Probing nanoscale limits of polarization switching and controlling electronic properties in devices combining carbon nanotubes and ferroelectrics

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Thesis

Reference

Probing nanoscale limits of polarization switching and controlling electronic properties in devices combining carbon nanotubes and

ferroelectrics

BLASER, Cédric

Abstract

La présente thèse est consacrée en grande partie à l'étude des interactions entre les nanotubes de carbone et les couches minces ferroélectriques. D'une part, l'effet de champ ferroélectrique peut moduler de façon réversible et non-volatile la densité de porteurs de charge de nanotubes semi-conducteurs. D'autre part, des nanotubes mis sous tension électrique peuvent être utilisés comme une source de champ électrique extrêmement concentrée pour modifier localement la polarisation d'une couche mince ferroélectrique sous-jacente. Ces effets ont été explorés dans cette thèse dans le cadre de la création de transistors à effet de champ ferroélectrique formés par des nanotubes semi-conducteurs déposés sur une couche mince ferroélectrique de Pb(Zr0.2Ti0.8)O3. De tels transistors sont non-volatils tout en permettant une lecture non-destructive. L'utilisation de nanotubes de carbone comme électrode a permis la création et l'étude de domaines ferroélectriques de dimensions nanométriques avec une durée de vie supérieure à 1,5 an.

BLASER, Cédric. Probing nanoscale limits of polarization switching and controlling electronic properties in devices combining carbon nanotubes and ferroelectrics . Thèse de doctorat : Univ. Genève, 2014, no. Sc. 4711

URN : urn:nbn:ch:unige-418765

DOI : 10.13097/archive-ouverte/unige:41876

Available at:

http://archive-ouverte.unige.ch/unige:41876

Disclaimer: layout of this document may differ from the published version.

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de la Matière Condensée

Probing nanoscale limits of

polarization switching and controlling electronic properties in devices combining carbon nanotubes and

ferroelectrics

THÈSE

présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de Docteur ès Sciences, mention Physique

par

Cédric Blaser

de

Langnau im Emmental (BE)

Thèse n^◦4711

GENÈVE Imprimerie Harder

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Résumé

La présente thèse est consacrée en grande partie à l’étude des interactions entre les nanotubes de carbone et les couches minces ferroélectriques. D’un diamètre de l’ordre du nanomètre et d’une longueur généralement de l’ordre du micromètre, les nanotubes de carbone sont de petits tubes exclusivement composés d’atomes de carbone. Suivant leur configuration atomique exacte, les nanotubes peuvent avoir un comportement électrique métallique ou semi-conducteur. Depuis que Sumio Iijima les a caractérisés en 1991, un grand nombre de chercheurs ont commencé à étudier leurs étonnantes propriétés électriques, mécaniques, optiques et même thermiques.

La ferroélectricité, qui a été découverte en 1920 par Joseph Valasek dans le sel de Rochelle, consiste pour un matériau à avoir deux états stables de polarisation, un champ électrique externe permettant de passer de l’un à l’autre. En associant à ces états les valeurs 0 et 1 d’un bit d’information, on peut envisager la conception de mémoires informatiques basées sur ce phénomène physique. Durant le XX^ème siècle, de nombreux matériaux ferroélectriques ont été découverts, la plupart avec une structure bien plus simple que le sel de Rochelle. De plus, les techniques de croissance de couches minces permettent aujourd’hui de réaliser des films ferroélectriques avec une précision atomique.

On distingue deux types d’interactions entre les nanotubes de carbone et les couches minces ferroélectriques. D’une part, l’effet de champ ferroélectrique peut moduler de façon réversible et non-volatile la densité de porteurs de charge de nanotubes semi-conducteurs. D’autre part, des nanotubes mis sous tension électrique peuvent être utilisés comme une source de champ électrique extrêmement concentrée pour modifier localement la polarisation d’une couche mince ferroélectrique sous-jacente. Des travaux antérieurs ont montré que la croissance directe d’un de ces matériaux sur l’autre avait pour conséquence

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des propriétés physiques dégradées. C’est pourquoi l’approche retenue pour le présent travail est une croissance séparée suivie d’un assemblage des dispositifs expérimentaux par enduction centrifuge à partir d’une suspension aqueuse de nanotubes de carbone. Cette méthode a l’avantage d’éviter tout processus nécessitant une température élevée et n’expose pas les nanotubes à d’autres étapes de fabrication, l’idée étant d’étudier un système aussi intact et propre que possible.

L’influence de la ferroélectricité sur les nanotubes semi-conducteurs peut être étudiée par des mesures de transport électrique de transistors à effet de champ avec un canal formé par un nanotube et un oxyde de grille constitué d’une couche mince ferroélectrique. L’effet de champ ferroélectrique de ce matériau modifie fortement la conductivité des nanotubes semi-conducteurs qui y sont déposés, ce qui influence la relation courant–tension des dispositifs. Tout comme les transistors à effet de champ habituels avec un oxyde de grille diélectrique, ceux qui sont basés sur un matériau ferroélectrique peuvent être ouverts ou fermés, selon le champ électrique appliqué. Tandis que les transistors standard perdent leur état s’ils ne sont plus sous tension, ceux avec un oxyde de grille ferroélectrique peuvent conserver leur état grâce à la persistance de la polarisation et à l’effet de champ ferroélectrique.

Ce fonctionnement a été observé en réalisant des transistors à base de nanotubes semi-conducteurs avec des oxydes de grille composés de Pb(Zr0.2Ti0.8)O3 (qui est ferroélectrique) d’une part, et de SrTiO3 (qui est diélectrique) d’autre part.

La comparaison des deux types de dispositifs a permis de mettre en évidence ce qui relève de l’effet de champ ferroélectrique et ce qui est dû à des déplacements transitoires de charges à l’interface entre les nanotubes et les oxydes de grille. La non-volatilité des deux états des transistors à base de Pb(Zr0.2Ti0.8)O3a pu être démontrée, tout comme la possibilité de lire ces états de manière non-destructive.

