appropriate tool to calculate the current through the system. The transition rate through a single junction, between an initial state in the source |s to a final state in the island |i can be calculated using the Fermi Golden rule

Γ s→i = 1 h | s|H|i | 2 δ(E s -E i -∆E). (2.5)
Where H is the tunnel Hamiltonian defined in [START_REF] Averin | Coulomb blockade of single-electron tunneling, and coherent oscillations in small tunnel junctions[END_REF] and | s|H|i | 2 = |T| 2 is the square modulus of the transmission coefficient T. To obtain the total tunneling rate, we have to consider the density of states (DOS) of the electron on each side of the junction G s and G i and the Fermi-Dirac functions of the electrons in the source F(s) and of the empty states on the island (1 -F(i))

Γ(∆E N ) = -∆E N e 2 R ts 1 -e ∆E N k B T , (2.6)
where R ts is the tunneling resistance of the source tunnel junction, given by:

R t = h e 2 |T| 2 G s G i (2.7)
For N tunnel junctions in series, this equation is valid for each of the tunneling events in between elements. In this case we would have to consider that the junctions might not have equal resistances. The SET can be characterized with by its normal total tunneling resistance R t = R ts + R td . Where R ts and R td may be different in case of asymmetric junctions, for instance due to different dimensions.

To calculate the total current, we account for all the tunneling processes, this is an electron going from source to island, from island to drain, from where

C s V s + C d V d + C g V g + C env V env = ∑ j=s,d,g,env C j V j is
a displacement charge induced by the polarizing electrodes and the environment. The island, in principle neutral, may still have a certain background charge Q bg when the potential of the electrodes is set to zero. An additional term Q 0 accounts for the presence of an excess charge storage on the island. This charge can increase if for example an electron tunnels from source to drain. The following calculations are adapted from [START_REF] Fuhrer | Phase coherence, orbital and spin states in quantum rings[END_REF].

The electrostatic energy U (N) needed to charge the island with N excess electrons at a fixed V j is:

U (N) = -eN 0 V 0 (Q 0 ) dQ 0 = e 2 N 2 2C ∑ + eN Q bg C ∑ -∑ j=s,d,g,env C j C ∑ V j . (2.10)
To completely describe the system, the quantization energy of the electrons in the island ∑ N i=1 ǫ i should also be included1 in the total energy E (N):

E (N) = U (N) + N ∑ i=1 ǫ i = e 2 N 2 2C ∑ + eN Q bg C ∑ - n ∑ j=s,d,g,env C j C ∑ V j + N ∑ i=1 ǫ i (2.11)

Electrochemical potential

The energy E (N) can be seen as the total free energy of the island and takes into account both the electrostatic energy stored on the island and the work done by the voltage sources. The change of the free energy in a thermodynamic system when an electron is added or removed is known as the electrochemical potential µ(N) = ∆E (N) . The use of µ is very convenient to combine electrostatic effects and tunneling in a simple model [START_REF] Likharev | Single -electron transistors: Electrostatic analogs of the dc squids[END_REF].The electrochemical potential of the island can be written as the energy needed to add an extra electron. For instance, when it goes from N -1 excess electrons to N,

µ N = E N -E N-1 = e 2 C ∑ N - 1 2 + e Q bg C ∑ - n ∑ j=s,d,g,env C j C ∑ V j + ǫ N , (2.12)
where the term ǫ N is the energy difference between quantum levels occupied by electrons on the island due to confinement. For large island sizes (> 10 nm (as it is the case in this work) it can be considered that the DOS of the electrons in the metal form a continuum of states and ǫ N is very small and negligible compared to the electrostatic contribution.

As in a typical band semiconductor picture, the electrochemical potential in the source and drain leads can be considered as the Fermi level of the electrons µ s = µ 0 -eV s and µ d = µ 0 -eV d for a certain applied voltage V s source and V d drain respectively. Here µ 0 is the reference electrochemical potential when the source and drain connected in thermodynamic equilibrium (V s , V d = 0). When a physical system is not in equilibrium, it will try to minimize its free energy. In the SET, this means that an electron moves from areas with a higher electrochemical potential to areas with a lower one. Quantum tunneling is the physical mechanism. The difference in the µ level in the electrodes and the island gives then a straightforward idea of the tunnel probability and thus the current. This description can be represented by the electrochemical potential diagrams as those shown in figure 2.3.

The different electrochemical potential thresholds for µ N expressed by Eq. 2.12 can be represented as diagrams with the discrete charging levels when filling the Nth excess electron level on the island. A consequence of eq. 2.12 is that the energy difference between two charging levels remains constant with respect to any of the electrodes voltages. This constant is known as addition energy E add ,

E add = µ N+1 -µ N = e 2 C ∑ . ( 2

.13)

With this model, the favorable situations for tunneling are easily found and the influence of the electrostatic effects on the total tunneling probability. As seen in the previous section, a tunneling process involves the transition of an electron between a initial to a final state. To have a finite probability, an empty final state should be energetically accessible for the electron (Fermi Golden Rule). These diagrams give an idea of the energy range where available states are accessible and thus of the different tunneling rates for each electron transfer.

In Fig. 2.3 for a certain source, drain and gate voltage, the system starts with N -1 electrons on the island: the island electrochemical potential is µ N-1 . From this situation, when an extra electron tunnels to the island, the only possibility for the system is a final state with electrochemical potential µ N . To cross one of the tunnel junctions, both charge states needs to be energetically accessible to the electrons in one of the electrodes. Additionally, in order for the Nth electron to have a probability to leave the island through the other junction, it has to be able to reach the Fermi level of the opposite electrode. If these two conditions occur, both tunnel junctions can be crossed by the electrons. An energetically favorable situation appears which is summarized by the following equations

µ s ≥ µ N ≥ µ d or µ d ≥ µ N ≥ µ s . (2.14)
With a large enough bias V sd , one of these conditions can be achieved as it is represented in Fig. 2

.3 (A) and (C). In this case, the island changes between

N -1 and N electrons every time a new electron tunnels out the island and another takes its place. This is represented with shaded white and blue area filling the level µ(N) in Fig. 2.3. A single electron current I SET (V sd , µ N ) can flow. The larger the difference µ sµ d = eV sd , the larger the number of available states from source to drain, and hence the larger tunnel probability and SET current. We would then expect a higher current in Fig. 2.3 (A) than in Fig. 2.3 (C). On the opposite, represented in figure 2.3 (B), the bias may be not large enough for an electron to reach a final state in the level µ N . This is the condition for the Coulomb blockade. The N -1 electrons on the island are blocking the others to tunnel in.

Stability diagram

I SET (V g , V sd )
In Fig. 2.3 two situations are highlighted: when decreasing the bias V sd , one can pass from a conducting situation (A) to a blocking state (B). Alternatively, by tuning the electrochemical potential levels of the island, even at a small bias, a conducting situation (C) can be recovered from the blocked state (B). According to Eq. 2.12, this can be done by varying the gate potential V g . Since µ(N) = µ(N, V g ) the combination of both effects for different charge states builds what is known as stability diagram of an SET, seen in Fig. 2

.4 (A).

In order to build such a diagram, a situation where the drain electrode is connected to ground so V d = 0 and the source to a potential V s = V sd is considered (see Fig. 2.2. For V sd = 0, the starting situation will be µ s = µ d = µ 0 which is represented as green circles in Fig. 2

.4 (A).

For positive bias, V sd > 0, this is the top half of the stability diagrams, the electrochemical potential of the source goes below that of the drain. Then, in order to have a tunneling current between source and drain we would need the condition,

µ s = µ 0 -|e|V sd ≤ µ N ≤ µ d = µ 0 (2.15)
Using eq. 2.12, these two inequalities can be solved for the bias V sd =

V N sd corresponding to an initial situation where there are N electrons on the island, For the left inequality, we get the the group of solutions,

V N sd ≥ + C g C ∑ -C s V g + C env C ∑ -C s V env - e N -1 2 C ∑ + C s - Q bg C ∑ + C s + µ 0 C ∑ C ∑ + C s = = C g C ∑ -C s V g + c(N) (2.16)
The equality corresponds to an alignment of the source electrode with the µ N level in the island. With respect to the gate electrode, we see that the solution is a line with a positive slope,

C g C ∑ -C s (2.17)
which would correspond to the dashed red line in figure ??. On this line island has N -1 electrons but one extra coming from the source is energetically allowed, thus enabling the transport. A positive bias below this line will not give enough energy to the electron to tunnel into the island. This defines the grey regions in the diagram corresponding to the Coulomb blockade, for instance (E) in Fig. 2.4.

For the second inequality we have the region defined by,

V N sd ≥ - C g C s V g - C env C s V env + e C s N - 1 2 + Q bg C s + µ 0 C ∑ eC s = - C g C s V g + c(N) (2.18) 40
Chapter 2. The instrument This is a straight line as a function of the gate voltage with a negative slope

C g C s (2.19)
In this case, the equality corresponds to an alignment of the the electrochemical potential of the island µ N with that of the drain electrode. This is represented as a blue dotted line in Fig. 2.4 (A).

Decreasing the gate potential, we pass from being aligned with the source to be aligned with the drain. Both lines intercept in the starting position, light green circle in Fig. 2.4 V sd = 0. The gate value for which this happens is called degeneracy point (D) in Fig. 2.4. At this position no blocking state can be reached by just changing the bias.

Periodicity with respect to the gate voltage

As it can be seen in the diagram, Fig. 2.4 (A), the latter calculation gives the same results for any extra electron added or removed from the island, defining parallel dotted and dashed lines which cross at the light green circles. The diagram also accounts for possible asymmetries between the source and drain capacitances, which result in different slopes for the dotted and dashed lines. Solving eq 2.18 for two consecutive degeneracy points N and N + 1,

V N G (V sd = 0) = C s C g C env V env C s - e C s N - 1 2 - Q bg C s + µ 0 C ∑ eC s V N+1 G (V sd = 0) = C s C g C env V env C s - e C s N + 1 - 1 2 - Q bg C s + µ 0 C ∑ eC s (2.20)
We get a distance between two degeneracy points in gate voltage,

V N G (V sd = 0) -V N+1 G (V sd = 0) = - e C g N - 1 2 + e C g N + 1 2 = e C g (2.
21) which is a constant with respect to the gate capacitance C g , which means that the diamonds are equal and thus equivalent for any N excess charges on the island. The same calculation for negative bias describes the bottom half of the diagrams. The continuation of the limiting lines of the inequalities gives a particular diamond shape to the Coulomb blocked regions. For that reason they have been called Coulomb diamonds.

Addition energy

In the diagram, we see that the Coulomb blockade is restricted below the crossing points of the dotted and dashed lines (dark green circles). In these points, the source is aligned with for example the Nth island level and the drain with the (N + 1)th level (situation (B) in Fig. 2.4). We analyze this case for a positive bias for the alignment of V N+ sd , the positive slope for the limiting solution for an alignment of µ N , and V (N+1)sd , the negative slope of the limiting solution of the drain electrode aligned with µ N-1 :

V N+ sd = - C g C ∑ + C s V g - C env C ∑ + C s V env - e N -1 2 C ∑ + C s - Q bg C ∑ + C s + µ 0 C ∑ C ∑ + C s V N+1 sd = - C g C s V g - C env C s V env - e C s N + 1 - 1 2 - Q bg C s + µ 0 C ∑ eC s (2.22)
This imposes the condition

V N+ sd = V (N+1)- sd = V sd . After extracting -C s C ∑ +C s we get; V sd - C s C ∑ + C s V sd = V sd 1 - C s C ∑ + C s = -e N -1 2 C ∑ + C s + -e N + 1 2 C ∑ + C s V sd C ∑ C ∑ + C s = e C s + C ∑ V sd = e C ∑ (2.23)
For negative bias, the same calculation leads to

V sd = e C ∑
. This shows that the height of the Coulomb diamonds with respect to the zero bias axis is a constant related to the addition energy of the system:

E add = e 2 C ∑ (2.24)

Action of the bias voltage V sd

From Fig. 2.4 (A), it is possible to anticipate the characteristic SET current behavior with respect to variations in the bias potential at a fixed gate voltage V g , this is keeping µ N constant. Such characteristic is depicted in the graph at the right part of Fig. 2.4 (A). The dark and light blue curves correspond to a bias sweep at the gate position sketched by the dark and light blue lines cutting the diagram respectively. In the dark blue curve, the current remains blocked as long as µ N does not fall in between µ s and µ d (situation (E)). This occurs in the grey regions of the diagram until a certain bias V (N+1)sd for which the electrochemical potential of one of the polarized electrode aligns with that of the island (crossing between dotted line and dark blue line in Fig. 2.4 (G)). At higher bias, the tunnel probability and thus the current increase. The current will ideally increase (T = 0) with a constant slope related to the tunnel resistance R t , as the density of states is constant for metals. The current behaves differently for the light blue curve since the gate voltage corresponds to a degeneracy point (green circle). In the latter case, no Coulomb blockade is possible at any bias. However around the degeneracy point, the resistance will increases to a bigger value with respect to the normal conducting state in the dark blue curve. Since the bias is small, the charge transfer is slower (around 4 times slower). This current will ideally tend to the normal value for bigger bias inside the green diamonds.

Action of the gate voltage V g

At the top part of Fig. 2.4 (A) three I SET (V g ) curves (green, orange and red) are represented corresponding to three different bias values, depicted by the position of the green, orange and red lines in the diagram respectively. For the green curve, corresponding to V sd ≈ 0, the current remains blocked for any gate voltage except in a region around the degeneracy point (light green circle) where periodic peaks appear. This situation is depicted in Fig. 2.4 (D) and (E). In the case of the orange curve (V sd > 0) the conductive region is larger than in the previous case. Since the bias is also larger, more final states are available and hence the current peak will also be larger.

A special situation occurs when the bias voltage is as big as the addition energy, eV sd = µ(N)µ(N -1) = E add (red curve). In this case there is always an µ N charge level available for tunneling and no Coulomb blockade appears for any gate voltage. Another degeneracy point arises when the limiting lines for N excess electron meets those of N + 1 or N -1 (dark green circles and situation (B) in Fig. 2.4). In these points (dark green circles), the tunneling probability is higher since two electrostatic charge states are available. The system can oscillate between N -1, N and N + 1 electrons in the island. This results in periodic maxima of current.

Superconducting SET

In this work, the SET are made of aluminium. This metal becomes superconducting below a critical temperature T c . This temperature is usually around 1.2 K [START_REF] Cochran | Superconducting Transition in Aluminum[END_REF]. As described by the Bardeen-Cooper-Schrieffer (BCS) theory, below such temperature, due to the electron-photon interactions with the lattice of the material (Type I), the electrons condense into Cooper pairs (see for instance [START_REF] Lévy | The BCS Theory of Superconductivity[END_REF]). These are bosonic particles consisting of two electrons with equal and opposite momentum and spin. When the system is in the superconducting state, Cooper pairs are bounded in the ground state inside the superconducting gap, ∆ (T). At T = 0, the gap is related to:

2∆(0) = 3.528k B T c (2.25)
Where ∆ and hence T c depend for instance on the thickness of the granular Al film [START_REF] Yu | Dependence of the critical temperature and energy gap of the thickness of the superconducting Aluminium films[END_REF]. For energies higher than this gap, Cooper pairs break into two individual electron and hole-like quasiparticles which, following a semiconductor picture occupy available states above or below the superconducting condensate. This can be represented by two quasiparticle bands starting at an energy ∆ from the Fermi level of the superconductor (set in E = 0) with a density of states:

N (E) ∝ |E| √ E 2 -∆ 2 (2.26)
It is possible to use Fermi statistics to calculate the occupation of these quasistates. At zero temperature (k B T ≪ ∆) quasiparticles fill completely the bottom band while the top one remains empty. We see that outside the gap, the quasiparticle occupation in the electrodes gives an image similar to that of the metallic SET but with a different DOS.

Single quasielectron tunneling

For an SET in the superconducting state, these quasiparticles can be treated as electrons for energies above the gap. The tunneling rate from occupied states to empty states is also defined by Fermi's Golden rule as in the case of the SET in the normal state. However we should not forget that these electrons can never fall inside the gap region of any of the electrodes since no quasi-electron states are available. To complete a tunneling event from one lead to the island, the energy of the quasi electrons should exceed that of the available states of the island. The latter requires an energy at least of 2∆ with respect to the island. The top of the filled band in the superconducting

Other tunneling phenomena in S-SET

The tunneling of Cooper pairs through the junctions is also possible due to the phase difference in the superconducting wave function between the electrodes. This is known as Josephson Effect [START_REF] Josephson | Possible new effects in superconductive tunnelling[END_REF] and results in a supercurrent crossing through the junctions. In the diagram of Fig. 2.5, this process only happens when the Fermi energies of two of the electrodes are aligned (this is inside the gap). This implies two bound electrons crossing the junctions from source to island and island to drain. In this case, we should observed additional current peaks in the gap region of the superconducting S-SET stability diagram. Since it implies an equivalent transfer of charge of 2 e, the peak is detected with a twice as large period with respect to that of single quasielectrons tunneling [START_REF] Amar | 2e to e periodic pair currents in superconducting Coulombblockade electrometers[END_REF]. Another possible tunneling event, also happening inside the gap consists of one Cooper pair tunneling through one junction and two quasiparticle tunneling through the other junction. [START_REF] Fulton | Observation of combined Josephson and charging effects in small tunnel junction circuits[END_REF][START_REF] Maassen Van Den | Combined single-electron and coherent-Cooper-pair tunneling in voltagebiased Josephson junctions[END_REF][START_REF] Nakamura | Quantitative analysis of Josephson-quasiparticle current in superconducting single-electron transistors[END_REF][START_REF] Hadley | 3 e tunneling processes in a superconducting singleelectron tunneling transistor[END_REF]. This process is known as double Josephson quasiparticle tunneling.

The characteristic curves for a S-SET transistor depend on the relative magnitudes between four energies in the system [START_REF] Hadley | 3 e tunneling processes in a superconducting singleelectron tunneling transistor[END_REF]: Thermal (k B T), charging energy E ch , the superconducting gap (∆) and the Josephson energy E J . The last energy scale can be estimated using,

E J = hI c 2e (2.27)
where I c is the critical current that can be obtained from the Ambegaokar-Baratoff relation [START_REF] Ambegaokar | Tunneling Between Superconductors[END_REF],

I c = π∆ 2eR t (2.28)
To clearly see tunneling events implying Cooper pairs, it is required E J at least in the order of E ch . For this reason, considering eq 2.28, the Josephson effect is less likely to happen for junctions with higher tunneling resistance R t . As it will be presented in the chapter 4, the relation of the characteristic energies of the SETs fabricated in this doctoral is E J ≪ E ch ≈ ∆. This limits considerably the Cooper pair tunneling. Hence no intra-gap S-SET features will be analyzed further in this doctoral work.

The behaviour of the SET or S-SET as a sensor is quite similar as long as the bias voltage leads to the single quasielectron transport regime in the superconducting case. As a comment, some studies [START_REF] Song | Advantages of superconducting Coulomb-blockade electrometers[END_REF] reveal that in fact the S-SET has some advantages with respect the normal SET. Due to the singularities of the quasielectron DOS there is a steeper charge transfer function from blocking to conducting states. This results in larger charge sensitivity of the S-SET with respect to the SET in normal state having both comparable noise (see section 2.2.1). Furthemore, the Coulomb blockade remains up to higher operating temperatures ( T c ) in the S-SET than in the normal states. In our instrument we would use the SET in superconducting or normal state depending on the requirements of the sample. The superconductivity of Al will be completely suppressed at 0.5 T resulting in a normal state SET at larger values. In chapter 4 we show results scanning with an S-SET with no magnetic field.

Principle of Scanning SET Microscopy

At a certain bias, the tunnel probability and thus the SET current is governed by the gate voltage since it has a direct effect on the electrochemical potential of the island I SET = I SET (µ N ), as seen in eq 2.12,

µ N = e 2 C ∑ N - 1 2 + e q bg C ∑ + e C s V sd C ∑ + e C g V g C ∑ + e C env V env C ∑ = e C ∑ q SET
(2.29) When the SET scans above a sample at a fixed gate voltage, two other terms in eq.2.29 may vary: A background charge q bg ,and a capacitive coupling with external environmental sources, in this case, the sample C env V env = C sample V sample . One can imagine that, if these components also vary, the SET current will react in a similar way as it does with the gate voltage. This is the idea behind the scanning SET microscopy. When we approach the SET close to a sample, and we scan above it, the local electrostatic interaction between the SET and the sample is different at each point of the scanning which is manifested as a change in the SET current.

The variation of the total charge of the SET q SET (x, y) can be written as,

δq SET (x, y) = δ(C sample V sample ) = = δC sample (x, y)V sample + C sample δV sample (x, y) (2.30)
The first term of the equation accounts for changes in the local capacitance C sample (x, y) with respect to a certain potential source V sample . An example is depicted in Fig. 2.6 (A), where the scanning is performed above a polarized electrode with a certain inclination with respect to the SET. The second term gate electrode close to the SET island and it will be discussed in detail in the Chapter 4

SET sensitivity

A seen in the previous sections, the SET current reacts quickly to the presence of electric field sources places close to the island. These changes in the current are in fact coming from variations in the island charge due to that electric field. Over a charge period the current variation is large with a period related to the local capacitance between the sample and the island. This means that inside a period, a current difference comes from a fraction of a single electron shift in the island charge. Since the devices was experimentally probed, the community realized that this shift could be as small as ∼10 -4 e/ √ Hz [START_REF] Kuzmin | Single-electron charging effects in one-dimensional arrays of ultrasmall tunnel junctions[END_REF][START_REF] Geerligs | Tunneling time and offset charging in small tunnel junctions[END_REF][START_REF] Zimmerli | Noise in the Coulomb blockade electrometer[END_REF] and the SET could be exploit as a very sensitive electrometer [START_REF] Lafarge | Direct observation of macroscopic charge quantization[END_REF]. Many theoretical and experimental studies were done in the nineties to understand, analyzing and improving the smallest signal level that can be detected with instrumentation in terms of the induced charge on the island. This determines the sensibility limits of the SET as a charge detector. [START_REF] Korotkov | Single-Electron Transistors as Ultrasensitive Electrometers[END_REF][START_REF] Korotkov | Intrinsic noise of the single-electron transistor[END_REF] derived an analytic expression for the minimal fundamental output thermal/shot charge noise (q x ) min of the SET. This was done in the limit k B T ≪

e 2 2C ∑
, for a bias close to the Coulomb blockade region (edges of the Coulomb diamonds) and a for self-capacitance C i of the source of noise very low with respect to its coupling with the SET C 0 , this is

C 0 ≫ C i . (q x ) min ≃ 5.4e   k B T e 2 0.5C ∑   1 2 R t 2 0.5C ∑ ∆ f 1 2 (2.32)
where ∆ f , is the output bandwidth of the SET including the filtering and amplification for a low frequency measurement range ( f < 0.1 -1kHz). This arises from the stochastic character of the tunneling process and by the discreet nature of charge carriers. For even smaller temperatures,

k B T ≪ R Q πR t e 2 C ∑
, (with R Q ∼ = πh 2e 2 = 6.5 kΩ), the noise would be dominated by quantum fluctuations and co-tunneling which are not included in the previous equation. The equation 2.32 is for that reason considered as the intrinsic fundamental quasi-classical noise. In the instrument that is presented in this thesis, typical values are C ∑ ∼ 100 × 10 -16 F, R t = 500 kΩ and T = 100 mK. This gives a floor noise value of 10 -6 e/ √ Hz.

Unfortunately, this represents only a theoretical threshold. In a real experimental set up, the SET current is also sensitive to random background charge fluctuations in the SET environment. A source of noise arise from the redistribution of trapped charges in the tunnel barriers or/and in the substrate near the SET island and other impurities in the different interfaces and (native) oxidized surfaces on the electrodes and substrate as reported in [START_REF] Zimmerli | Noise in the Coulomb blockade electrometer[END_REF][START_REF] Manninen | Background charge fluctuations in SET-transistors[END_REF][START_REF] Likharev | Single-electron devices and their applications[END_REF]. In both metal and superconducting state SET [START_REF] Hergenrother | Fabrication and Characterization of Single-Electron Tunneling Transistors in the Superconducting State[END_REF] it manifest with a 1 f dependence which dominates over the intrinsic thermal/shot noise on the low-frequency 1-100 Hz range. Even though the origin of this noise is still not totally understood [START_REF] Li | Low-frequency charge noise in suspended aluminum single-electron transistors[END_REF] many studies suggested that it increases with larger island size and the surface of contact between the island and a dielectric substrate [START_REF] Verbrugh | Optimization of island size in single electron tunneling devices: Experiment and theory[END_REF][START_REF] Zorin | Background charge noise in metallic single-electron tunneling devices[END_REF][START_REF] Zimmerman | Modulation of the charge of a single-electron transistor by distant defects[END_REF][START_REF] Bouchiat | Quantum fluctuations of the charge in single electron and single Cooper pair devices[END_REF] and also depends on the quality of the tunnel junctions oxidation. In a scanning SET, the presence of the sample will also add a contribution to the noise. This behavior sets the minimal noise level above the intrinsic value at least a factor 10-100 at low frequencies [START_REF] Zimmerli | Noise in the Coulomb blockade electrometer[END_REF]. Reducing this noise by an effort in the fabrication step makes possible SETs sensitivities of 2.5-7 × 10 -5 e/ √ Hz at 10 Hz [START_REF] Krupenin | Noise in Al single electron transistors of stacked design[END_REF][START_REF] Visscher | Fabrication of multilayer single-electron tunneling devices[END_REF] respectively. As a conclusion, in order to reduce the noise of the SET as much as possible and understand the sources of noise, a good control of the SET fabrication is needed. This highlights the importance of a controlled and reproducible SET probe fabrication. We will see in the next sections that it can be achieved by means of a standard lithographic fabrication of the SET.

SET on-chip probe fabrication 2.3.1 Motivation

The implementation of the SET on a scanning probe imposes the challenge of engineering a way to fabricate it close to a sharp tip. This eases the approaching of the device to a sample at sub micrometer distances. As depicted in Fig. 2.7 (A) one option to do so is to manufacture the device at the apex of a pulled glass fiber. This technique has been used in Yacoby's group instrument [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF] for the SET probes and in Weizmann group [START_REF] Finkler | Self-aligned nanoscale SQUID on a tip[END_REF] for µSQUIDS probes. In both microscopes, the sensitive part of the device (island of the SET or SQUID loop) is fabricated exactly on top of the fiber apex as can be seen in Fig. 2.7 (B) for the case of µSQUIDS probes. This ensures that the SET is the first contact point with the sample after the tip-sample approach. In addition, the particularity of the SET is the presence of two nanometer size tunnel junctions, this is, a metal-insulator-metal superposition. The size of the junctions defines the transport characteristics of the SET. Thus, there is angle, say of α = 5 deg between the substrate edge and the sample, the SETto-sample distance would be as short as z = 430 nm. The challenge of using an on-substrate approach was to find a way to fabricate the probe as close as possible to the edge of the wafer in order to reduce as much as possible z and keeping the SET safe during the process. The on-substrate fabrication of lithographic scanning probes has been achieved for instruments such as Hall probe microscopes, [START_REF] Chang | Scanning Hall probe microscopy[END_REF][START_REF] Keyser | Scanning Hall probe microscopy of vortex patterns in superconducting micro-squares[END_REF][START_REF] Dede | Scanning Hall Probe Microscopy (SHPM) Using Quartz Crystal AFM Feedback[END_REF][START_REF] Shaw | A scanning Hall probe microscope for high resolution, large area, variable height magnetic field imaging[END_REF], and µSQUIDS [START_REF] Veauvy | Scanning µ-superconduction quantum interference device force microscope[END_REF][START_REF] Hasselbach | High resolution magnetic imaging: MicroSQUID Force Microscopy[END_REF][START_REF] Hykel | MicroSQUID Force microscopy in a dilution refrigerator[END_REF] probe microscopes. In the following sections we detail the different technologies required and the steps that haven been followed during this doctoral work in order to achieve a batch fabrication of an SET probes very close to the edge of a commercial Si-SiO 2 substrate.

E-beam lithography and shadow evaporation

To fabricate an SET it is necessary to deposit thin metallic films in welldefined nanometer size areas. A thin metallic film can be deposited with a specific pattern on top of a substrate by means of a lithographic mask. For this purpose, e-beam lithography mask fabrication appears as a very convenient choice. In e-beam lithography the mask template consists of different kinds of polymer resists spin-coated on top of a substrate. The electrons of the e-beam are able to weaken the bonds of the polymer only in the exposed regions. The weakened zones of the resists are removed afterwards with a solvent which develops the final lithographic pattern. In order to deposit a thin metallic film, the mask is later introduced in an e-gun evaporator that generates a metallic flow towards the mask and only reaches the surface of the substrate on the regions that have been patterned.

As sketched in the SEM image in Fig. 2.8 (A), adapted from ref [START_REF] Lecocq | Novel E-beam lithography technique for insitu junction fabrication: the controlled undercut[END_REF], when a cavity is dug in the resist, the same pattern can be deposited in several regions of the substrate by projecting the metallic flow with different angles. This concept is called shadow evaporation and is key for the obtention of small tunnel junctions. A resist bilayer, where the bottom resist is less resistant to the e-beam bond breaking than the top one, is necessary to fabricate the cavities. In our case this is done with a top layer of PMMA (light red in Figs. 2.8 (B) and (C)), and a bottom layer of PMMA diluted in MMA (light red). In the the case of low acceleration e-beams voltages (20 keV) as seen in Fig. 2.8 (B) when the electrons interact with the substrate after going through the resist bilayer, they are backscattered to regions close to the initially exposed region (the green region representing the exposed zone). For that reason, the resists are actually exposed to larger regions than those defined by the original pattern. This is called proximity effect. Since the bottom layer is weaker than the top one, this effect affects mainly the bond-breaking of the bottom layer, hence resulting in a cavity after development.

At the Neel Institut, a high acceleration 80 keV voltage e-beam lithographic machine is available, anB5 from NANOBEAM LTD. With such voltage, as seen in Fig. 2.8 (C), the electrons are backscattered deeper in the substrate and they emerge far from the original pattern. Therefore no undercut appears. This is depicted by the well-defined left flank of the patterned resist seen in the bottom image in Fig. 2.8 (C). The final pattern (that should correspond to the exposed region represented in dark green) is developed very precisely in both the bottom and top resist. However, in principle no cavity is created. It is worth pointing out that a second treatement is needed in oder to obtain the required undercut. This is the controlled undercut technique (CUT), which was developed by ref [START_REF] Lecocq | Novel E-beam lithography technique for insitu junction fabrication: the controlled undercut[END_REF]. This method consists in exposing regions of the resists with a e-beam doses (less current) which is small enough to leave unaffected the top resist layer but strong enough to weak the bonds of the bottom one. This is represented by the light green region in Fig. 2.8 (C). In this way, after development, only the top layer remains an a cavity is created underneath the exposed region depicted in light green as seen at the bottom figure in Fig. 2.8 (C).