Pour de potentielles applications industrielles de ce type de mémoires, il sera crucial d’optimiser la qualité de l’interface entre les deux matériaux, afin de limiter au maximum les effets parasites observés durant ces travaux exploratoires.

L’action des nanotubes de carbone sur les couches minces ferroélectriques a été étudiée en appliquant des pulses de tension aux nanotubes. Dans cette configuration, ces derniers servent d’électrode permettant d’inverser localement la polarisation de la couche de Pb(Zr0.2Ti0.8)O3sur laquelle ils reposent. En utilisant des pulses de tension de seulement 10µs, des domaines ferroélectriques filiformes de moins de 20 nm de large ont pu être formés dans une couche de Pb(Zr0.2Ti0.8)O3de 270 nm d’épaisseur et sont restés stables durant plus de 1,5 an. La taille de ces domaines n’excède pas la moitié de ce qui est prédit par des modèles thermodynamiques, ce qui illustre l’importance de la stabilisation des parois de domaines par des défauts structurels dans le Pb(Zr0.2Ti0.8)O3. Ces domaines induits par des nanotubes ont été comparés à des domaines induits par d’autres géométries de champ électrique telles que des pointes de microscope à force atomique ou des électrodes déposées par photolithographie. Des simulations ont montré que le champ électrique généré par les nanotubes était le plus intense localement, mais le plus faible à plus de 10 nm de la source du fait de sa très rapide décroissance. Cela est en bonne adéquation

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RÉSUMÉ

avec les observations : parmi les trois géométries de champ électrique considérées, les nanotubes permettent en effet d’induire des domaines ferroélectriques avec les pulses de tension les plus courts et génèrent les domaines les plus petits avec les pulses les plus longs.

La dernière partie de cette thèse approfondit l’étude de la polarisation à l’échelle nanométrique et en particulier les effets de l’environnement sur l’inversion de cette polarisation dans des couches minces de Pb(Zr0.2Ti0.8)O3. En effet, il a pu être démontré que la taille des domaines ferroélectriques circulaires induits par des pointes de microscope à force atomique dépend non seulement de la durée et de l’amplitude des pulses de tension appliqués, mais également du taux d’humidité, et ce de manière linéaire. Comme le rayon des domaines ainsi induits peut varier du simple au double suivant l’humidité ambiante, toute étude quantitative de croissance de domaines ferroélectriques devrait tenir compte de cet effet. L’origine du phénomène est liée au ménisque d’eau qui se forme aux conditions ambiantes par condensation capillaire autour de la pointe de microscope à force atomique. Les dimensions de ce ménisque augmentent avec le taux d’humidité, mais le mécanisme exact qui explique la linéarité de la taille des domaines jusqu’à des rayons de plusieurs centaines de nanomètres reste à déterminer.

Une variation du protocole expérimental, consistant à appliquer non pas un pulse de tension unique mais une série de pulses beaucoup plus courts, a permis d’étudier plus spécifiquement la phase initiale de formation des domaines ferroélectriques.

Les expériences ont révélé qu’à ce stade, le taux d’humidité joue un rôle mineur alors qu’il a un impact beaucoup plus important durant la phase ultérieure de croissance latérale. Le même protocole a été utilisé avec des séries de pulses dix fois plus courts que ce qui est nécessaire pour avoir un effet individuel observable.

Il a été possible d’induire des domaines stables avec moins d’une vingtaine de ces pulses. Un temps de relaxation étonnamment long de l’ordre de 100 ms a été mis en évidence en variant le délai inter-pulses et attribué à des changements associés à la formation et à la stabilisation d’un noyau critique de polarisation inversée dans la couche mince ferroélectrique.

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CHAPTER 1 Introduction

1.1 General introduction

Since the seminal paper by Sumio Iijima in 1991 [1], carbon nanotubes (CNTs) have attracted a huge interest from the scientific community. Researchers all over the world started to study these tiny carbon structures and discovered exceptional mechanical, electrical, chemical, optical, and thermal properties [2,3]. For example, CNTs are stronger than steel and conduct electricity better than copper at equivalent dimensions. Moreover, their revelation and subsequent development, including integration into devices, came at a particularly opportune moment, as technology moved from the “micro” to the “nano” world at the end of the last century.

Ferroelectricity has been known since 1920, with the discovery of a stable and switchable polarization in Rochelle salt [4]. For many years, this phenomenon remained only a scientific curiosity, until its discovery in BaTiO3 [5–8], which has a much simpler structure than Rochelle salt and could be produced relatively inexpensively in ceramic form on a large scale. Ferroelectric ceramics were then used in various industrial and military applications like transducers, sensors or actuators during the second half of the 20th century. More recent developments in thin film technology, yielding quasi atomic level control and single-crystal quality, make ferroelectric materials a valuable candidate for data storage.

Potentially multifunctional systems can be obtained by combining CNTs with ferroelectric materials. For example, ferroelectrics deposited around CNTs could provide actuation at the nanometer level. Another avenue is to fabricate transistor devices, in which the CNT acts as the channel, and the ferroelectric as a non-volatile

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already-polarized gate dielectric. In such devices, ferroelectric field effect doping could allow local and reversible gate architecture with significantly greater effective fields available to modulate CNT charge carrier density. Meanwhile, CNTs can act as local electric field sources for nanoscale polarization switching at resolutions beyond those available with standard atomic force microscopy tips.

A possible application of such a multifunctional device could be data storage.

From the first magnetic hard drive in 1956 (Fig.1.1(a)) to the much smaller modern devices (Fig.1.1(b)), magnetic data storage experienced an increase in density of almost eight orders of magnitude. The associated size decrease of individual ferro- magnetic domains was pushed to such an extreme that the intrinsic domain wall thickness has begun to become the limiting factor. However, whereas a ferromag- netic domain wall has an average thickness on the order of 50 nm, a ferroelectric domain wall has a thickness on the order of only 1 nm. This could make ferroelectric materials the base of future high density data storage devices, and CNTs with their intrinsic nanometric size the tool of choice to control the polarization at this scale (Fig.1.1(c)).