SET on-substrate fabrication

The Niemeyer-Dolan technique [START_REF] Niemeyer | Observation of large dc supercurrents at nonzero voltages in Josephson tunnel junctions[END_REF][START_REF] Dolan | Offset masks for lift-off photoprocessing[END_REF], also called the double angle shadow evaporation technique, is a lithographic method to create nanometer-size overlapping thin-film structures in well defined areas of a substrate, using one single lithographic mask. It is based on the cavity concept presented in the last section through which an evaporation mask is created where the top resist forms a suspended bridge above an undercut cavity, as seen in Fig. 2.9 (A). An insulating substrate surface is needed in order to fabricated the SET. A commercial Si wafer coated with a 100 nm thermal SiO 2 layer serves as the substrate for the SET fabrication. Using the suspended bridge resist mask, a tunnel junction consisting in a Al-AlOx-Al, superposition can be done in the following way: A metal evaporation in a α direction through the top patterned region (Fig. 2.9 (B)) deposits an Al metal layer underneath the suspended bridge. Then the metal is afterwards oxidized creating a thin AlOx layer which will act as a tunnel barrier. A second evaporation (Fig. 2.9 (C)) in an opposite direction β deposits another Al layer overlapping the previous structure to finally obtain the tunnel junctions. Each of the fabrication parameters have an effect on the transport characteristic of the SET and thus on its performance as an electrometer. For instance, the larger the area of the tunnel junctions the larger the charging energy and the lower the junction resistance. The advantage of the Niemeyer-Dolan technique is that one gets total control of the SET geometry. In fact, due to the large acceleration voltage of the e-beam used in this work, the lithography template can be developed with a very high resolution. As a result, one can design a mask that anticipates very accurately the surface and position of the metallic films that will be deposited. As seen in Fig. 2.9 (B), the latter only depends on the evaporation angle α and the total thickness of the resist bilayer

d 1 + d 2 .
In Fig. 2.9 (D) the mask to fabricate the tunnel junctions is presented. The white regions correspond to the developed pattern for the source (S'), drain (D') and gate (G') electrodes and the island (I'). The red regions represents zones where the top layer resist remains after development. As depicted in Fig. 2.9 (E), during the first angle evaporation, the source (S), drain (D) and gate (G) metallic electrodes are deposited on the regions represented in blue. The deposited metal film is place at a distances 1 = (d 1 + d 2 )tgα from the pattern. The second evaporation deposits the island on top of the source and drain electrodes as represented by the cyan regions in Fig. 2.9 (F)). As a result, two small tunnel junctions are created.

The latter process is done using an e-gun evaporator from the company PLASSYS. Inside, the sample is placed on a sample holder which can be tilted with respect to the metal flow. The instrument measures the thickness of the evaporated metallic films with nm precision. As seen in the front view of the tunnel junctions and island presented in Fig. 2.10 (A), the second metal layer (∼ 40 nm) has to be thicker than the first one (∼ 20 nm) to ensure the overlapping. This machine also permits an in situ oxidation of the metal in a well controlled O 2 atmosphere. The oxidation is a critical step in the fabrication since the tunneling rate of the junctions decreases exponentially with the barrier thickness. For AlOx it is usually in the order of 1 nm. Small changes in the oxidized layer provoke large changes in the junctions resistance so in order to ensure a certain reproducibility of the process an effort is done to keep always the same O 2 pressure and oxidation times in each operation.

As a last comment, when the two evaporations are completed, and before taking the sample out from the evaporator, an oxidation of the entire structure is done again under a controlled atmosphere. The intention is to protect the metal from irregular native oxidation in laboratory conditions, which may trap charged particles close to the SET and affect the eventual sensitivity.

After the evaporation, the resist is removed from the substrate using a strong solvent in a process called lift-off leaving only the metal deposited on the substrate surface. The result can be seen in the SEM image in Fig. 2.10 (B). The source, drain and gate electrodes resulting from the first angle evaporation have been coloured in blue. The island deposited in the second evaporation appears in light cyan. The regions of overlap are coloured in orange and represent the tunnel junctions. It can be observed an area of the tunnel junctions in the order of 50 nm 2 . The usual charging energies for junctions of such order of magnitude are in the range 3 -5 K in units of temperature. Hence, in order to characterize the SETs cryogenic temperatures are needed to fulfill the SET requirements E ch ≫ k B T. The size of the island is a second important parameter since it is the sensitive point of the SET to the external charges. The bigger the island the smaller the expected lateral spatial resolution since the measurement would be less localize. It can be seen that a good control of the island geometry is achieved with the lithographic fabrication of SETs. In this case, the dimensions of the top surface are in the order of 50 × 300 nm 2 . The evaporation process represented in Figs. 2.9 (A),(B) and (C) would correspond to a cross section of the resist along the white line (1) represented in Fig. 2.10 (B). Since in the region of the source, drain and gate patterns there is a large density of exposed regions, a wide cavity is created underneath the top resist due to proximity effect. This results in a second deposition of the source, gate and metal electrodes which can be seen in cyan in Fig. 2.10 (B). On the contrary, the island pattern is far from the others. For that reason no cavity exists on the right part of the island pattern and thus the metal is deposited on the walls of the bottom resist layer for one of the evaporation. Therefore, such metal is later removed with the lift-off. With this we avoid a residual floating metal film close to the island which may produced unwanted charging effects.

The success of the SET fabrication requires a extensive training and trial in order to find a reproducible recipe. In Figs 2.10 (C) and (D) two examples of the most common issues that have to be faced are presented. In (C) due to a small misalignment of the substrate inside the e-gun evaporator, the resulting scanning stage. Hence, this increases our understanding of the SET properties, selecting the most appropriate fabrication parameters and correcting error in the recipe. A reproducible fabrication of SETs has been achieved, which reduces the number of bad surprises when implementing the SET on a tip in scanning conditions.

Another advantage of a having a gate is that once in scanning mode, it gives full control of the SET transport even in the presence of large electric fields. The closer the gate electrode is to the island, the bigger the capacitance and hence the control of the charging state of the island. This last feature enables a linear read-out of the electrostatic interaction between the SET and the sample, as will be discussed in Chapter 4.

One-by-one SET tips on the corner of a Si chip

The NANOBEAM used for the e-beam lithography allows to pattern in one same lithographic step both the small SET patterns, where detail is needed (tunnel junctions, island..), and the large electrodes and bonding pads required to make the electrical connections. All this is done with flawless stitching between structures. The NANOBEAM also enables the patterning of large substrate surfaces with high precision alignment and focus. As a result, a batch fabrication of SETs tips is conceivable. Certainly, it would be necessary to separate the SETs in individual chips leaving the SET island very close to the edge of each single substrate to obtain a sharp tip.

There are two main methods in order to cut a Silicon substrate: a diamond scriber, available in the NanoFab clean room at the Neel Institute and a diamond dicing machine, in PTA Grenoble.

The scriber consists of a sharp diamond tip that scratches straight lines on the Silicon surface. This weakens the Si atomic structure enabling the cleavage of the substrate in smaller pieces. If this is done along the atomic planes of the Si, the resulting pieces will have very clean and sharp edges. However, if the cutting path differs from the atomic plane of the Si, the flanks of the new chip are affected by chipping and irregularities may appears some µm away from the edges. If the SET is placed close to one of these edges, it would be surely damaged. For this reason, in order to use the scribing machine after the SET fabrication, the lithography has to be perfectly aligned with the Si atomic planes. This is not a simple task.

On the other hand, the dicing machine makes micrometer deep and large trenches in the Si surface along the desired direction. This is a rather invasive resist was spin-coated. With this procedure, as seen in Fig. 2.11 (B), one of the corner of each cut chip has a uniform resist distribution at the edges (no multicoloured region), since it belonged previously to the center of the wafer.

The main difficulty encountered when patterning the SET close to the chip corner is to set the origin of the lithography with respect to such corner. Localizing the angle of the chip with the e-beam in SEM image mode would already expose the resist, thus making more difficult the success of the SET lithography 2 . Instead, a routine was engineered to align the e-beam with respect to the flanks of the wafer. This permits to figure out the exact position of the corner without exposing it. Due to the fact that the resist at the edges was weaker than expected, and due to the difficulties encountered to maintain the edges clean in the latter steps of the process (evaporation, development..), it was possible to fabricate the SET only some 5 to 10 µm away from the corner as seen in Fig. 2.11 (D). With this recipe, the SETs can only be evaporated one by one, hence losing the batch fabrication possibilities given by the on substrate technology. For these reasons the mentioned fabrication approach was not pursued and other possibilities were explored.

Mesa approach

The problems described earlier have been addressed in the development of other lithographic probes [START_REF] Veauvy | Scanning µ-superconduction quantum interference device force microscope[END_REF][START_REF] Hasselbach | High resolution magnetic imaging: MicroSQUID Force Microscopy[END_REF][START_REF] Hykel | MicroSQUID Force microscopy in a dilution refrigerator[END_REF][START_REF] Shaw | A scanning Hall probe microscope for high resolution, large area, variable height magnetic field imaging[END_REF] using what is called a mesa approach. The concept can be summarized in the following phrase: If it is difficult to fabricate the device close to a sharp edge, the device can be instead lithographed far from the edge and then the edge redefined close to the device. This is represented in Fig. 2.12. Starting from the situation in Fig 2 .7 (B), by removing a part of the substrate (dotted lines) a second edge is created that will be closer to the probe than the previous one. However, in order for the new edge to become the contact point with the sample, the sides of the removed area have to be of the same order of magnitude as seen in Fig. 2.12. If such condition is fulfilled, with the required small angle, the edge of the mesa is the contact point with the sample. This reduces substantially the device-sample distance compared to the configuration shown in Fig 2 .7 (B). The mesa approach requires then a controlled etching process that will remove the desired part of the substrate. The objective would then be to find the most appropriate way to etch the substrate as close as possible to the device while keeping a sharp and clean edge. use of simple lithographic resist masks to define the etching regions. Such process is available in the clean room with a SPTS Xactix e2 machine.

Since the etching process presented here is isotropic, the design of the etching mask should anticipate the lateral etching distance so the resulting edge is defined as close as possible to the SET. As seen in Fig 2 .15 (A) the etching mask can be perfectly aligned with respect to the previous SET lithography, by means of same gold alignment marks. After the mask is developed, the thermal SiO 2 layer (100 nm)is removed in the regions where the Si will be etched by using a CHF 3 reactive ion etching (Fig. 2.15 (B)). Immediately, the array of SETs is introduced inside the XeF 2 machine. As depicted in Fig. 2.15 (C) it will be as deep as it is large. The etching selectivity of the process leaves intact the SiO 2 and the resist so the etched area will progress underneath both of them.

The XeF 2 is a gas that etches the Si in a series of expansion cycles into a chamber holding the sample. Before starting the Si etching process, some precycles clean the native oxide that may have appeared on the substrate surface after the SiO 2 etching. The strength and speed of the Si etching depends in parameters such as the gas expansion pressure, the time of each cycle and the relative surface to be etched. Finding a good compromise between all these factors took a big part of the thesis. One advantage of our etching mask was that both the resist and the SiO 2 are transparent, hence the position of the etched region could be verified after each cycle by optical microscopy. In Fig 2 .15 (E),(F) the light orange region corresponds to unetched Si, SiO 2 and resist while dark orange corresponds to etched regions. In (E) we see that the etched region has progressed to an intermediate position between the SET tip and the original patterned trench. In (F) the etching process was stopped very close to the SET island. After the etching, the floating SiO 2 and the resist is removed in an ultrasound bath using the lift-off solvent as represented in Fig. 2

.15 (D).

The SEM image in Fig. 2.16 (A) shows the result after the lift off. As it can be observed, a large trench has been etched in between the individual SET probes. The SET ends in a mesa very close to the edge of the walls formed between the trench and the mesa, as seen in the magnified SEM image in ig. 2.16 (B). The trench width is of the order of 30 µm and the mesa is placed around 10 µm above it.

With the XeF 2 etching process, an SET-to-edge distance of the order of 1 µm is routinely achieved for all the probes in the same substrate. As an example in Fig. 2.17 (A) we show a SEM image of an SET only 500 nm away 2.5. Key points of the chapter 69 With this concept in mind, we have combined high resolution e-beam lithography and etching processes to develop a new method of manufacturing an SET very close to the corner of a small Si chip (∼1 µm). The final result is a batch fabrication of 10 to 16 SET probes at a time, on a routine basis, within a period of 2-4 days. Typical tunnel junctions areas are in the order of 50 × 50 nm 2 and the island length in the order of 300 nm. The charging energy is in the order of 3-7 K, which obliges to use the SET at cryogenic temperatures, as it is explained in the next chapter. We will also explain how, thanks to the sharp tip of the substrate, we can use a distance control such as the tuning fork to bring the SET at sub-micrometer distances from sample.

Chapter 3 Environment

Description of a scanning SET microscope

In order to implement a local probe microscope it is necessary to combine several techniques. First, the SET probe should be placed in the immediate vicinity of a sharp tip. The next challenge is to be able to approach the tip at a controlled distance to the sample, and scan it over the sample to form an image. For this purpose, a set of piezoelectric elements purchased from Attocube are integrated into the scanning stage. Two piezoelements are used for the coarse displacement in the X and Y direction. The sample region of interest in the X, Y plane can be selected rapidly. Another pizeoelement allows for coarse motion in the Z direction. In addition, a scanning module provides a fine control in the three X, Y, Z directions of space. The X, Y plane motion is able to cover a maximum field of 30 × 30 µm 2 field and the vertical z-displacement for the tip-sample approach can cover up to 15 µm, with sub-nanometer resolution. This vertical motion require a mean to sense the surface with high resolution to control the tip-to-sample distance. This is critical to prevent crashing the SET into the sample. A distance sensor is incorporated on the tip to control the vertical motion. In some scanning probes, an electrode is placed close to the tip forming an STM tip: For example, on-chip Hall probes [START_REF] Chang | Scanning Hall probe microscopy[END_REF][START_REF] Keyser | Scanning Hall probe microscopy of vortex patterns in superconducting micro-squares[END_REF] often use a tunnel current to control the distance. However, this approach is limited to clean metallic surfaces. Since we plan to study oxide surfaces, we focused on AFM based control using a tuning fork, which was introduced in near field microscopy by K. Karrai in Refs. [START_REF] Karrai | Piezoelectric tip-sample distance control for near field optical microscopes[END_REF][START_REF] Karraï | Piezo-electric tuning fork tipsample distance control for near field optical microscopes[END_REF]. This technology is implemented at the Neel Institute to control on-chip µSQUIDs [START_REF] Veauvy | Scanning µ-superconduction quantum interference device force microscope[END_REF][START_REF] Hasselbach | High resolution magnetic imaging: MicroSQUID Force Microscopy[END_REF][START_REF] Hykel | MicroSQUID Force microscopy in a dilution refrigerator[END_REF]. The new generations of Hall probes microscopes [START_REF] Dede | Scanning Hall Probe Microscopy (SHPM) Using Quartz Crystal AFM Feedback[END_REF] and [START_REF] Shaw | A scanning Hall probe microscope for high resolution, large area, variable height magnetic field imaging[END_REF] also use this method. Similar to the deflection of an AFM cantilever, the resonance frequency of a tuning fork is shifted due to nm range interactions between the sharp tip, glued to one of the tuningfork prongs, and the sample surface. This frequency is the reference for the approach. The set-up required to control the tip-sample distance using the shift in the resonance frequency has been already implemented in our lab for an existing scanning Hall probe microscope. It consists in the read-out of the resonance frequency shift through a phase lock-loop (PLL). Integrating the SET substrate on the tuning fork also presents some technical difficulties and potential mechanical and electrostatic risks and requires a final verification of the SET operation at room temperature once this probe assembly is mounted on the microscope. All these steps are described in this chapter.

The control of the microscope requires a great deal of electronics and an extensive software. Fortunately, much of it is now available commercially. A Nanonis SPM5 controller with its complete software package was purchased and controls the piezo-positioning, the piezo scanner and the vertical positioning using the tunning fork as well as a Ditter piezo for mechanical excitation of the tuning fork, useful for the vertical control. It has also various safety and scanning modes, all exploited in this experiment.

The SET can only be operated at temperatures T ≪ E ch /k B below the charging energy. Therefore, the complete microscope, including the scanning stages are part of a cryogenic system. For a charging energy E ch of 5 K, temperatures below 2 K are adequate. However, lower temperatures open greatly the range of accessible physical problems. With a very compact design, the microscope could be mounted on our Kelvinox 100 (Oxford Instruments) dilution refrigerator which fits inside a high magnetic field superconducting magnet capable of reaching 18 T. The bottom temperature measured on the microscope is of the order of 40 mK.