Figure 1.1: (a)IBM 350 disk storage unitintroduced in 1956, having the size of two big fridges and storage capacity of 5 million characters.(b)Modern hard drives are able to store 1000 billion characters on a single platter of 3.5” diameter.(c)Artist’s view ofIBM’s “Millipede” project, which uses multiple atomic-force-microscopy tips to read and write data in the surface of a thermo-active polymer.

Ferroelectric data storage could benefit from a similar multi-probe technique, especially when combined with CNTs to reach even higher densities. Reproduced from [9–11].

Transistors are another device that underwent a phenomenal miniaturization since the first prototype in 1947 (Fig.1.2(a)), whose size was measured in centime- ters. In contrast, current lithography techniques pattern transistors with dimensions of around 10 nanometers (Fig.1.2(b)). However one limitation that affects most of the transistors produced over the last seven decades is the fact that if they lose their external power supply, they also lose their resistive state by capacitive discharge, and thus their data (Fig.1.2(c)). Recently developed ferroelectric memories are non-volatile but not yet competitive with standard memories based on conventional silicon based transistors. One of their flaws is that they require a destructive read process, meaning that to read the value stored in a single memory cell, one has to overwrite it, measure the resulting current pulse and restore the initial value if it has changed during the procedure. Transistors combining CNTs and ferroelectric

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1.2 The present study

Figure 1.2: (a)Replica of the first transistor, invented at Bell Labs in 1947.(b)Diagram of a modern processor with several billion transistors on a single chip of a few cm², illustrating the miniaturization that happened since 1947.(c)Aerial view of New York from November 2012, revealing the power outage after Hurricane Sandy. During such an event, all conventional computer memories lose their data, whereas non-volatile memories would have been ready to resume operation as soon as the power came back. Reproduced from [12–14].

material could potentially provide a way to make more competitive non-volatile memories while providing a simpler, more direct read-out scheme.

1.2 The present study

In this thesis, I develop the research I performed on different fundamental aspects of combined CNT-ferroelectric devices, without forgetting about the practical effects that have to be mastered to turn a physical property into a useable technology. While working on the minimum domain size reachable in a ferroelectric thin film, I also studied the stability of these domains. Everybody will agree that a storage device losing its data after a few hours is of no use. I showed that domains as small as 9 nm – which would allow significantly higher storage densities than current hard drives – are stable for at least 1.5 years. For the transistors combining CNTs and ferroelectric thin films, I investigated not only the ferroelectric field effect as it may be described in textbooks, but also the charging effects that happen in actual devices, because only by taking into account both phenomena could they serve as the future non-volatile memories described above.

This thesis is organized as follows. In Chapter2, I will present the most important characteristics of CNTs, their synthesis methods, how they can be imaged, and how they can be used to build transistors. In Chapter3, I will describe ferroelectricity and present the ferroelectric materials I used in this thesis. I will also introduce the principles governing the switching of the polarization, as well as the ferroelectric field effect and how it can be used to build transistors and multifunctional devices in combination with CNTs. In Chapter4, I will then present the experimental setup and the techniques used in this thesis. Electrical transport measurements through CNTs will be reported in Chapter5, with a comparison of field effect transistors based on CNTs using either a ferroelectric or a dielectric perovskite oxide as the gate material. I will also detail low temperature measurements which highlight the

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one-dimensional behavior of the CNTs. In Chapter6, I will focus on controlling polarization at the nanoscale, using CNTs, atomic-force-microscopy tips and macroscopic planar electrodes as different electric field sources. Finally, in Chapter7I will go into more depth on the switching dynamics of atomic-force-microscopy tips induced ferroelectric domains. In particular, I will elucidate the impact of the environment and the effect of subcritical pulses to investigate the initial nucleation process.

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CHAPTER 2 Carbon nanotubes

2.1 Structural properties

Carbon nanotubes (CNTs) can be considered either as very long molecules composed exclusively of carbon atoms or as one-dimensional tubular crystals. Isolated carbon atoms have four valence electrons in 2s, 2p_x, 2p_y, and 2p_zatomic orbitals.

When they form CNTs, the orbitals 2s, 2p_x, and 2p_yhybridize into three equivalent sp²orbitals. This results in a hexagonal lattice with strong covalentσbonds between adjacent carbon atoms and 2pzorbitals formingπbonds perpendicular to the lattice [15].σbonds are responsible for the mechanical properties of CNTs whileπ bonds are responsible for their electrical properties.

CNTs are characterized by their number of walls. While some, calledsingle- walled carbon nanotubes(SWCNTs) are formed out of a single sheet of carbon atoms, as shown in Fig.2.1(a), other nanotubes calledmulti-walled carbon nanotubes(MWCNTs) are formed by multiple concentric sheets of carbon atoms, as shown in Fig.2.1(b), each sheet being arranged in a hexagonal lattice like SWCNTs.

Whereas most SWCNTs have a diameter of around 1 nm, MWCNTs can have diameters of up to 100 nm. The spacing between their different sheets is close to 3.41 Å, which is the spacing between individual graphene layers in a graphite crystal [3,16].

For some applications, describing CNTs by their diameter, their length and their number of sheets is precise enough. In other cases, it is necessary to describe the precise atomic arrangement forming the hexagonal lattice of each sheet. This is usually done using the chiral vector [3]. Starting with the two base vectorsa1

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Figure 2.1:Schematic representation of(a)a SWCNT and(b)a MWCNT.

Figure 2.2:Graphene hexagonal lattice with the lattice vectorsa1anda2. The chiral vectorc=8a1+4a2

of the (8,4) tube is shown with the 4 graphene-lattice points indicated by circles; the first and the last coincide if the sheet is rolled up. Perpendicular tocis the tube axisz, the minimum translational period is given by the vectora=−4a1+5a2. The vectorscandaform a rectangle, which is the unit cell of the tube, if it is rolled alongcinto a cylinder. The zig-zag and armchair patterns are highlighted in red.

Adapted from [3].

and a₂ as shown in Fig.2.2, one forms the chiral vector c = na₁+ma₂. The hexagonal lattice is then rolled up in such a way that the chiral vector becomes the circumference of the tube. A nanotube of specific atomic configuration is often described by the components (n,m) of this vector. As shown in Fig.2.2, one can also define a chiral angleθwhich varies between 0° and 30°, given the six-fold rotational symmetry of the lattice. Some configurations are special. Nanotubes with (n,0) havingθ=0° are called “zip-zag” because of the pattern along their

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2.2 History, synthesis, and imaging methods

Figure 2.3:Schema of the atomic structure of(a)a ziz-zag and(b)an armchair nanotube. Reproduced from [17].

circumference (see Fig.2.3(a)) and nanotubes of type (n,n) havingθ=30° are called “armchair” for the same reason (see Fig.2.3(b)). Zig-zag and armchair tubes are achiral, in other words their image in a mirror is the same as the original, in contrast to generic (n,m) tubes withn,mwhich are said to be chiral.

Usual dimensions for CNTs range from 1 to 100 nm in diameter and up to a few micrometers in length. Some extreme sizes have been reported, like a 55 cm long CNT [18]. In fact, it seems that there is no fundamental upper limit for the length of nanotubes; it all relies on big enough equipment and stabilizing the right parameters during the whole growth time. The thinnest freestanding SWCNT has been observed with a diameter of 4.3 Å [19]. Thinner CNTs with diameters down to 3 Å have been reported [20,21], but they were not freestanding as they were grown inside other CNTs.

2.2 History, synthesis, and imaging methods

2.2.1 Discovery of carbon nanotubes

The discovery of CNTs is often attributed to Sumio Iijima. While it is true that his seminal paper from 1991 [1] generated a tremendous interest from the scientific community, as proven by the currently over 22 000 citations, the history of CNTs is older, as demonstrated in a 2006 editorial [22] from the journalCarbon. The first mention of “hair-like carbon filaments” for electric lighting made from thermal decomposition of carbon-containing gas was reported in a patent [23] as early as 1889, but the resolution of optical microscopes available at that time was not sufficient to demonstrate the nanometer-scale of the produced fibers. Significant progress could be made only after the invention of the transmission electron microscope (TEM) whose first commercial versions were produced bySiemensin 1939. This instru- ment was used by Radushkevich and Lukyanovich [24] in 1952 to show the oldest known evidence of tubular carbon structures with a diameter on the order of 50 nm,

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what is today called a MWCNT. The publication in theRussian Journal of Physical Chemistryduring the Cold War did not help this scientific knowledge to propagate widely, and so the discovery remained largely unnoticed. Other evidence of CNTs was published [25,26] in the Western world but their interest never reached the impact of the 1991 paper [1] by Iijima. What really boosted the research on CNTs was, on one hand, the theoretical predictions of exceptional electronic properties of SWCNTs [27–29] in 1992, and on the other hand, the actual discovery of SWCNTs independently by Iijima and Ichihashi [30] and Bethuneet al.[31] in 1993. The reason why Iijima made it into the history books is probably because he coined the name “carbon nanotube”. Indeed, in a time where these carbon structures were first called “helical microtubules of graphitic carbon” [1], “fullerene tubules” [27],

“graphene tubules” [28], or “graphitic microtubules” [29], he was the first to call them “carbon nanotube” in the title of his third paper [32] on the topic in 1992, and the name stayed.

2.2.2 Synthesis methods

Unlike C60fullerenes which can be found in the form of traces in interstellar dust, CNTs were never detected in a natural state either in space or on Earth [33]. How- ever, they can be synthesized via multiple techniques. The first to be used were electric arc discharge [1,34] and laser ablation [35,36], and both relied on the sublimation of solid graphite. In the electric arc discharge process schematized in Fig.2.4(a), two high-purity graphite electrodes are moved together under an inert atmosphere and a voltage is applied until a stable arc is achieved. The average temperature in the inter-electrode region is extremely high (on the order of 4000 °C) and therefore carbon is sublimated and the anode is consumed. The material then deposits on the cathode and the reactor walls, and forms a variety of carbon allotropes including fullerenes, graphitic sheets, CNTs, and amorphous carbon [17,33].

In the laser ablation process schematized in Fig.2.4(b), a graphite target is vaporized by a laser under inert atmosphere at temperatures close to 1200 °C. An inert gas flow sweeps the ablated particles away from the high-temperature zone and deposits them on a water-cooled collector where the CNTs can grow. In both the laser ablation and the electric arc discharge methods, addition of transition metal in the graphite target or anode promotes the growth of SWCNTs. However, scale-up limitations of both techniques due to the limited volume of CNTs they can produce stimulated the development of gas-phase-based synthesis methods which are more adapted to large-scale processing [17,33].

Such methods include gas-phase catalytic growth from carbon monoxide [37]

and chemical vapor deposition (CVD) from hydrocarbons [38,39]. The latter is nowadays the most used method due to its versatility. CVD relies on the catalytic decomposition of carbon-containing gases (such as methane, ethylene, or acetylene) in a flow furnace (see Fig.2.4(c)) at atmospheric pressure, together with an inert process gas (such as argon or nitrogen). A substrate covered with metallic catalyst

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2.2 History, synthesis, and imaging methods

Figure 2.4:Schematic representation of devices used for(a)electric arc discharge,(b)laser ablation, and(c)chemical vapor deposition. Reproduced from [33].

particles is inserted in a quartz tube and heated between 500 and 1100 °C depending on the protocol [33]. As with the previously described synthesis methods, various allotropes of carbon can be formed, but with a careful optimization of the experimental parameters such as gas composition, gas flow, temperature, and catalyst choice, high purity and high yield of either SWCNTs or MWCNTs can be obtained.

The diameter of the grown CNTs can be controlled by an appropriate choice of the size of the metallic catalyst particles. The continuous supply of carbon in this method enables the growth of very long CNTs, such as the ones described at the end of Section2.1. Plasma-enhanced CVD [40,41] uses a plasma generated by a strong electric field to synthesize well aligned arrays of CNTs with uniform diameter, length, straightness, and site density. With the right parameters, CNTs can even grow vertically like a forest, as will be shown later in Fig.2.7(b).