SETs are very high impedance devices, very sensitive to radiation. A minimal constraint is a vanishing spectral intensity at frequencies above E ch /h. This is not simple to achieve, as the radiation coming from the Helium bath does not fulfill this constraint. Furthermore, all electrical wires and coaxes coming from room temperature can potentially carry sufficiently intense radiation to kill the SET Coulomb blockade. Shielding and filtering are therefore a critical part of the experiment, much easier to achieve in a dilution machine than on a 1 K system.

Our set-up is designed to work in two scanning modes

• an SET current mode, for which the input is the SET current while scanning the sample.

• A feedback mode in which the SET current is kept constant with a PID regulation using the on probe gate voltage as a control parameter.

Scanning stage 73

These different SET measurement modes are integrated into the Nanonis SPM controller and its imaging software.

In this chapter, we describe the physical implementation of the SET probe with the mechanical positioning sensor and the overall integration into the Kelvinox dilution refrigerator, including the specific shielding and filtering required for a proper operation of an SET microscope.

Scanning stage

A scanning stage is used to position the sample relative to the probe and to scan the image. A 3D design in Fig. 3.1 (A) shows the different parts. The piezo positioners 1 (5) controlling the coarse motion in the X and Y directions are rigidly mounted on the bottom assembly flange. They use a stick-slip [116] scheme to incrementally shift the sample in the X and Y directions. A saw-tooth pulse is applied by a high-voltage source which increments the position by a known distance. With a periodic series of pulses, a linear motion takes place. The modules can travel up to 3 × 3 mm 2 across the sample's surface. The tuning fork (6) and the SET ( 7) are placed on this bottom assembly. The vertical z-positioner 2 and the scanning module 3 (2) are mounted on the top part of the assembly where the sample is placed. In Fig. 3.1, a photo of the complete assembly is shown. The assembled SET-substrate on its tuning fork is seen in a zoomed-in photo at the center. As can be seen, large openings are made in the cell holder (Phosphorous brass) to verify that all the elements are correctly placed after bonding the wires to the sample and the SET.

The piezo modules are driven by an Attocube high voltage amplifier unit which is under the global control of the Nanonis TM SPM5 Controller 4 . The bloc diagrams and the detailed description of the experimental control are described in Sec. 3.6.

1 Attocube ANPx51. 2 ANPz101. 3 Attocube ANSxyz100. 4 Nanonis controller from SPECS TM Zurich.

Dilution refrigerator 75

Dilution refrigerator

Coulomb blockade is essential to the operation of the SET used in this work and requires low temperature (below 2 K) for a 5 K charging energy E ch . Furthermore, a sufficient degree of shielding from blackbody radiation is most conveniently achieved in a low temperature environment. For this reason, the scanning stage of the microscope was designed to fit in a Kelvinox 100 dilution refrigerator. This insert fits inside a superconducting magnet which will be useful for most future projects. We briefly explain describe the principle and operation of this particular dilution refrigerator. The schematic diagram of the dilution refrigerator is shown Fig. 3.2. A dilution refrigerator is capable of reaching mK temperature by exploiting the latent heat of dilution of 3 He into 4 He. Hence a mixture circulates continuously and removes the latent heat of dilution in the 3 He -4 He mixture. This cools down the microscope and the experiment. The main elements of the refrigerator fit inside a vacuum chamber which isolates them from the main Helium bath. This internal vacuum chamber (IVC) fits inside a cryostat placed inside a dewar filled with liquid 4 He at 4.2 K. The main elements inside the IVC are: the 1-K pot which is used to condense the 3 He -4 He mixture, a still to evaporate the 3 He from the dilute side of the 3 He -4 He mixture and a mixing chamber where the dilution of 3 He takes place between two distinct 3 He -4 He phases, one with a high concentration of 3 He and the other one with a small concentration. The still and the mixing chamber are connected to each other through a series of heat exchangers where the enthalpy of the concentrated 3 He phase is reduced by the flow of the dilute phase which travels from the mixing chamber and to the still. This particular machine has a continuous heat-exchanger and two discrete heat exchangers.

There are three vacuum circuits in this dilution refrigerator. The first one pumps the IVC, and is used during the initial cooldown of the refrigerator. After cooling down, the IVC is isolated and the active charcoal placed inside maintains the vacuum. The second circuit pumps continuously the 4 He from the 1K pot (inside the IVC) to pin its temperature below 1.2 K. A syphon feeds 4 He from the 4 He bath into the 1K pot through a needle valve. The last and main circuit circulates the 3 He -4 He mixture (in fact mostly 3 He by pumping the still (0.7 K) which contains a dilute mixture of 3 He -4 He. At this temperature, the vapor pressure of 4 He is very low and it is mostly 3 He which is removed from the still. This 3 He gas is re-injected through a condensing line after purification through a liquid nitrogen trap followed by a 4 He trap (integrated to the Oxford insert). This injection pressure should be sufficient to liquify the 3 He on the 1K pot (27 mbar at 1.2 K). This now liquid 3 He is cooled further on the still, in the continuous and discrete heat exchangers. On the last heat exchanger, the temperature is 2.4 times that of the mixing chamber.

Below a tri-critical point at 0.86 K, a 3 He/ 4 He mixture phase separates into a dilute and a concentrated phase. In the mixing chamber, the two phases are present. The dilute phase has ≈ 8% 3 He in the 4 He and is at the bottom while the concentrated phase (mostly 3 He) is on top. The injection of 3 He in the concentrated phase and the removal of 3 He from the dilute side at the still level, maintain an osmotic pressure through the interface between the two phases. This drives the dilution of 3 He from the concentrated side into the dilute side, absorbing a latent heat of dilution. The cooling power depends on the flow of 3 He through the interface, or equivalently to the quantity of 3 He which circulates. This is the reason why, the still is heated most of the time to increase the 3 He vapor pressure and hence the circulation.

This machine is relatively easy to operate. It only requires to have a circulation as high as possible in the initial stages of the cooldown to avoid the trapping of liquid 4 He bubbles inside the heat exchangers which reduce the 3 He flow rate through them and reduce the cooling power.

The cooling power of a dilution refrigerator follows a T 2 law. The maximum cooling power of this machine, at the highest possible circulation rate does not exceed 70 µW at 100 mK. It is enough to have a bottom temperature of the order of 40 mK measured on the mixing chamber. However, all the piezo elements dissipate some energy, to various degrees which energy is released to the dilution refrigerator. The dissipation of positioning piezos in the x, y and z directions is large when they are active but the dissipation stops once the coarse motion is finished. The piezo scanners describes hysteric cycles which are normally very small and thus the dissipation is minimal. The power dissipated depends mainly on the scanning speed, and the scanning field of interest. When scanning is started at the base temperature (around 50 mK), the temperature starts increasing and converges at a maximum value, normally not much higher (usually around 100 mK.) and remains stable for the rest of the scanning.

High magnetic field

The insert of the dilution fridge is designed to fit inside a coil installed in the dewar providing magnetic fields up to 18 T. The coil becomes superconducting at low temperatures and can be used in a persistent mode for static magnetic fields. Sweeping the magnetic field is done relatively slowly to avoiding eddy current dissipation.

Shielding and filtering

SET's are high impedance devices, very sensitive to voltage noise. The voltage noise is largest for high impedance lines and high temperature: the Johnson voltage noise (in root mean square (RMS) is

≪ v ≫= 4Rk B T∆ f (3.1)
where R is the resistance and ∆ f the bandwidth of the measurement set-up. Ar room temperature it becomes

≪ v ≫= 0.13 R∆ f nV (3.2)
i.e. 50 nV for 100 Ω and a 1 kHz bandwidth. This is the reason why the coaxial lines used to bias the SET and polarize its gate are a couple of ohms, thus reducing the Johnson noise by an order of magnitude. At low temperature, the Johnson noise decreases, and filtering which typically requires larger impedance and some capacitance can be used. The main filtering element are thermo-coax one meter long cables, with low thermal conductivities containing MgO powder which absorbs all the high frequency noise. This scheme was first used in ref [START_REF] Zorin | The thermocoax cable as the microwave frequency filter for single electron circuits[END_REF] for single electron devices and filters very efficiently noise above 10 -100 MHz. π-Filters initially used in the distribution box on the top of the cryostat were for the most part discarded as they deteriorated the noise performance of the SET current amplifier. Another critical element is the shielding from the blackbody radiation. In addition to the condition on temperature, the SET cannot be exposed to photons with an energy comparable to E ch (typically of the order of 3

-7 K × k B ) hω ≪ E ch = e 2 2C ∑ (3.3)
Since the IVC is at 4.2 K, it radiates photons with energies far larger than E ch .

As an example to illustrate the importance of the shielding, we can consider Wien's displacement law to calculate the wavelength λ max of the peak of the spectral black-body radiation at 4 K:

λ max ≈ 2900 µm K 4 K ≈ 800 µm (3.4)
which is equivalent to a photon energy of:

E ph k B = hω k B = h 2πc k B λ max ≈ 18 K ≫ e 2 2k B C ∑ (3.5)
Therefore, the SET would never work without the shielding. In addition to a copper shield mounted on the 1K pot, the most critical copper shield is mounted on the "100 mK" plate (between the continuous and discrete heat exchangers). All orifices used to pass cabling and coaxes on the 100 mK plate are also obstructed. Similarly, there are shields inserted inside the pumping lines. Finally the scanning assembly itself acts to some extend as a Faraday cage for long wavelength thermal radiation. With these precautions, very nice Coulomb diamonds could be observed with little rounding near the thresholds.

Another requirement is to minimize the thermal load to the dilution unit. This can be done by choosing materials with low heat conductivity and by implementing an appropriate thermal anchoring to the intermediate temperature stages. As schematically represented in Fig. 3.3, 24 constantan wires and 24 a CuNi clad superconducting NbTi wires were also available for the thermometry, and the high voltage control of the piezo elements. Depending on their actual use, superconducting or constantan wiring was used. For the connections to the piezoelectric elements (requiring a pair of wires for each element) twisted pairs of superconducting wires followed by twisted pairs of copper wires for the connection to the piezo elements were used to minimize the resistance of the connections. This is required to implement a correct stick-slip movement of the positioners. A too high resistance of the piezoelectric connections will increases the time constant of the slip-cycle τ = RC making it too slow and making impossible the coarse movement. A wire resistance per piezoelement lower than 10 Ω is obliged. The resistance of the superconducting wire in our the set-up is R = 2 Ω at low temperatures. Additionally, some care was taken to avoid electric field leakage into the scanning region. electrodes and the drain current is measured keeping the local gate V G also at a fixed value. The amplified SET current (converted to a voltage V ( I SET by the trans-inductance amplifier) is taken as an input for the Nanonis controller while scanning as represented by the dark green line in Fig. 3.7. The Nanonis controller records the SET current map above the sample and displays the measurement in real time.

Feedback mode

We have implemented a second readout of the SET signal that we call on probe gate feedback mode, or simply feedback mode. The objective of this mode is to keep the SET current stable at a certain value I 0 by using the on probe gate voltage V G . This can be done by means of a PID feedback loop included in the HF2LI Zurich Instruments. The interest of this mode will be discussed in detail in the next chapter. Here we explained the general concept and the optimization of the method. This device takes the current I SET as an input and compares it to the reference value I 0 . Using a control parameter (in this case the gate voltage V G ) the error signal I SET -I 0 is swiftly minimize. The on-probe gate voltage V G feedback is very effective because of its strong effect on the SET current as seen in the chapter 2. The performance of the PID has to be optimized to the time response of the system. This is correcting the error faster than the scanning time between two points. In order to do so we have analyzed the transfer function of the experimental set-up with an spectrum analyzer also included in the HF2LI instrument. The system fits to a first order low pass filter model with a typical 20 dB per decade decay slope and a cut-off frequency of 10 Hz, mainly limited by the Femto I/V amplifier. The gain of the SET current with respect to the gate voltage variations was of the order of 10. The HF2LI PID include an algorithm to propose suitable PID parameters with respect to our device under test (DUT) model. The chosen model and the proposed parameters were tested and optimized. In this way, a fast (bandwidths from 20 Hz to 60 Hz) and stable PID loop has been achieved. In the feedback mode the Nanonis controller uses the gate voltage value provided by the PID electronics to keep the SET current constant as the input and records it as 2D map (pink line and image in Fig. 3.7).

Topographic control: AFM with a tuning fork

Readout of the tuning fork response

For other local field microscopes, miniature quartz tuning forks (commercially conceived for digital watches synchronization) proved to be a very effective method to monitor the tip-sample distance control and was adapted to this SET microscope. It is a piezoelectric mechanical resonator which characteristic frequencies depend on the geometry, i.e. the dimensions of the prongs but also on external forces, and damping coming from the environment. This is the property used to transform the device in a very sensitive force sensor. In a piezoelectric material, a mechanical displacement causes a corresponding charge polarization. When a mechanical excitation creates an asymmetric charge separation between the prongs. By an appropriate configuration of two electrodes along both electrodes, the tuning fork is discharged creating a current. To collect such current, in our set-up one electrode is connected to ground and the other to a femto DCLPA-200 trans-impedance amplifier (gain of 10 6 and a low pass filter with a cut-off frequency of 200 kHz) which amplifies the current as seen in Fig. 3.7. In our set-up the tuning fork is excited mechanically by means of one of the piezo X, Y positioners underneath. This is done by means of voltage control oscillator (VCO) included in the Nanonis controller. The tuning fork electrical response amplified by the femto is measured as an input in the Nanonis controller as sketched by the blue lines in Fig. 3.7.

Phase lock loop

When a tip, placed on one of the prongs, is approached close to a surface, it experiences Van der Walls and other surfaces forces that change the resonance condition of the tuning fork. This is similar to what deflects a cantilever in an AFM microscope. In our case, the tip is the edge corner to the SET substrate. By measuring the change in the resonance conditions (frequency, phase, quality factor), we know the proximity of the sample as measured by the SET tip.

The best way to control the height is by using a phase lock loop (PLL). A fast and continuous tracking of the resonance frequency shift when approaching the prong of the tuning fork to the sample or while scanning can be achieved in this way. The main elements of the PLL are shown in the diagram of Fig. 3.7. The tuning fork is initially excited at a reference resonant frequency f 0 through the VCO and the controller demodulates the response at this frequency with a lock-in detection (mixing the input with the reference and a low pass filter (LPF)).

Due to the tip-sample interaction, the tuning fork response may be out of phase with respect to the reference excitation. This phase is corrected by adding a compensation frequency with a feedback loop, f 0 + δ f (t) (by means of the VCO) until the two phases are again equal, which occurs with a final frequency shift ∆ f . The Nanonis controller includes an algorithm specially thought for AFM purposes [START_REF] Gildemeister | Construction of a dilution refrigerator cooled scanning force microscope[END_REF] which allows to optimize the PLL to the specific tuning-fork resonance. The advantage of the PLL, is that ∆ f is proportional to the force sense by the tuning fork, which depends on the distance. Hence, this means that at a certain ∆ f , the tip is placed at a very specific distance from the sample. A topographic study can be done by using ∆ f as the error function of the height with another PID feedback loop. This means keeping a constant ∆ f by changing the height with respect to the tip-sample interaction: the readout is the topography. Yet, the topographic operation with our scanning probe substrate is dangerous since the Si-SiO 2 is rigid. Therefore, there is a big risk of scratching the sample surface and eventually also break the SET with ripped pieces of the sample (see appendix B). For these reasons, the scanning SET measurements are always done at constant tip-to sample distance with the tip some nanom above the sample. In this setup, the PLL is mainly used during the approach procedure (section 3.8.4) and to protect the tip while scanning. The concept for the approach is simple: a small reference ∆ f (typically 1 Hz) is chosen as the contact distance between the tip and the sample. The z scanner moves the tip towards the sample while the PLL measures continuously the frequency shift. When the PLL detects a frequency shift over the reference, the approach stops and the tip is fast retracted some hundreds of nanometers. In addition to that, the PLL was usually left active during the scanning. In this way, if for instance there is an unexpected topographic feature (see appendix B) and the frequency shift overcomes the reference value, the tip is instantaneously retracted to protect the tip from crashing or scratching.