2.2.3 Imaging methods

Transmission electron microscopy (TEM) is the technique which allowed Radushke- vich and Lukyanovich to publish the oldest known image of a CNT in 1952 [24].

Unfortunately, the resolution at that time was not sufficient to resolve the multiple walls, contrary to the images published by Iijima in 1991 [1] and reproduced in Fig.2.5(a). Carbon walls are almost transparent for the electron beam when they are crossed perpendicularly, so only the side walls where the electrons need to go through many atoms are clearly visible. Fig.2.5(b)shows a bent bundle of hexagonally packed SWCNTs [42], highlighting their cross-section. More recent aberration-corrected high-resolution TEMs are able to at least partially image the hexagonal carbon lattice between the two walls [21], as shown in upper half of Fig.2.5(c). Given the quality of the image, comparison with a simulation of what should be observed (shown in the lower part of Fig.2.5(c)) is nonetheless needed.

The first atomically resolved images of CNTs [43,44] were done using scanning tunneling microscopy (STM) at cryogenic temperatures. As STM could resolve simultaneously both the atomic structure (as shown by Figs.2.6(a,b)) and the elec-

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Figure 2.5: (a)TEM images of MWCNTs where the number of walls is clearly visible.(b)TEM image of a bundle of SWCNTs which are parallel to the electron beam, highlighting their cross-section.(c)Top:

Aberration-corrected high-resolution TEM image of a double-wall carbon nanotube where the hexagonal lattice is partially visible. Bottom: simulation of the upper image based on the atomic configuration of the observed nanotube. Adapted from [1,42,21].

tronic density of state [45], the predicted [27–29] relation between the (n,m) atomic configuration and the electronic properties of SWCNTs could be experimentally confirmed. However, the analysis is not as straightforward as one could naively expect. Although the chiral angleθand the diameterDcan be unambiguously related to the (n,m) configuration of the SWCNT, the relations between the experimentally determined valuesθexpandDexp, and the real valuesθandDare rather subtle due to the fact that STM images are a convolution of the tip shape and the sample [46], like any scanning probe microscopy technique. Room temperature STM measurements of SWCNTs have also been reported [47], but as one could expect, thermal noise renders blurred images as illustrated by Fig.2.6(c).

Both TEM and STM have rather strong requirements on what can be imaged. In TEM, only extremely thin samples can be studied and in STM, the substrate needs to be conductive. A less constraining technique is scanning electron microscopy (SEM) which works with an electron beam as does TEM, but uses the backscattered electrons to form the image. The technique does not reach the same resolution as the two previously mentioned, but is more appropriate to image large CNT structures directly on the substrate where they were grown, as shown in Fig.2.7. Atomic force microscopy (AFM) is another technique used to image CNTs and to measure their lengths and diameters. As it is the main technique that was used during this thesis, it is described in more detail in Section4.3in the chapter on experimental techniques.

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2.3 Electronic properties

Figure 2.6: (a)Low-pass filtered STM image with atomic resolution of a SWCNT exposed at the surface of a bundle at 77 K. The tube axis is indicated with a solid, black arrow, and the zig-zag direction is highlighted by dashed lines. A portion of a two-dimensional graphene layer is overlaid to highlight the atomic structure.(b)Atomically resolved STM image of an individual SWCNT at 4.2 K. The lattice on the surface of the cylinder clearly identifies the tube chirality. The dashed arrow represents the tube axis Tand the solid arrow indicates the direction of nearest-neighbor hexagon rowH.(c)Atomic resolution STM image of an armchair SWCNT at room temperature. The hexagons highlight the underlying lattice of the tube wall. Reproduced from [44,43,47].

Figure 2.7: (a)SEM image showing a high density of entangled CNTs formed by the electric arc discharge synthesis method.(b)SEM image of a forest-like CNT growth using the plasma-enhanced CVD method. A site density of about 10⁷tubes/mm²was estimated. Reproduced from [34,40].

2.3 Electronic properties

Numerous theoretical studies predicted the exceptional electric properties of SWCNTs [27–29]. To access them, the nanotubes need to be connected to electrodes in order to perform electrical transport measurements. This can be done in a number of ways, including the deposition of nanotubes onto previously patterned electrodes or the deposition of the nanotubes on a substrate and the subsequent patterning of electrodes at positions located by scanning electron microscopy.

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In both cases, the electrodes are called “source” and “drain” in analogy with the standard field effect transistor terminology (more details will be given in Section2.4) and a third electrode called “gate” located below the SWCNTs (see Fig.2.8(a)) is often used to electrostatically induce a carrier density modulation in the tube. In metallic SWCNTs, conduction is almost unaffected by additional charge carriers (see Fig.2.8(b)), while in semiconducting nanotubes, conductance is very strongly modified by this modulation (see Fig.2.8(c)). Theoreticians have predicted that electrical properties of SWCNTs could rival, if not exceed, the best metals or semiconductors known [48,49]. This led to a major interest by both universities and industrial laboratories as most of them have now at least one group studying these properties.

Figure 2.8: (a)Schema of a typical setup for electrical transport measurements of SWCNTs.I_sdsource- drain current as a function ofV_gdgate voltage for(b)a metallic and(c)a semiconducting SWCNT.

V_sd=10 mV in both cases. Adapted from [50,15].

To explain why SWCNTs – which are all formed exclusively of carbon atoms arranged in a hexagonal lattice – can exhibit such very different electrical properties, it is necessary to understand their parent material, graphene, from which they are constructed. Graphene, which is a two-dimensional material based on the same hexagonal carbon lattice, has a very particular electronic band structure. Unlike ordinary metals with a partially filled conduction band and unlike ordinary semiconductors with a bandgap between a full valence band and an empty conduction band, graphene is called a “zero-bandgap semiconductor” or a “semi-metal” because of its filled valence band in direct contact with its empty conduction band, as shown in Fig.2.9. In most directions, electrons moving at the Fermi energy (those that have an energy exactly at the intersection of the valence and the conduction band) are backscattered by the lattice, but in some special directions, they can propagate freely as in a metal. These contact points between the two bands are called Dirac points [48,51].

SWCNTs can be considered as a rolled up sheet of graphene and as such their band structures share some common features. For SWCNTs, the dimension along the tube axis can still be considered as infinite but the tube circumference is a fixed very small value. As a result, the wavevectorkk parallel to the tube axis is still continuous, butk⊥which is perpendicular to the tube axis is strongly quantized, as

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Figure 2.9:Band structure of graphene with a zoom on one of the Dirac points. Adapted from [51].

the possible values are restricted to

k⊥p =2πp/C (2.1)

where pis an integer andCthe circumference of the tube. If one of the allowed values of k⊥ passes exactly through a Dirac cone of graphene (as illustrated in Fig.2.10(a)), the SWCNT will have a linear dispersion relation and no bandgap (see Fig.2.10(b)), and therefore be metallic. In all other cases, a gap between the conduction and the valence band will open and the SWCNT will be a semiconductor (see Fig.2.10(c)). Which case will be realized depends on the (n,m) atomic configuration of the tube. Nanotubes wheren–mis a multiple of 3 will be metallic while others will be semiconducting with a bandgap

Eg≈0.7 eV/d, (2.2)

wheredis the tube diameter expressed in nanometers [50,15]. As often, reality is more complex than this simple derivation of the band structure of SWCNTs based on the one of graphene. Curvature, strain, or twists often have a second order effect which opens a small bandgap of usually less than 100 meV in otherwise metallic nanotubes. This is not big enough to turn such a SWCNT into an insulator but can be seen as a dip in a conductance versus gate voltage plot. Armchair nanotubes, due to their symmetry, are true metallic SWCNTs [15,52]. MWCNTs are predominantly metallic for two reasons. As they have many shells, they statistically also have metallic ones which dominate the electronic properties. Additionally, outer semiconducting shells with diameters of up to 100 nm have an extremely small bandgap, due to the inversely proportional relation described by Eq.2.2[16,53].

The first studies on individual metallic SWCNTs were reported in 1997 [54,55].

To understand their transport characteristics it is useful to employ the Landauer-

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Figure 2.10: (a)First Brillouin zone of graphene with conic energy dispersions at the six corners. The allowedk⊥states in a SWCNT are presented by dashed lines. The band structure of a SWCNT is obtained by cross-sections as indicated. Zoom-ups of the energy dispersion near one of the Dirac cones are schematically shown along with the cross-sections by allowedk⊥states and resulting 1D energy dispersions for(b)a metallic SWCNT and(c)a semiconducting SWCNT. Reproduced from [15].

Büttiker formula for conduction through 1D systems G=Ne²

hT (2.3)

whereGis the conductance,Nthe number of parallel channels, andT the transmission coefficient for electrons through the system. For SWCNTs,Nis 4 due to the spin degeneracy and the orbital degeneracy. The latter can be considered as left- and right-handed electron states spiraling down the tube [50]. In the optimal case,T is 1 and the maximal conductance is then 4e²/h=155µS, or equivalently the minimum resistance is 6.5 kΩ. In this case, the mean free path is longer than the tube length and the electronic transport is said to be ballistic. Although SWCNTs usually have a high quality crystalline structure, various kinds of defects are possible, like vacan- cies, pentagon-heptagon pairs, and impurities such as foreign atoms or chemicals deposited or formed on the surface [15]. In these cases,T will be lower than 1.

Imperfect contacts will add another contribution to the total resistance.

Several groups reported conductance values approaching the theoretical maximum, even at room temperature [56,57], but only for lowVsdsource-drain biases.

Indeed, at low energies there is almost no scattering in structurally perfect metallic SWCNTs, but aboveVsd ∼160 mV, optical and zone-boundary phonons can backscatter electrons [50]. This limits the current in metallic SWCNTs to∼25µA, which corresponds to a current density of 3×10⁹ A/cm² for a 1 nm diameter nanotube. This is nonetheless 6 orders of magnitude higher than the recommended maximum value for standard copper electric wiring.

Semiconducting SWCNTs were first reported in 1998 [58]. Based on the relation

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derived from band theory,Eg≈0.7 eV/d, semiconducting SWCNTs with diameters in the ranged =0.4–3 nm have a bandgap between 0.2 and 1.7 eV. Intrinsic semiconductors, with only excess electrons due to thermal fluctuations, would create ambipolar devices. This means that they could be made conducting either by a positive or a negative bias on the gate electrode. Semiconducting SWCNTs arep-type in ambient conditions because they have only holes as charge carrier and are turned conducting by the application of a negativeVgdgate bias and turned insulating by a positiveVgd. This behavior is due to the metallic electrodes and also to chemical species, in particular oxygen, adsorbed on the tube [49]. Using special treatments, SWCNTs have been turned ton-type, highlighting their sensitivity to the chemical environment and their possible use as sensors [59]. A broader application of semiconducting SWCNTs is as channels in field effect transistors as we will see in Section2.4. This is motivated by the high mobilities that can be reached in clean nanotubes. Values ofµ∼15 000 cm²/Vs at room temperature and even over 100 000 cm²/Vs at 50 K have been reported [60]. Such high values are largely due to the lack of electronic surface states in SWCNTs as opposed to lithographed bulk semiconductors. These electronic states which arise when a three-dimensional crystal is interrupted by a surface are a constant challenge in device miniaturization as they are responsible for degraded operating properties.

SWCNTs solve the problem in an elegant manner: as a rolled up cylinder of a two-dimensional material, there is neither a 3D lattice meeting a surface, nor an edge [48,49].