Selecting a resonance frequency

Commercial tuning fork are design with a asymmetric resonant mode with a peak which is exactly 2 15 Hz = 32 768 Hz in commercial conditions (inside a vacuum case). As we will be presented in next section, in order to use it for AFM purposes, the tuning fork is glued on a sample holder and manipulated. Furthermore, the probe's Si-SiO 2 substrate is glued on one of the Chapter 3. Environment prongs. In this whole operation and particularly due to the extra mass added to the prong, the peak of the resonance frequency shifts to a lower value with respect the one in commercial conditions. In fact, for the same reason, new resonant modes are created and typically some amplitude peaks appear in a range from 25 to 30 kHz. In order to implement a PLL it is necessary to find a reference resonance condition f 0 with a large enough quality factor Q.

The Nanonis oscillator is able to sweep the frequency of the mechanical excitation over a large frequency ranges to localize the resonance peaks. The red and blue curves in Fig. 3.8 (A) shows the tuning fork responses in frequency and phase respectively for an excitation of 50 mV in the piezo possitioner x. The tuning fork is inside the scanning stage, with the probe glued on a prong and at 100 mK. Two resonance peaks, one at f 2 = 27.8 kHz and another at f 1 = 28.36 kHz, can be distinguished. From the phase drop at the resonance frequency the Nanonis controller calculates the quality factor. At low temperature and in cryogenic vacuum, we usually have peaks with Q > 1000 which have been proved to be good enough to implement an efficient PLL loop. Once an appropriate frequency peak is found, a more narrow frequency swept is done to find a more accurate value for the quality factor and the resonance frequency as seen in Fig. 3.8 (B) around the frequency f 1 = 28.36 kHz showing a quality factor of Q = 1307. As a last comment, it has to be noted that the resonance frequency is frequently affected by the voltage pulse applied during the coarse x and y movement. The blurred curve in Fig. 3.8 (B) shows the new resonance condition after a coarse movement of some steps in the y direction. The resonance frequency has shifted 2 Hz with respect to the previous value but, what is more remarkable is the fact that the quality factor has suffered a degradation. For this reason, after each coarse displacement, a new calibration of the tuning fork resonance is always carried out.

Control of the displacement and of the scanning plane

After approaching the SET probe to the sample, the next step is scanning the SET probe above the sample. For that purpose, the Nanonis controller provides a number of software driven functions to define the scanning field the scanning speeds, etc. One of these functions is extremely important: the possibility of tilting the scanning SET-probe plane with respect to the sample's surface. As explained in the chapter 2, the substrate of the SET probe has to form a small angle with respect to the sample's surface in order to use the bias U 0 is chosen (typically 1 mV) between the red points. When the needles polarize the source and drain electrode, a drop in the potential occurs due to the extra resistance of the junctions. From this drop in potential, the total resistance R T J can be computed as,

R 300 K = 470 kΩ × U T J U 0 -U T J (3.6)
The SET recipe was optimized to have junctions resistances from 200 kΩ to 1 MΩ. For this reason, SETs with resistances out of this range are discarded. A smaller resistance usually implies either the presence of a single junction or a shortcut. No voltage drop on the contrary means an open circuit between source and drain. In this way, one can store only the SET probes that are more likely to work. Having an stock of SET probes is crucial since it will increase the possibilities of success in the next stages.

Mounting the SET probe on a tuning fork

To adapt a commercial quartz tuning fork to the SET substrate, it is extracted from its housing very carefully to avoid damaging the tuning fork. The tuning fork is then glued to the sample holder as shown in Fig. 3.11 designed to make the SET and tuning fork electrical contacts. To fix one of the prongs to the sample holder, an epoxy glue compatible with cryogenic temperatures is used. The original leads from the tuning fork have magnetic properties which can present problems when using high magnetic fields: they are removed by heating the welding contacts and pulling out with tweezers. They are substituted with Cu wires soldered to the electrical contacts ((1) in Fig. 3.11 (A)). in Fig. 3 Once this is done the on-chip SET can be glued on top of one of the prongs of the tuning fork. The contact needs to be robust to avoid displacements during the bonding step, to guarantee the resonant properties of the device and to avoid tearing if a contact with the sample occurs. These conditions are satisfied with a small quantity (apex of a toothpick) of epoxy glue. In Fig. 3.11 (B) a photo after the micro-bonding of the sample holder can be seen [START_REF] Heger | Combined X-ray and neutron diffraction study of K 2 [Pt(CN) 4 ] X 0.3 .3H 2 O with X = Br, Cl(KCP) between 31 K and room temperature[END_REF] where the S, D and G contacts, the tuning fork and the contacts (2,4), the SET on chip (3), the tip of the microbonding machine (5) and the Aluminum bonding wires (6) which join the SET pads to the respective contacts microscope while making a preliminary approach. In Fig. 3.12 (C) we can see an optical microscope image after the sample approach, where the bonding wires of the SET and the sample can be seen through their reflection on the Si-SiO 2 substrate. We observe the SET chip (2) and the tuning fork (1) at the bottom, and their reflection on the sample (1') and (2') at the top.

One critical parameter is the angle between the probe substrate and the sample. The edge of the probe substrate can only be used as an AFM tip of there is an angle with respect to the sample's surface. However, due to the mesa approach used in the fabrication (terrace around the substrate) and since the SET-probe is not exactly at the edge, such angle has to be small. This will reduce the final SET-to-sample distance. In principle, the plane of the sample SET and sample-holders are designed to be parallel. When we glue the tuning fork to the sample holder we need to pay attention to keep the top surface of the prong also parallel to such plane. This is done by keeping it straight until the Epoxy dries. However, it may still lean a little. In addition the on-chip SET may acquire a certain inclination during the gluing step. If the tilt is in the wrong direction, the approach of the SET would be impossible. Also, the bonding wires may touch the sample's surface if the angle between the SET corner and the sample is too small. To prevent this, the size of the probe chip was chosen long enough to guarantee some space for the bonding wires as represented in Fig. 3.12 (A). With the reflection as reference, it is possible to verified the angle as seen in Fig. 3.12 (B). We have some margin to correct the angle by tilting the bottom scanning stage module by unscrewing slightly one side of the support flange. From this image, using cylindrical projections, one can also estimate the angle (see Fig.A.1 in appendix A). Typical angles are between 3 and 5 deg.

Approach procedure

After the manual adjustment of the SET position above the sample, the z positioner is retracted to some hundreds of µm before placing the scanning stage on the dilution fridge. This is necessary to avoid crashing the sample and the SET probe due to thermal deformations induced by changes in temperature. Once the dilution is cooled down, the tip is approach to the sample. For that we keep the PLL on as described in section 3.7.2, while the z scanner sweeps the sample towards the SET probe all the distance range (15 µm). If no frequency shift is detected, the sample is retracted again to the initial position. Afterwards, the z piezo positioner makes a certain number of steps. The approaching distance achieved with the coarse movement should be shorter than the maximum z scanner distance range. Then the z scanner sweeps again all the distance range, starting from a point closer to the sample. The process repeats until the reference frequency shift is finally detected. Usually around 1500 coarse steps in the z direction are needed to overcome the tip-to-sample distance after the room temperature retraction. Once this is done, the SET probe is ready to scan.

Key points of the chapter

In this chapter, we have presented the experimental set-up of our instrument and how to prepare it for scanning. A compact scanning stage has been built adapted to work in a dilution cryostat at temperatures below 100 mK and with high magnetic fields, up to 18 T. We have wired appropriately the dilution for each of the elements of the microscope (piezoelectrics, tuning fork, SET probe and sample...). Special attention has been paid to the shielding and filtering of the sample and SET lines, specially to protect the SET from high frequency electromagnetic noise sources that can take the SET out of its operating regime, E ch ≫ k b T. As a result, a low-noise set-up has been achieved. At the same time we have implemented two circuits that are used to read the SET signal: The current that circulates through the SET and a feedback signal through a PID loop used to keep the SET current constant. The central element of our set-up is a multipurpose Nanonis controller that reads the scanned signals, controls the movement of the piezoelectrics and pilots the signal of a tuning fork with a PLL to control the scan distance. Setting the instrument into operation is not a trivial task and has been challenging. It involves a lot of manual work taking care of a SET probe which is extremely sensitive to the electrostatic environment. Thanks to having a large stock of SETs we have found a rigorous routine to glue the SET probe over the tuning fork, adjust it over the sample and implement the scanning stage in the cryostat and bring the SET safely to the sample with a high probability of success. In the next chapter, some results of scanning SET measurements are presented, confirming the good performance of our instrument.

Chapter 4

Electric field imaging

Introduction

After installing the SET in the microscope, mounting the microscope on the dilution fridge, the cool-down to cryogenic temperatures done, the instrument is ready for scanning probe microscopy. In a first step it is necessary to characterize the performance of the SET with the Coulomb Diamonds diagrams I(V sd -V g ) presented in Chapter 2. In this way, the main properties of the SET can be extracted: The charging energy, the capacitance of the source and drain junctions C s , C d and the capacitive coupling between the SET island and the on-probe gate C g . The Al SET has been used without magnetic field, hence in the superconducting state. In a first place the transport characteristic for the SET probe are presented. At a certain bias voltage V sd , the SET current varies periodically with respect to the gate voltage, which induces a continuous polarization charge on the SET island. Additionally, charges induced by the SET-sample electrostatic interaction also tunes the SET current. Hence, tracking the SET current variations while scanning above the sample gives information on the spatial distribution electrostatic of the interaction between the SET and the sample. Using this Coulomb diamonds diagram, it is possible to find the most appropriated transport conditions for the scanning. In order to demonstrate our instrument capabilities, we show a number of scans close to a sample that was specially designed to probe well-known electric field distributions. It consists in metallic sub-micrometer large electrostatic gates with different geometries and spacing that can be polarized with different potentials. This permits to estimate the limits in charge sensitivity and in spatial resolution. In principle, between maximum and minimum amplitude of a current oscillation, the response of the SET current behaves almost linearly with respect to external potential variations. In the case large field variations, the SET current completes several periods, one each time an extra charge is induced on the island. This implies a non-linear response.

SET current images are presented showing this kind of behavior. Despite the non-linearity, many aspects of the electrostatic potential distribution above the sample are revealed by a SET current image. In addition, by image analysis, a linear reconstruction of the electrostatic potential variations can be done revealing the local capacitance between the SET and the sample, and the spatial distribution of the background potential.

A novel scanning method has been developed in order to obtain a linear SET response with respect the external field variation. This is the feedback scanning mode. The idea is to compensate the new charges induced on the island while scanning the sample with the on probe gate voltage. This is done in real time while scanning by means of a PID feedback regulation on the gate voltage. The feedback images presented contain very valuable information of field distribution coming from the sample. The measured profiles have been compared to finite elements simulations confirming that the feedback mode yields a linear measurement of the electrostatic field distribution.

In the last part of the chapter, an unexplained effect observed on a large number of images is discussed: The SET current oscillations lose amplitude close to metallic regions of the sample and at short SET-to-sample distances. It is a "quenching of the Coulomb blockade".

SET characterisation

In this section, the experimental transport curves of the SET are presented. The measurements are carried at T = 100 mK and B = 0 T in scanning conditions before the tip-sample approach. Under these conditions, the Al is in the superconducting state: we have an S-SET.

As described in chapter 2, measuring the SET current I SET as a function of the bias voltage for different on probe gate voltages results in the SET transport Coulomb diamonds diagrams shown in Fig. 4.1 (A). In the S-SET, a single electron current only flows at bias energies |eV sd | > 4∆. For lower bias, in white, we find the superconducting gap region 8∆ e , delimited by the two dashed yellow lines. In this region, no single quasi-electron current is allowed. Above this bias limit, two conducting regions appear, in blue for negative and in red for positive current values. In this regime, quasi-electrons are able to tunnel from source to drain as a single charge tunneling process. The diagram shows the main characteristic of the SET: the current is governed by diamonds. From this cuts, we identify the charging energy E ch and the superconducting gap ∆. A criteria to set the frontier between the blocked and conducting regimes in superconducting junctions is to consider the first inflection point in the I SET (V sd ) curve [START_REF] Douglass | Energy Gap Measurements by Tunneling Between Superconducting Films. I. Temperature Dependence[END_REF][START_REF] Meservey | Energy Gap Measurements by Tunneling between Superconducting Films. II. Magnetic Field Dependence[END_REF]. This is equivalent to the first max- imum in the G SET (V sd ) curve. Selecting these maxima for all the gate voltage range (green dots) the contour of the Coulomb diamonds are reconstructed as seen in Fig. 4.1. The dashed yellow (resp. black) vertical lines of these cuts set the bottom (resp. top) of the Coulomb diamonds for positive and negative bias. The difference in bias between the yellow and black dashed lines correspond to the height of the Coulomb diamonds: this is the addition energy of e × e C ∑ = e × (600 ± 50) µeV. With this, it is possible to calculate the total capacitance of the island, C ∑ = (270 ± 20) aF and the charging energy,

E ch = e 2 2C ∑ = (300 ± 30) µeV. In units of temperatures, it is E ch k B = (3.3 ± 0.
3) K which confirms that the regime E ch ≫ k B T is well achieved. The difference in bias between the two yellow vertical lines corresponds to 8∆, where the Al superconducting gap ∆ = (230 ± 10) µeV. The order of magnitude of this value is in good agreement with the literature for thin granular Al films at 100 mK [START_REF] Douglass | Energy Gap Measurements by Tunneling Between Superconducting Films. I. Temperature Dependence[END_REF][START_REF] Meservey | Energy Gap Measurements by Tunneling between Superconducting Films. II. Magnetic Field Dependence[END_REF][START_REF] Walmsley | Superconductivity of very thin aluminium films[END_REF][START_REF] Court | Energy gap measurement of nanostructured aluminium thin films for single Cooper-pair devices[END_REF]. The source and drain capacitances are extracted from the negative and positive slopes of the Coulomb diamonds contours (see Chapter 2), giving a value of C s = (90 ± 20) aF an C d = (170 ± 10) aF respectively. This indicates asymmetric junctions. The total capacitance using these calculations gives C ∑ = C s + C d + C g = (263 ± 25) aF. As one would expect, it is in good agreement with the value extracted from the addition energy. The green dotted curve in Fig 4 .2 (A) corresponds to an I SET (V sd ) measurement at room temperature. The slope of such curve is the room temperature conductance G T=300 K = 0.029G 0 , marked with the same linestyle in Fig 4 .2 (B). This means a room temperature resistance of R t=300 K = 446.8 kΩ, which respects the requirement for the orthodox theory for Coulomb blockade, R t ≫ 22.4 kΩ. From the Ambegaokar-Baratoff relation [START_REF] Ambegaokar | Tunneling Between Superconductors[END_REF] (see eq. 2.27 in chapter 2) the Josephson energy is also estimated as E J = 1.6 µeV. This confirms a regime for the S-SET, E J ≪ ∆ ∼ E ch , where sub-gap charging features are very low in current and are not observed in the Coulomb Diamonds.

The characteristic Coulomb oscillations in I SET (V g ) are seen in figure 4.3 for a bias voltage of 0.8 mV in yellow, 1.0 mV in magenta, and 1.45 mV in black. We note the same periodicity for the three curves, with a period, p g = (50 ± 1) mV. This period corresponds to a change from N to N ± 1 in the number of electrons on the island. The extra electrons are induced successively through the on-probe gate capacitance C g which is determined from this period as C g = e p = (3.1 ± 0.2) aF. This means that C s , C d , ≫ C g , so C g can be neglected from the total capacitance.

At the bottom of the Coulomb diamond (yellow line), quasi-electrons can tunnel from source to drain in a small range of gate voltage around the degeneracy point (see chapter 2) where a current peak appears. The peak has a certain width due to thermal smearing of the density of states. When the bias supplies an energy equal or larger than the addition energy, (black line) tunneling takes place for any on-probe gate voltage (no Coulomb blockade). The current shows a maximum at the degeneracy point µ s = µ N+1 and µ N = µ d where two charging states are available for tunneling. The largest current I SET (V g ) amplitude occurs at 1.0 mV (magenta curve). This implies a higher current change for the same on-probe gate voltage variation and hence a higher sensitivity.