At room temperature, the one-dimensional nature of SWCNTs enables ballistic transport in metallic nanotubes due to a different scattering mechanism as compared to 2D or 3D materials. In a traditional metal, low energy lattice vibrations backscatter electrons through a series of small angle deviations which eventually reverse their direction. In metallic SWCNTs, this is impossible because only forward and backward propagation is allowed. Acoustic phonons which are the lowest energy lattice vibrations have not enough energy for a complete direction reversal of electrons, and optical and zone-boundary phonons only appear above an energy of∼160 meV, explaining the particularly long mean free path in metallic SWCNTs [49,50]. At low temperatures, their low dimensionality causes interesting quantum phenomena like single-electron charging or quantum interference (see Section2.5). Because the energy scales for these effects typically increase with decreasing system size, the “low-temperature” regime is often more readily reached in nanotubes than in other mesoscopic electronic devices [50]. In the case of single-electron charging, also called “Coulomb blockade” or “Coulomb oscillations”, the electronic states extend over the entire length of the nanotube which behaves like a single quantum dot [61]. A requirement to observe this phenomenon – which was reported since the first measurements on metallic SWCNTs [54,55] – is a high enough contact resistance so that the energy required for a single additional electron in the quantum dot is higher than the thermal fluctuationsk_BT, which explains why the effect is only observed at low temperatures. Only a change of theV_gd gate voltage can allow

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Figure 2.11:The conductance of a metallic nanotube at six different temperatures as a function of gate voltage. At low temperatures the conductance oscillates as individual electrons are added to the tube.

This indicates that the nanotube acts like a long and narrow quantum dot, with electronic states that extend over the entire length of the tube. The average conductance of the nanotubes slowly decreases as the temperature is lowered (see insert). Reproduced from [48].

Figure 2.12:Two-dimensional color plot of the differential conductance, dI/dVsd, versusVsd and negativeV_gdatT=4 K (black is zero, white is 3µS). In the black diamond-shaped regions, the number of charge carriers (yellow numbers) is fixed by Coulomb blockade. Adapted from [62].

additional electrons to tunnel into the nanotube, explaining the oscillations visible in the lowest temperature conductance measurement in Fig.2.11. So-called “Coulomb diamonds” appear when the differential conductance dI/dV_sdis measured as a function of both theV_sdsource-drain andV_gdgate-drain bias, as shown in Fig.2.12. The Coulomb blockade plot is then the cross-section withV_sd ∼0 [15]. In clean and low defect nanotubes, oscillations are regular and periodic. In SWCNTs presenting a shorter electronic mean free path due to chemical impurities or structural defects, the oscillations become much less regular [48].

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2.4 Carbon nanotube-based field effect transistors

2.4 Carbon nanotube-based field e ff ect transistors

Transistors are electric components that can be used as electrically controlled switches or as amplifiers. These semiconducting devices invented at AT&T’s Bell Labs in 1947 by Bardeen, Brattain, and Shockley are the building blocks of any modern electronic equipment [63]. Their importance was quickly noticed and got the inventors the 1956 Nobel Prize in Physics. In the 1960s and 70s, the most commonly used transistor type was thebipolar junction transistor, which is either formed of a thin layer ofp-type semiconductor between twon-type semiconductors (forming a n-p-n transistor), or a thin layer ofn-type semiconductor between two p-type semiconductors (forming a p-n-p transistor). These constructions with two p-n junctions have three terminals corresponding to the three semiconducting layers:

a base, which is connected to the central layer, a collector, and an emitter. A small current flowing between the base and the emitter can control or switch a much larger current between the collector and the emitter [64].

Figure 2.13:Schema of ann-type silicon-based FET.(a)With no applied voltage, the transistor is OFF.

(b)With an applied voltage on the gate terminal, ann-doped channel forms between the source and drain terminals and the transistor turns ON. Adapted from [65].

Another transistor geometry that captured almost all market shares since the development of integrated circuits is thefield effect transistor(FET). In contrast to a bipolar junction transistor where both electrons and holes are used for conduction, a FET uses either only electrons (n-type) or only holes (p-type) for conduction, which is why it is also calledunipolar transistor. The three terminals of a FET are labeled source, drain, and gate. Fig.2.13(a)shows ann-type FET in a modern integrated circuit where the transistor substrate is formed byp-doped silicon and the source and drain electrodes byn-doped silicon. The gate is electrically separated from the substrate by an insulating SiO2layer. In this configuration, the transistor is said to be OFF because no current can travel through bothnandp-doped regions.

Fig.2.13(b)shows the situation where a positive voltage is applied between the

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gate and the drain terminals. The electric field that is then produced generates a migration of electrons from the substrate to the region right under the gate, forming a negatively doped channel between the source and the drain electrodes which allows a current to flow and so turns the transistor to the ON state. In ap-type FET, the substrate isn-doped whereas the source and drain electrodes arep-doped, and so a negative voltage to the gate generates ap-doped channel which turns the transistor ON.

The first carbon nanotube-based field effect transistors (CNT-FETs) were reported at the same moment as the semiconducting behavior of SWCNT [58], and were recognized as a promising candidate to replace the standard silicon-based FET due to their intrinsic nanometric size. Early CNT-FETs were relatively simple in structure and performed far below the FETs based on silicon technology, which had over 50 years of technological head start [66]. SWCNTs were deposited from solution onto lithographically patterned noble metal electrodes that served as source and drain, the SWCNT being the active channel of the FET. Doped silicon served as the gate electrode, separated from the nanotube by a thick SiO2layer. These devices displayed ap-type behavior with a source-drain current modulated by several orders of magnitude, but ON currents far from the previously described theoretical maximum due to high parasitic contact resistances. They also showed high subthreshold slopes, defined asS =h

d(log₁₀I_sd)/dV_gd)i−1

, on the order of 1 V/decade, and thus requiring several volts to be turned ON or OFF.

After the initial pioneer studies showing the proof of concept, rapid improvement of the device characteristics were reported. Rather than relying on weak van der Waals forces for the contact between CNTs and electrodes, the nanotubes were deposited first and the electrodes patterned on top of them, yielding much lower contact resistances. Another approach that continued with the nanotube-on-top scheme was to thermally anneal the device to improve the metal–nanotube contact.

The use of a top gate, as shown in Fig.2.14(a), instead of the doped substrate required additional processing steps, but showed superior device performance like subthreshold slopes of only 130 mV/decade due to better gate-CNT coupling and brought the advantage of independent addressing of individual devices [66].