Maximum charge sensitivity

The maximum charge sensitivity occurs at the maximum slope of the magenta curve in Fig. 4.3. At this operating point in the oscillations, a small change in the external voltage induces the largest variation in the SET current. This can be expressed in terms of the inverse of the slope at such point as s = 0.0125 V nA -1 (The maximum slope of the curve is of the order of 80 nA V -1 ). When rescaled by the period, it can be expressed in units of number of electrons as s = 0.0125 V nA -1 0.05 V e -1 = 0.25 e nA -1 . A intrinsic noise exists in the experimental set up including the current transconductance amplifier. In DC measurements (cut off bandwidth of 10 Hz), the noise level is of the order of 1.6 pA/ √ Hz, with a strong low frequency noise contribution. The maximum sensitivity of this SET is then:

S max = 0.25 e nA -1 × 1.6 pA/ √ Hz = 4 × 10 -4 e/ √ Hz. (4.1)
This value is of the order of magnitude of those reported in literature (see Chapter 2) for DC measurements of the SET current.

As it has been proposed in [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF], the SET current variations can be written as a periodic function of the total charge induced on the island, q SET with a period of e. In Fig. 4.3, the dashed magenta curve corresponds to a fit to a sinusoidal curve of the SET current oscillations with the gate. In spite of slight asymmetries, the results shows that it is a valid in a first approximation Chapter 4. Electric field imaging and it can be written:

I SET (V g ) = A sin 2πq SET e = A sin 2π(q g + q 0 ) e = A sin 2π(C g V g + q 0 ) e (4.2)
Where q SET includes the charge induced through the gate q g = C g V g , and an extra fixed charge q 0 which account for the bias C s V sd contribution and any background charge q bg as well as a capacitive coupling to the environment q env = C env V env . As seen in chapter 2, variations in any of these terms tunes the electrochemical potential by changing the total charge on the SET island with a straightforward effect on SET current.

µ N = e 2 C ∑ N - 1 2 + e C s V sd C ∑ + e C g V g C ∑ + e C env V env C ∑ + e q bg C ∑ = e C ∑ q SET (4.3)
In the following, capital letter Q for instances Q SET = q SET e measure charges in units of e. The SET as a electrostatic sensor is a readout of the changes in q SET , by measuring the SET current variations I SET (q SET ) when scanning.

Sample

A test sample consisting of submicrometer large Al electrostatic gates was designed in order to demonstrate the operation of the instrument. The gates were designed by e-beam lithography and e-gun evaporated (50 nm) on a Si-SiO 2 substrate. The main idea of the sample design is to produce, wellknown electrostatic potentials emanating from specific electrodes geometries. For that, they are arranged in a way such that two consecutive gates can be polarized with a different potential V 1 or V 2 . This is sketched in the optical image seen in Fig. 4.4 (A) of a region of ∼ 100 × 70 µm 2 of the sample. Electrodes with different geometries where distributed in 30 × 30 µm 2 regions, the maximum scanning range of the microscope. In Fig. 4.4, two of these configurations are magnified with an SEM image. One has an hexagonal geometry (B) and in the other the gates are in an interdigitated disposition in the longitudinal and the transverse directions (C). As seen in (A), these zones are repeated several times and arranged in an array. This is done all around a sample's area of 2 mm 2 to ease the positioning of the SET above one of the gate arrangements. The different shapes, sizes and separations help to find the location of the SET above the sample after the approach and are used to test the resolution. In the following sections, different scanning SET images are presented above the two black square regions depicted in (B) and (C).

Reconstruction of the SET current images

From the SET current images, it is also possible to extract the electrostatic profile. The red curve in Fig. 4.6 (A) shows the SET current variations along the scan line sketched also in red in Fig. 4.5 (A) (starting from the bottom). The curve reveals several current oscillations with a similar period, meaning a nearly constant electric field gradient. However, not all the oscillations are complete, as it is the case at the positions marked with the vertical dotted lines. The colored circles give a reference for their location on the SET current map of Fig. 4.5 (A)). An explanation for this feature is sketched in Fig. 4.5 (B). The bottom curve represents the profile for an effective potential produced by the sample with a piecewise linear behavior as a function of the position (y). A larger slope implies a shorter period. Along a scanning line, one does not know whether the effective potential increases or decreases, since only periodic oscillations are measured. However, a change in slope does leave a signature in the current behavior (positions ( 2) and ( 3)). This would be equivalent to retracing the SET current oscillations as shown on the top right graph in Fig. 4.5 (B). In a continuous measurement along a scanning line, the result has to be a symmetric function (blue) of the initial curve (black) (2) 2 . This results in an incomplete oscillations as those highlighted in Fig. 4.5 (A) 3 . The magnitude of the potential variation can be estimated by counting the number of full SET current oscillations. For each oscillation of the SET current, an extra polarization charge of e is induced by the electrostatic field at the SET position. These correspondences between charge oscillation and polarization field allow to reconstruct the electrostatic profile sensed by the SET above the sample, as the one shown at the bottom of Fig. 4.5 (A). In this profile, the initial sign of the slopes is random, and the symmetric profile with respect the y axis would also work. This profile seems acceptable taking into account the symmetric distribution of the electrodes along the y axis. It shows very precisely a slope reversal in between the electrodes (white and cyan circles) and exactly on top of the electrodes (magenta, dark and light green circles). The current remains constant on top of the grounded electrodes: this is consistent with no changes in the effective potential over this region. with yellow arrows) along the (x axis) has changed from three in (B) to five in (A). This shows an increase of the the electric field gradient along this direction. The way the slope is added to SET-to-sample distance is sketched in Fig. 4.8 (D).

In Fig. 4.8 (C) the effect of a tilting x = 1°angle with respect to a fixed axis in the x direction is shown. The number of black contours has increased along the yaxis above the large left electrode from one in (B) to three in (C). This correspond to the additional slope in the scanning height along x axis shown on Fig. 4.8 (E).

It was impossible remove all the SET current oscillation even for large tilting angles 4 . The minimum number of contour lines was realized for the image shown in Fig. 4.8 (B), meaning the flattest scanning plane with respect to the sample plane. From these images, we learned that, no matter the scanning plane, the large number of oscillations on the right part of the image persist. This confirms that a large intrinsic electric field is produced by the sample in this region. Since all the electrodes are grounded, the field has to come from charges present in the insulating region of the substrate.

Effect of the distance

In Fig. 4.9 (A) to (F) several SET current images were taken above the 8 µm 2 sample's region shown on the SEM image, panel (G). The tip-to-sample distances from (A) to (F) are 200,300,400,500,600 and 700 nm. This region was selected because it includes the thinnest electrostatic gates, most appropriate to check the spatial resolution and performance. The electrostatic gates are all grounded during the measurements. At 700 nm (F), a black equi-current contour, i.e. a minimum in the SET current (pointed by a yellow arrow), surrounds a white equi-current region, at the maximum of the current range. The shape of this contour fits the electrodes geometry (marked with red dashed lines). Smooth current variations persists in between the red dashed lines as we scan closer and closer to the sample and match the position of the electrodes. This feature shrinks when the SET is closer to the sample, showing the improvement in spatial resolution. Also, the geometry of the sample appears better defined the closer we are to the sample. In fact, at a 200 nm distance (A), in the bottom left corner (green rectangle), the contours become square shaped matching the change in shape from transverse to longitudinal of the aluminum electrodes. This feature becomes rounded when scanning is so large that it is difficult to count them all. The evolution of the number of contours and their geometries confirms that the electric field originates from the sample. The presence of large gradients between the metal and the insulating regions were also visible in Fig. 4.5: the images are consistent with a charged area over the insulating regions.

Not only does the number of contours increases as the SET gets closer to the sample, but the shapes of the contours become also increasingly complex. For instance, the region surrounded by the dotted magenta line splits into smaller equipotential contours from F to A. In fact at 200 nm from the sample, small circular contours, which were not visible at a larger distance from the sample, are very clearly resolved. They may arise from inhomogeneities in the sources of the electric fields (for example a charged area with resist residues or small particles of aluminum...). This observation shows explicitly how the lateral spatial resolution improves as the distance from SET to sample is reduced. The size of this structures (∼ 350 nm) is an upper limit for the spatial resolution. This is of the order of the SET-to-sample distance and the SET-island size (300 nm). The high sensitivity of the SET in charge detection can be verified as smooth fields variations are still observed far from the sample, as well as the presence of small artifacts close the sample and the many oscillations observed around the grounded electrodes.

Reconstruction of the SET current images using the electrostatic gates

In [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF] a convenient method was proposed to reconstruct the electrostatic environment from the SET current images. As it has been previously presented, the current variations with respect to the gate voltage, I SET (V g ) can be modelled by a sine function. If the SET is at a fixed position (x, y) over of the sample, the same behavior is expected with respect to changes in the potential V 1 or V 2 coming from the sample gates which are coupled to the SET through their local capacitance C 1 (x, y) and C 2 (x, y),

I SET (x, y) = A sin 2πq SET (x, y) e = A sin 2π(q 1,2 (x, y) + q 0 (x, y)) e = = A sin 2π(C 1,2 (x, y)V 1,2 + q 0 (x, y)) e , (4.4)
where A is the amplitude of the oscillations. There are two contributions to the total charge q SET (x, y). (1) The polarization induced by the electrostatic gates q 1,2 (x, y) = C 1,2 (x, y)V 1,2 and (2) the initial background charge 4.5. Feedback scanning mode 123 large field variations, this induces non-linear periodic current oscillations in I SET (q SET ). At the same time, the SET current oscillations can be controlled by sweeping the on-probe gate voltage, q SET = q SET (q g ) as shown in Fig. 4.3.

The idea of the feedback mode is to compensate any new polarization charge on the island induced by the sample by an equal and opposite charge with the on-probe gate voltage. As seen in Fig. 4.13 (A), this means q SET is kept constant, effectively keeping the SET current constant along a scanning line. This is performed continuously in real time by a feedback PID loop (presented in chapter 3).

To do so, a SET current amplitude I 0SET (V sd ) is chosen as the reference for the feedback loop. It is the working point of the SET (blue-black point in Fig. 4.13 (B)). When the scanning starts, changes in the electrostatic environment induces a variation of the SET current to a new value I SET (t). The PID electronics compares the new I SET with the working point I 0SET (V sd ) and defines the error(t) = [I 0SET -I SET (t)]). With optimal PID parameters, the electronic device is able to correct the error by acting on the on-probe gate voltage ∆V g , the feedback control parameter. This should settle before the SET moves to its next position, i.e. the PID should be faster than the displacement speed. In this feedback mode, the output is the gate voltage ∆V g which maintains the SET current at its working point.

As sketched in Fig. 4.13 (B) an optimal working point is in the middle of a SET current oscillation (blue-black circle) where I SET (V g ) is quasi-linear and the slope

dI(I 0SET ) dV G
is the largest. This ensures the SET response will be linear with respect to the electric field which causes the SET current deviation. With a large slope at the working point, a smaller ∆V g is needed to compensate for the same error(t). This also provides a large gain for the PID loop, which implies a smaller error and a faster response. The SET bias V sd is chosen the get the broadest I SET (V g ) curves where compensation for larger SET current deviations from the working point is possible. The magenta curve in Fig. 4.3 corresponds to an optimal bias V sd . In the following, the bias is fixed to V sd = 1 mV.

With this method, the charge induced by the on-probe gate is calibrated since the capacitance C g to the island is measured from the gate charge period in ∆V g . One period is p = 50 mV and means one extra electron on the island. Hence, the total number of electrons induced is The insulating regions in between the electrodes appear in grey scale. A big compensation charge is needed on the right part of the image, which appears in a white scale. This results is in good agreement the large field gradient between the left and right parts of the image that was predicted from the big number of SET current oscillations visible in Fig. 4.5 (A). The image can be interpreted easily with a 3D map of the data that is included in Fig. 4.14 (B). In this graph, it can be observed a contrast between the regions with electrodes (light green and magenta circles) with respect to the regions without electrodes (cyan circle). In fact, the charge regime is substantially bigger above the insulating region at the right of the image electrodes and particularly low on top of the large electrode at the left part. As was inferred with the SET current measurements, this reveals a large electric field coming mainly from the insulating region of the sample. In Fig. 4.14 (C) a polarization V 2 = -100 mV is applied. It can be noted that an increase of compensation charge on top of the polarized electrode (light green circle). However, the variations in the image due to the electrodes polarization is difficult to recognize due to the presence of the large electric field gradients already detected without polarization.

Q g (t) = V g (t) period = V g (t)C g e ( 4

Origin of the background electric field

The presence of very large electric field coming from the insulating regions is surprising. These regions correspond to oxidized Silicon. In commercial Si-SiO 2 wafers, used as a substrate, a large amount of charges are trapped below the SiO 2 layer. The origin and the density of such charges was a hot topic in the sixties and seventies [START_REF] Atalla | Stabilization of Silicon Surfaces by Thermally Grown Oxides[END_REF][START_REF] Breed | Stable and unstable surface state charge in thermally oxidized silicon[END_REF]. The conclusions of these studies are summarized in ref [START_REF] Deal | Standardized Terminology for Oxide Charges Associated with Thermally Oxidized Silicon[END_REF]. These charges accumulate mainly at the Si-SiO 2 interface as sketched in Fig. 4.15 (A). Four different types of charges have been identified, usually expressed in net number of elementary charges N per unit area (number/cm 2 ): Positive fixed oxide charge (N f ) coming from structural defects in the oxide layer very close to the Si-SiO 2 interface. Mobile ionic charges (N m ) coming from ionic impurities. Oxide trapped charge (N ot positive or negative) due to holes or electrons trapped inside the bulk of the oxide. And interface trapped charges, (N it positive or negatives), located exactly at the Si-SiO 2 surface arising from structural or oxidation-induced defects, metal impurities or other defects due to bond breaking processes. difference in electrostatic field at the interface between the metallic and insulating regions which creates the contrast in the response. The position of the electrodes in both scanning SET images and in feedback mode become quite visible even though no voltage is applied on them.

Confirming the linear response of the SET

The compensation charge plot (Fig. 4.16) clearly shows the contrast in screening. This plot is along the cut (red line) in Fig. 4.14 (A), from bottom to top. The grey regions mark the position of the Al electrodes. The continuous red curve is extracted from the feedback image taken with both sample electrode grounded. The green curve is the reconstructed background charge q 0 (x, y) for an equivalent scanning line previously shown in Fig. 4.12 (A). The profile looks similar (except for the sign). The reconstruction from the SET current image and the image in feedback mode are equivalent. With the on-probe gate polarization the SET current is

I SET (x, y) = A sin 2π(C 1,2 (x, y)V 1,2 + q 0 (x, y) + q g ) e . ( 4.7)
When V 1,2 = 0, the condition q 0 (x, y) = -q g keeps the SET current constant. This explains the opposite sign between green and red curves in Fig. 4.14 (A). In Fig 4 .16 (A) The compensation charge above the polarized electrodes (light and dark green circles) is larger when V 2 = -100 mV (dotted red curve) compared to the curve for grounded electrode (continuous red curve). On the other hand, it is lower for V 2 = 100 mV (dotted red curve). A negative polarization enhances the background charges and the positive polarization decreases it. The background charges produce a negative potential which far exceeds than the effect of the gate polarization. This confirms the large density of charges present below the electrostatic gates. By contrast, no significant variations due to polarization are detected above the central electrode (magenta circle), which is always kept grounded.

In Fig. 4.16 (B) the compensation charge variations with respect to V 2 are presented for the scanning positions depicted with the light and dark green circles (above the polarized electrodes), magenta (above the central electrode) and cyan (above a position in between the electrodes). The curves are linear, with a negative slope at all the selected points (the color of the curves correspond to the color of the circles at each position). The linear behavior confirms that the feedback is compensating the linear term C 2 V 2 in eq. 4.5.3. Using a linear fit, the variations in Q g (x, y) = C 2 (x,y) e V 2 + Q 0 (x, y) can be polarized to the corresponding values in the abscissa axis. Comparing these curves with those in Fig. 4.16 (B) it can be noted that this time they meet close to the origin of coordinates. This confirms that the background has been removed. This result corroborates that the SET in feedback mode performs a linear read-out of the electrostatic features present in the sample. In addition to that, the fact that this analysis can be carried out shows very clearly the great charge stability during the scans.

Comparing the feedback images with simulations

Background charges

Figs.4.18 (A) and (B) are two feedback images taken over the 8 × 8 µm 2 region shown in Fig. 4.9 (G), with both electrodes grounded and a SET-sample distance of z = 1200 nm and 300 nm respectively. The electrodes can be seen in the black contrast of the images. The spatial resolution is significantly higher close to the sample and the 300 nm electrodes are very well resolved. In Fig. 4.9 (C) the dotted blue and red curve show the compensation charge profile along the scanning line shown in blue and red in Figs. [START_REF] Shenoy | Electronic Phase Separation in Correlated Oxides: The Phenomenon, Its Present Status and Future Prospects[END_REF].18 (A) and (B) respectively. At 300 nm (red curve), a dip in the feedback response occurs in between the electrodes. The profile just shows a maximum between the electrodes when scanning further away from the sample (1200 nm, blue curve). The simplicity of the geometry in this area makes it ideal to compare the measurements with a simulation of the electrostatic environment with a background layer of charges underneath the sample.

A finite elements simulator is used to calculate the potential variations over the sample. We specify the interdigitated electrode configuration according to the image Fig. 4.4 (C) (see appendix D). The electrodes placed on top a 100 nm dielectric layer are set to ground. The permittivity of SiO 2 is ǫ r = 3.9. On the bottom surface of the SiO 2 layer, a surface charge density σ = n = 5 -10 11 e cm -2 is introduced, the value quoted in reference [START_REF] Vitkavage | An investigation of Si-SiO <sub>2</sub> interface charges in thermally oxidized (100), (110), (111), and (511) silicon[END_REF]. The potential is defined with respect to a ground plane placed some µm below the sample. With these boundary conditions, the calculated potential profile looked very similar to the experimental results for a broad range of surface charge σ. In Fig. 4.18 (C) the dashed curves show the simulated potential for a σ = 1.67 × 10 -4 C m -2 (equivalent to n = 10 11 e cm -2 ) along the experimental scanning line. It was observed that the simulated potential profile for this charge density fits to the experimental curve at 1200 nm and was therefore taken as a reference. At 300 nm the simulated profile also follows the experimental curve relatively well except for the relative minima observed in between the electrodes. The origin of this minima is not clear. Note that the relative scales between the curves far and close to the sample in both experiments and simulations are comparable. From left to right, the same relative increase of the maxima is observed in the experiment and the simulations. This indicates a weaker screening of the electric field in the right part of the image. Looking to the sample image in Fig. 4.4 (C), we see that the largest electrodes are placed on the left and make a larger contribution to the screening. This effect is most obvious in the simulated potential through all the sample sections and illustrated in appendix D. This gives confidence that the model used is a good approximation for electrostatic field produced by the sample. It also confirms that the large SET response observed comes mainly from the background charges trapped at the Si-SiO 2 interface. In a first approximation, the variations of the charge induced on the SET q 0 by the background charges can be expressed as,

δQ g ∝ δQ 0 (x, y) = C 0 δV 0 (x, y) e , (4.8)
where C 0 accounts for a fixed capacitance between the SET and the plane of background charges while V 0 is the effective potential created by the surface charges at each position of the SET. Such potential obviously changes above and in between the electrodes. As a last comment, in the experimental profile measured very close to the sample (dotted red curve in Fig. 4.18 (C)), differs significantly from the simulated profile above the electrodes. In fact, the feedback loop has a low sensitivity in this region for a reason which is analysed in section 4.5.7.

Polarized electrodes

The feedback mode images in Fig. 4.19 (A) and (B) are obtained by subtracting an image taken when V 1 , V 2 = 0 from another image in which V 2 = 0 and V 1 = 100 mV, at z = 1200 nm and 300 nm respectively. The polarized electrode V 1 is the central one in Fig. 4.18 and appears in white contrast in the images. The blue and dotted curves in Fig. 4.19 (C) show the compensation charge profile along the scanning line drawn in blue and red in Figs. 4.19 (A) and (B) respectively. After the subtraction operation, the background charge is removed and the relative charge compensation maxima between the electrodes have disappeared.

The blue and red dashed curves in Fig. 4.19 (C) are the simulated potential along lines matching the experimental scans, at z = 1200 nm and 300 nm respectively. The simulation takes the central electrode at 100 mV while the other two are grounded. The background charge is now set to zero. The experimental and simulated curves are quite similar. Far from the sample (blue curves) the curves are parallel. Close to the sample (red curves) similar trends are seen in the experiment and the simulation, with an inflection point in the middle between the electrodes. The relative magnitude in potential variation seen in the experiment and the simulation match relatively well.

As explained in the previous sections, when V 2 is applied to the electrode 2, the SET measures the changes in the local capacitance C 2 (x, y),

δQ G = δC 2 (x, y)V 2 .
(4.9)

The variation in the C 2 (x, y) magnitude is proportional to the potential variations above the sample without the presence of the SET. This confirms that the SET in feedback mode makes a direct measurement of the electric potential without much distortion up to geometrical factors.

Charge sensitivity in feedback mode

With the addition and subtraction of images, it is possible to recreate an electrostatic image where no electric field comes from the SiO 2 layer. That is, a neutral sample with no polarization on the electrodes and an Si-SiO 2 interface charge set to zero. In this situation, the feedback compensation charges should stay constant. Any deviation may be attributed to the floor noise in the charge detection in feedback mode.

The procedure to build such zero charge image is shown in Fig. 4.20 (A). By adding a feedback image (a) where the electrode polarization is V 2 = 100 mV and V 1 = 0 mV with an image (b) in which the polarization is V 2 = -100 mV and V 1 = 0 mV, the resulting image (c) is equivalent to a polarization V 2 = +100 -100 mV = 0 mV and V 1 = 0 mV. Since two images are added, then the effect of the background charges is also duplicated. For that reason, it is necessary to subtract from (c) twice the image with zero polarization V 2 , V 1 = mV. A final zero charge level image (d) can be obtained.

The previous analysis was performed by using feedback scans above the interdigitated gates (Fig. 4.18) at a SET-to-sample distance of z = 1200 nm. The histogram presented in Fig. 4.20 (B) shows the distribution in number of pixels of the image, in a small range of compensation charge (in number How this translate in SET current is shown in Fig. 4.23 (A). The bias is fixed at V sd = 1.3 mV, and the SET current is measured as a function of gate voltage for tip-to-sample distances of 900 nm (black), 150 nm (green) and 50 nm (blue) respectively. The value bias is shown as the white horizontal line in Fig. 4.22 (B),(C) and (D). The smaller the SET-to-sample distance is, the smaller the oscillation amplitude in the SET current. The oscillation occurs always around the same mean value as seen in Fig. 4.21. In Fig. ?? (A), the period of the oscillations decreases at shorter SET-to-sample distance. The period is inversely proportional to the capacitive coupling between the SET and the sample, C sample . In the standard electrostatic model of the SET (Chapter 2), the charging energy is C g (900 nm) ≈ 4 from z = +900 nm to z = +50 nm. Using Eq. 4.12: this change would account for a 10% change in charging energy between 900 and 50 nm. From the Coulomb diamonds, it is obvious that the changes in charging energy are considerably larger. For SET-to-sample distances d smaller than the island size, a parallel plate capacitor model may be appropriate C sample ∝ ǫ r A d where A is the island area. This matches the last points on the curve. Extrapolating with a linear fit at closer distances7 (dashed green line in Fig. 4.23 (B)), we see that C sample never exceed 20% of C s + C d . Hence, the quenching of Coulomb blockade cannot be explained within the SET standard model. Another tunneling mechanism, which is insensitive to the SET electrostatic energies, is present. Cotunneling seem the most obvious candidate. The tunneling rate seems to be reinforced by the proximity of the surface.

E ch = e 2 2C ∑ = e 2 2(C s + C d + C g + C sample ) , ( 4
Thermal effects can be ruled out, since the SET is always superconducting and the charging energy extracted from the Coulomb diamonds in Fig. 4.22 (B) is E ch = (7.4 ± 0.6) K ≫ k B T. While it is difficult to completely rule out photon-assisted processes, all electric lines are thoroughly filtered and at low temperature, the SET is only exposed to the 50 mK blackbody radiation.

The consequence for SET microscopy is clear. When the SET current oscillations die out, the SET can no longer be used as an electric field sensor, at least as done in this chapter. The PID loop has difficulties, because of the reduced effect of the gate electrode on the SET current. If the PID set point is I 0 , the average SET current, no gate voltage is needed to compensate, but no electrostatic information can be extracted. When imaging, we noticed that the quenching is maximum not exactly on top of the electrodes but in the region where the largest electric field gradients occur as shown in Fig. 4.21 (A), even at relatively large distances far from the sample. This may be a clue to understand the mechanism.

Similar observations of the Coulomb diamonds quenching have been reported in [START_REF] Huefner | Scanning gate microscopy measurements on a superconducting single-electron transistor[END_REF], where a scanning a gate microscopy was used to image an SET. A scanning gate produces relatively large electric fields and gradients over the SET, and this may be analogous to the background charges at the Si-SiO 2 interface in our test sample. In any event, this is a novel effect not previously considered in the SET and Coulomb blockade literature.

Key points of the chapter

In this chapter, the SET current I(V sd , V g ) in the scanning conditions were characterized at low temperature. The Coulomb diamonds are very stable with a charging energy 50 times larger than the temperature. Its optimal charge sensitivity is S max ≈ 4 × 10 -4 e/ √ Hz. The SET periodicity and the Coulomb diamonds allow a full characterization of all the capacitances. Next, electric images were presented in the SET current mode. In this mode, gates and bias are fixed and the SET current is plotted over the test sample. It consists in submicrometer Al electrostatic gates deposited on top of a Si-SiO 2 substrate. With the large electric field coming from the Si-SiO 2 interface, many SET oscillations are observed on the images. It is nevertheless possible to image the background charges and the local capacitances by recording several current images as a function of electrode voltages. They reveal a high lateral spatial resolution (300 nm or better) and excellent maps of the local capacitances.

Next, a novel way to linearize the SET response, the feedback mode, was demonstrated. It consists in compensating the charges induced on the island by means of an on-probe gate voltage. A PID feedback loop controls the gate voltage and keep the SET current constant at a reference value I 0 . The feedback is sufficiently fast to be used in real time while scanning. Direct imaging the electric field above the sample the sample have been shown and compared to finite elements simulations. Even in the presence of large electric fields, the feedback mode has proved to be so stable, that addition and subtraction of images could be dome quite reliably. This has enabled the isolate the effect of the background Si-SiO 2 interface charges, from the electric field generated by the sample electrodes. In the feedback mode, we estimated a sensitivity of the order of δQ n = ±0.009 e. It becomes possible to detect very small polarizations 4.5 mV even at large (1200 nm) distances. At present the lateral spatial resolution is in the order of the island size 300 nm.

At very short distances from metallic surface, a quenching of the Coulomb blockade was observed. This reduces the SET sensitivity as the SET current amplitude decreases accordingly. This effect cannot be explained within the standard model for the Coulomb blockade. Additional research will be carried to precisely identify the origin of the effect.

Chapter 5

Conclusion and perspectives

In this manuscript we have presented a new scanning SET microscope. The scanning stage is mounted on a dilution refrigerator. With this instrument, electric imaging can be done at temperatures below 100 mK and magnetic fields up to 18 T. The results presented are quite encouraging and open up many new research avenues. Compared to Yacoby's group and Weizmann Institute's microscopes, it uses a standard on-chip fabrication technique. The method used to fabricate the SET at the extreme corner of a silicon chip is a new and original process developed by the group. An on-chip gate adds a considerable flexibility to the instrument. With such gate, a new feedback mode, described in chapter 4, allows a linear imaging of surface electric field. The instrument has a high charge sensitivity, Q n = ±0.009 e, even at large SET-sample distances, and very good lateral resolution of the order of the SET island size (300 nm). In addition, a strange phenomenon was found which origin still has to be revealed in the future: the quenching of the SET current oscillations. It occurred while scanning very close to the sample and in regions close to the metallic electrodes.

The novelty of our instrument, comes from the fabrication process for SET probes. It differs greatly from the previous successful and unsuccessful attempts to SET microscopy and fulfils our main goals: making an accessible instrument. The main challenge in making a nanometric SET close to a sharp tip is overcome by a lithographic technique and an etching process inspired in part by the technology used in some µSquids and Hall probe microscopes. This technology is based on the used of the edge of a substrate as in atomic force microscopy (AFM) sensors for the distance control. With a batch fabrication processing, a large number of probes could be made in a short period of time. They can be stored, shared and eventually commercialized. We summarize our main contributions and outline some potential improvements.

Control of the SET characteristics

The SET probe is made using standard lithographic processes and the Niemeyer-Dolan double angle shadow evaporation technique. This gives an excellent control of the geometry and the transport characteristics of the SET. The tunnel junctions areas are 50 × 50 nm 2 and a charging energy between 3 and 7 kelvin. The island size is of the order of 300 nm in length and limits the lateral spatial resolution of the instrument. The optimization of the SET probes can be done independently of the scanning microscope. We highlight two points of improvement:

• The resolution of the e-beam lithography (NanoBEAM) allows to make smaller tunnel junctions with larger charging energies. With smaller junctions, the SET microscope could be operated in 4 He cryostats. For samples which do not require extreme cryogenic temperatures, it is a simpler implementation and allows for quicker experimental turnover. Smaller tunnel junctions are however more fragile and more sensitive to electrostatic discharges. This last point has been specially addressed during the thesis. With a rigorous protocol in the SET probes could be mounted with a very high success. Within the same protocol, the use of weaker junctions appear feasible.

• The size of the island could also be reduced considerably and a 100 nm or better spatial resolution seem within reach. It would also be interesting to have SETs with different island sizes to understand precisely what sets the limits in spatial resolution.

On-probe gate electrode

Lithography permits the implementation of a gate electrode acting on the island. Capacitance and other island parameters can be determined from the bias-gate voltage Coulomb diamonds. In addition, with an on-probe gate, a PID feedback loop can be used to make linear electric field maps in feedback mode. The resulting images look very promising and could be checked against other physical systems.

In the future,

• the invasiveness of the on-probe gate potential needs to be studied. There may be an electrostatic effect on the sample.

Batch fabrication of probes

Batch fabrication of the SET probes is an asset in probe microscopy: SET probes are fragile and changes in oxydation parameter allow to fabricate SETs with the most appropriate tunnel resistance. This has been possible by combining lithographic and etching processes, particularly the Si isotropic etching using a XeF 2 chemical reaction. With the mesa approach used to fabricate the SET probe (chapter 2) we were able to make a trench close to the SETs which protects them during the cutting process of the individual probes. After the separation of the probes, the SET is usually 1 µm from the edge of the wafer in a plane some micrometers above a terrace that surrounds the probe. 10 probes can be fabricated in a short lapse of time of about 2-4 days. We used substrates 1 cm 2 in size as they are easily manageable. Several fabrication batches can be processed in parallel, and a large number of probes can be stocked. Future developments could involve,

• larger wafers with more probes: the machines used for lithography, evaporation, etching and the cutting process are compatible with 4 inches wafers.

• The mesa approach protects the sample during the displacement and manipulation of the SETs probes. This is ideal for sharing probes.

• During this doctoral thesis, a good control of the XeF 2 etching process has been achieved, but the reproducibility is not perfect due to the number of parameters involved. There is still room for improvement in the SET-to-edge distance variability (currently in the 200-500 nm range).

SET probe on Commercial Si-SiO 2 substrates

The SET probes are fabricated on commercial Si-SiO 2 substrate as the one that we used for the sample studied in the chapter 4. This type of substrates has a large number of technologies adapted to them since it commonly used in nanoelectronics. Some further remarks: Yet, the presence of the Si-SiO 2 interface charges has to be considered. The experimental studies showed that these charges were very stable as no jumps in the Coulomb diamonds were observed, to the point that image subtractions could be reliably done. The effect of background charges can be modelled as a fixed background which adds to the other charge sources. Some further remarks:

• For the SET, the fixed charges add an offset on the gate voltage which is not an issue since the SET current is periodic. More problematic would be mobile charges that would induce charge jumps.

• A separate issue is the effect that the background charges on the probe would have, say on a 2DES and would mimic a local gate shifting the Fermi level. With a careful experimental study, precise answers can be given for the specific problem of interest.

• We have seen that the background charges were effectively screened by metallic electrodes. With this in mind, it is worth to explore the possibility of including a ground plane underneath the SET to shield the sample from any background charges.

Tip-to-sample distance control and navigation

The tip-sample distance is controlled very precisely using a tuning fork. To do so, the edge of the probe substrate is used as an atomic force microscope (AFM) tip to bring the SET to distance as small as 50 -100 nm from the sample and to keep it safe from crashing. Additionally, the sensitivity of the SET is sufficient to detect the potential created by the electrodes from afar. Therefore, it is not difficult to navigate from a given electrode to find the structure of interest. Compared to an AFM or where a local interaction with the probe is required, it is simpler to localize elements and navigate over a sample with the SET microscope.

• Since the SET is not exactly at the edge of the probe substrate, a small offset exists between the AFM contact point and the real SET-to-sample distance. The capacitive coupling between the SET island and a sample electrode depends on the SET-to-sample distance and can be extracted from the Coulomb Diamonds. From the capacitance, a distance calibration can be done as in Yacoby's instrument: in this way, no tipto-sample contact is necessary. A SET-to-sample capacitive model as a function of distance need to be analyzed to get more accurate values for the offsets between the SET and the AFM contact point.

• When operating the microscope, the offset between the SET and the AFM contact point is an advantage. The SET is protected even when a sudden contact occurs while scanning. The probe is quite robust allowing continuous scanning during weeks with no major problems.

• As it is the case in Weizmanns Institute's instrument in ref [START_REF] Honig | Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3[END_REF], the distance control system can also generate topographic images of the sample. However, since in our case, the Si substrate is rigid, this operation may easily scratch the sample surface. Furthermore, rest of the sample ripped by the probe may put in big danger the SET.

High magnetic field

The results presented in the manuscript were performed with the SET in the superconducting state (B = 0 T). However, some measurements were also made in a magnetic field (adjusting the working point). Above B = 0.5 T, the SET shows typical "normal" Coulomb diamond, and this does not affect the SET performance as a sensor even at the largest fields (18 T). The performance of the piezoelectrics and the tuning fork at such large fields have not been thoroughly studied.

Perspectives: An ideal tool for 2D localization and metal-insulator transitions

The motivation for a scanning SET microscope was the study of the electronic properties of electronic systems. Yacoby's group instrument has already demonstrated the exceptional potential for two-dimensional electron systems (2DES) and other electronic phases. The next step is to exploit the microscope performance in this context. Adapting the instrument to measurements of the electronic compressibility should be extremely useful for 2DES (see appendix E). It is defined as

κ = 1 n 2 dn dµ (5.1)
where n is the average carrier density and µ the chemical potential. In a localized phase κ (insulating-like states) decreases and may even drop to zero for Mott insulators, and is large in metallic phases, where screening works. Electronic transport measurements provide information on the global behavior of a material, while the local compressibility gives a spatial map of an electronic property which changes a lot depending on the nature (metal or insulating) of the electronic system. In other words, the ability to measure the compressibility opens up a number of potential experimental studies. • Coulomb blockade device are unaffected by large magnetic fields. It can be used to drive an electronic system through different electronic phases (quantum Hall and fractional quantum Hall states).

• Furthermore, with the multiple filtered connections available on our sample holder, it is relatively simple to use back-gates and multiple electrodes necessary in Hall measurements. We can in fact simultaneously image a sample designed for transport measurements.

2D electron systems and topological insulator

The compressibility is crucial to the understanding of 2D electron systems such as graphene where inhomogeneous electronic phases coexists, for instance because of disorder. We can also study the evolution of the localization of 2DES as a function of magnetic field and carrier density. This makes this instrument ideal for investigating quantum Hall phases in 2DES. the dimension 2 is critical for Anderson localization. This means that inhomogeneous phases naturally arise due to disorder. Spatially resolving the metallic and insulating patterns in 2DES is in fact some kind of "holly grail" to localization in two dimensions. We can follow the redistribution of compressible and incompressible region when a quantum Hall plateau is reached. It should be possible to observe the edge channels in a quantum Hall state while the scanning SET may have difficulties to resolve the individually (a 5 to 50 nm resolution would be needed depending on the system and the magnetic field strength). For three dimensional topological insulators [START_REF] Brüne | Quantum Hall Effect from the Topological Surface States of Strained Bulk HgTe[END_REF], the nature of the electronic surface states and their evolution as a function of magnetic field is at present not well understood. For example, the quantum Hall state with n = 0 has a non-trivial edge state composed of electrons and holes. Using a scanning SET to image the conduction channels on the topological insulator top and bottom conducting surface of a slab would be quite interesting as coupled stated may appear as a function of thickness.

Metal Insulator transistor

In the chapter 4, we were able able to image a sample with metallic and insulating region in one single picture using the same experimental parameters. Hence, the instrument is quite adapted to metal-insulator transition studies.

More concretely, tracking the evolution of the compressibility, this is the localization, gives a direct view of the local electronic states and allow to follow the transition as a function of tuning parameter study with great precision. For instance in a Mott insulator the transition to metal is singular [START_REF] Imada | Metalinsulator transitions[END_REF] and such singularity should leave a signature in the compressibility. The scanning SET is also an ideal tool to explore the inhomogenous electronic phases that sometimes occur in the vicinity or at a metal-insulator transition either for intrinsic (charge density waves or stripe phases), or extrinsic regions (for example inhomogeneities in the stoichiometry and or disorder). The spatial scale of the different insulating or metallic regions seems accessible for the resolution of our scanning SET.

Current status of the project

This new instrument at the Neel Institute opens up new collaborations for 2DES studies and have already started. For instance, the studies in highmobility graphene performed in Sacepe's Group [START_REF] Zimmermann | Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices[END_REF][START_REF] Veyrat | Low Magnetic Field Regime of a Gate-Defined Quantum Point Contact in High-Mobility Graphene[END_REF]. As stated, a future objective for this project is to refine the instrument and carry out compressibility measurements. To this end, we performed preliminary images on an hBN-encapsulated graphene flake made by Benjamin Sacepe group at the Institute for transport studies. The images are shown in Figs. 5.2 (A),(B), (C) and (D). The electrostatic contrast between the insulating substrate and the electrodes is quite high even at distance far from the sample ∼ 1 µm. It is easy to navigate over the graphene flake (the contours of the electrodes and of the graphene flake are depicted in green dotted lines). In Fig 5 .2 (E) a SET current image zooms in the region of the graphene flake. It is rotated with respect to the feedback image shown (D). The electrodes are placed in a Hall mesa for transport measurements (Fig. 5.2 (E)).

General conclusion

The scanning SET microscope presented in this work has performances similar or higher to the existing machines. We can project ourselves toward the numerous possibilities offered by this instrument for electronic systems. There is still room for improvement to make the instrument even more powerful, particularly by adding compressibility measurement for 2DES. Our instrument is an advance in Scanning SET microscopy and is opened to future collaborations with other research groups.

Appendix D

Simulation of the electrical potential above the sample

Using a finite elements simulator, an equivalent situation to that deduced from the experimental measurements has been simulated. In It is clear that the potential is screened by the grounded electrodes and emanates in between. The potential is larger at the left part since there are thinner and less compacted electrodes hence a smaller screening than at the right part of the map. This is in good agreement with the experimental results.

  .11 (B). The source (S), gate (G) and drain (D) contacts are welded to 150 µm-diameter Cu wires (right part of the photo 3.11 (B)) which ends in the intermediate holder seen in Fig. 3.3.

. 5 ) 4 . 5 .

 545 Feedback scanning mode 125 colormap in Fig. 4.14 (A). The black contrast of the image corresponds to the Al electrodes (red dashed lines).

  .12) where C s and C d are the source and drain capacitances, C g is the on-chip gate capacitance and +C sample is the SET-electrode capacitance. At shorter distance, C sample increases and a decrease in charging energy is expected. On the other hand, all other capacitance stay unchanged. C s = (50 ± 10) aF, C d = (70 ± 20) aF and C g have been calculated from the Coulomb diamonds in Fig.4.22 (B)(900 nm distance) resulting in a total capacitance of C ∑ = (125 ± 10) aF.On Fig.4.23 (B), the capacitance C sample is plotted as a function of the inverse of the distance of the tip-to-sample distances of +50, 100, 150, 200, 250, 300, 350, 400 and 900 nm, with an additional offset estimated to 100 nm. The blue, green and black experimental points correspond to the C sample obtained for the current oscillations shown in Fig.4.23 (A). On this graph, C sample increases by a factor C g(50 nm)

5. 1 .

 1 Perspectives: An ideal tool for 2D localization and metal-insulator transitions 149

  Fig. D.1 (A) the simulated geometry is presented. The real size and separation between electrostatic gates has been reproduced with detail. The metallic gates are placed on top of a substrate, on top of 100 nm dielectric layer. The bottom plane of the dielectric layer has been defined as a surface density of σ = 10 × 10 10 e cm -2 and is the source of electric field. The bottom plane of the substrate is considered as a grounded plane. The equivalent region that has been scanned would correspond to the inside part of the black square. The curves presented in Fig.4.18 and Fig.4.19 are the potential along the red line at different heights only for the zone inside the scanned region. The electric potential along such line can be seen in Fig. D.2 at different heights. Fig. D.1 (B) shows the electric potential in a plane 300 nm above the sample.

Fig. D. 2 (

 2 A) shows the simulated electric potential along the red line depicted in Fig. D.1 for the background surface charges simulation. The red, green, blue and cyan curves shows the potential profile at a distance of 200, 600, 1200 and 1800 nm. It can be observed how the profile flatters when the distance is increased. The scanned region would correspond to the three electrodes painted in grey. The curves in Fig. D.2 (B) shows the simulated electric potential at different heights 200,600 and 1200 nm for red, green and blue, with a polarization of V 2 = 100 mV (see Fig.D.1 (A)) and no background charges. It can be observed that the potential difference on top of electrodes

The term ∑ N i=1 ǫ i can be used in this way if one assumes that the kinetic energies of the electrons ǫ i are independent on the number of electrons on the island, neglecting electron exchange, screening and correlations effects which is known as constant interaction model[START_REF] Beenakker | Theory of Coulomb-blockade oscillations in the conductance of a quantum dot[END_REF][START_REF] Kouwenhoven | Mesoscopic Electron Transport[END_REF][START_REF] Kouwenhoven | Reports on Progress in Physics Related content Non-sticking drops Non-sticking drops[END_REF]

With the NANOBEAM it is not possible to make single SEM image, thus with a minimal exposure, as in the case of other lithography machines(LEO 1530 (FESEM)

The same behavior is expected by switching the on gate probe voltage from decreasing to increasing and vice-versa in Fig.4.3 in continuous measurements in time.

A similar analysis has been done for the critical current images in scanning SQUID[START_REF] Hykel | Magnetic imaging of unconventional superconductors by scanning SQUID microscopy[END_REF].

The tilting angles where limited to the range -2.5°to 2.5°to avoid damaging the sample with the tip.

The obtained slope is (600 ± 300) aF nm, where the error takes into account the max and min slopes with considering the error bars. This results involves an equivalent "area" of the SET island of A = slope ǫ 0 = (70 000 ± 30 000) nm 2 = 2.6 -6.6 × A SET . The result is in the order of magnitude of the SET tops surface (A SET ≈ 300 × 50 = 15 000 nm 2 ) but overestimates it. This is acceptable considering that we are in the limits of the parallel planes approximation.

Acknowledgements

Appendix A SET probe-sample angle estimation

As describe in chapter 2 and 3, the edge of the SET substrate is used as an AFM tip. The angle is a critical aspect to ensure a correct tip-to-sample approach. Also it limits the minimal SET-to-sample distance since the SET is not exactly at the edge of the substrate which acts as the AFM tip. For that it is essential to be sure that it is suitable for the approaching procedure before the implementation of the scanning stage in the cryostat. As seen in Fig . A.1 (A) it is possible to observe the SET substrate through the windows of the scanning stage once it is incorporated on top of the sample. Furthermore, when the SET substrate is at a certain short distance from the sample, it is reflected on the Si-SiO 2 surface. Such reflection permits to verify if an angle exists between the SET substrate and the sample. However, the windows are not exactly facing the flanks of the substrate and the image shows an unknown The vanishing point of the image. Thus, the angle cannot be calculated straightforward. Since the dimensions of the substrate are well-known (700 × 2800 nm) it is possible to estimate the angle from a approximate geometrical projections. With this, using autoCAD, the image was reconstructed in 3D as shown in

Appendix C

Series of SET current images used for the analysis in section 4.4.2

In this appendix, the set of SET current images used for the analysis of section 4.4.2 is presented. The scan is performed above the same region of the sample, and starting from the same point. It can be observed that the color contrast changes for different polarizations of V 2 in many position of the sample. The equicurrent contours evolve in a way which is not easy to relate to the change in the polarization of one of the electrodes. With the analysis proposed in [START_REF] Yoo | Scanning single-electron transistor microscopy: Imaging individual charges[END_REF], considering the variations from all the images, information about the local electrostatic changes in each of the pixels of the image can be extracted. This enables the reconstruction of the local capacitance and the spatial distribution of the effect coming from the background charges.

Appendix E

Compressibility measurement set-up

In this appendix the intention is to give a general idea of the experiemntal setup that have been used in other works [START_REF] Yacoby | Electrical imaging of the quantum Hall state[END_REF][START_REF] Martin | Observation of electron-hole puddles in graphene using a scanning single-electron transistor[END_REF][START_REF] Benjamin E Feldman | Unconventional Sequence of Fractional Quantum Hall States in Suspended Graphene[END_REF][START_REF] Honig | Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3[END_REF] for the measurement of the compressibility of electronic states on 2D electron systems (2DES).

where n is the average carrier density and µ the chemical potential.

The SET measures the changes in the surface potential of a 2DES. This is related to the electrochemical potential µ which can be tuned through a voltage applied to the 2DES V 2D ∝ µ. At the same time, the density of carriers n, related to the Fermi level of the 2DES can be tuned by the action of a backgate V BG ∝ n. A variation in the back-gate voltage, produces a change in the SET current, induced by the local electric field. When the SET is placed above a compressible 2DES region, the electric field is screened locally, reducing the SET current variation compared to the unscreened field over incompressible regions. Thus, the compressibility is a measurement of the electric field transparency through the 2DES and can be related to dµ dn which is the inverse of the compressibility.

Fig.E.1 illustrates how this is done and is a simplified overview of the set-up presented in refs [START_REF] Yacoby | Electrical imaging of the quantum Hall state[END_REF][START_REF] Benjamin E Feldman | Unconventional Sequence of Fractional Quantum Hall States in Suspended Graphene[END_REF]. The idea is to applied an AC modulation V 2D (ω 2D ) ∝ µ(ω 2D ) at a frequency ω 2D and a second AC modulation V 2D (ω 2D ) ∝ n(ω BG ) with a different frequency ω BG . The strength of the contributions of each of the modulations to the SET current variations can be tracked using a lock-in readout of the signal. It is δI SET δV 2D (ω 2D ) for the signal coming from the 2DEG and δI SET δV BG (ω BG ) for the signal coming from the back- gate. The signal in ω 2D is used as a calibration to transform the transparency