Figure 2.14: (a)Schematic cross-section of a top-gated CNT-FET. Electrical characteristics of a back- gated CNT-FET(b)before and(c)after passivation with a PMMA layer. Adapted from [66,67].

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2.4 Carbon nanotube-based field effect transistors

Another undesirable feature of early devices was a significant hysteresis in the Isdsource-drain current versusVgd gate voltage characteristics when the back gate was swept back and forth [67], as shown in Fig.2.14(b). The size of the hysteresis depended on various parameters like sweeping speed, sweeping amplitude, and relative humidity, and thus was an issue for transistor applications in a logic circuit.

Nonetheless, as suggested by Fuhreret al.[68], it could be used to create memory devices. Several mechanisms explaining this hysteresis were proposed, but Kim et al.[69] showed in an extensive study that the dominant effect is the adsorption of water on the surface of the SiO2 substrate. Water molecules with a built-in dipole can serve as charge traps and cannot be entirely removed by pumping in vacuum because they are weakly chemically bonded to the SiO₂. Heat treatment above 200 °C in vacuum strongly reduced the hysteresis, but the improvement disappeared as soon as the device was exposed again to ambient air. A more durable passivation of the CNT-FET could be obtained by spin-coating a thin layer of poly(methyl methacrylate) (PMMA), a common lithography resist. The device was then additionally baked at 180 °C for 12 hours. After this passivation process, the hysteresis completely disappeared, as shown in Fig.2.14(c), at least for lowVgd

gate voltages and low relative humidities [69].

Currently, it is possible to realize single sub-10 nm CNT-FET which outperform the best competing silicon devices [70]. The challenge is now more in the assembly of individual CNT-FET into more complex circuits in order to attain improved functionalities. The presentation of ring oscillators [71] and logic gates [72] were certainly milestones on the road to the first CNT-based computer which was presented in 2013 by Shulakeret al.[73]. To realize their computer composed of 178 CNT-FETs connected in a maximum of seven stages of cascaded logic, they used an array of highly aligned CNTs and a clever imperfection-immune design methodology [74]. Indeed, the difficulty in building complex CNT-based circuits is to overcome the unavoidable imperfections, both in the positioning of the nanotubes and in their intrinsic properties. To achieve this, the authors used between 10 and 200 CNTs to form each CNT-FET, the average behavior being less sensitive to defects than an individual CNT would have been. They also etched away mispositioned CNTs and removed any metallic CNTs from the array by electrical breakdown. In this technique, all semiconducting CNTs are turned OFF by an appropriate bias voltage while a large enough current is made to flow through the metallic CNTs to destroy them by Joule heating. The result might not be able to compete with current silicon-based processors, but it is the first demonstration of a CNT-based computer that runs programs and is programmable. From a technological point of view, it is very interesting that the device works with no per-unit customization, being thus compatible with very large-scale integration processes.

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2.5 Universal Conductance Fluctuations

At room temperature, the wave-like behavior of electrons is mostly invisible, due to phase randomization caused by all kinds of inelastic scattering. This induces a phase coherence length much smaller than the sample size, too small to have any experimental incidence. As the temperature is lowered, and especially in low dimensional systems, the phase coherence length can reach the same size as the sample. The wave character of electrons becomes thus experimentally observable and gives rise to interesting phenomena. In mesoscopic electric channels of sizeL, with a phase coherence lengthLϕof the same order of magnitude asL, but much bigger than the elastic mean free pathle, one has to consider the sum over all possible scattering trajectories through the sample. The various paths do not ensemble average to zero due to quantum interference effects, thus giving a correction to the conductanceG[75–79]. The interferences are very sensitive to variations of global variables like the external magnetic field or the electron density, driven by the gate voltage. A measurement of the conductance as a function of such global variable yields thus reproducible, aperiodic fluctuations which are sample specific. The particular oscillatory pattern of the fluctuations, which depends on the exact scatterer positions, can be considered as a fingerprint of the sample. More surprisingly, it has been shown that atT =0 and forL_ϕ>L, the root-mean-square amplitude of the fluctuationsδG_rmsis on the order ofe²/h, regardless of the exact size of the sample and the degree of disorder. This is why the phenomenon is called Universal Conductance Fluctuations(UCF) [80,81]. IfL_ϕ<L, the amplitude of the fluctuations is reduced belowe²/h. This can happen for several reasons, for example if the temperature is too high, the sample not clean enough, having thus too many inelastic scattering centers, or if the voltage across the channel is too high, triggering additional scattering mechanisms like optical phonons. In all those cases, the channel effectively subdivides intoL/Lϕuncorrelated phase-coherent segments connected in series, reducing the amplitude of the fluctuations by a ratio (Lϕ/L)^3/2[76].

As discussed in Section2.3, conduction through clean metallic CNTs is ballistic.

However, in CNTs of lower crystalline quality or on rough substrates promoting adsorbed impurities, the conduction is diffusive and the conditionLleis met. As a result, UCF could be observed in bundles of MWCNTs [82], individual MWCNTs [83], and individual SWCNTs [57]. Due to the reduced size and dimensionality of CNTs, UCF can be observed at higher temperature than in other systems, up to 100 K. Fluctuations are generally similar during the same thermal cycle, but completely different after the system has been heated to room temperature and cooled down again. This is a strong indication that at least part of the scattering centers are not intrinsic to the CNTs, but related to environmental factors like adsorbed chemical species [57].

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CHAPTER 3 Ferroelectric materials

3.1 Ferroelectricity

3.1.1 Polarization

When exposed to an external electric field, matter can react differently. If there are free electrons, the material is a conductor and its electrons will move around in such a manner as to create an internal electric field of the same magnitude but opposite direction as the external one, resulting in a total electric field which is zero inside the conductor. If there are no free electrons, the material is an insulator, also called a dielectric. Although all atoms are electrically neutral, the electric field will still have an effect, as the nucleus and the electrons both have a well-defined electric charge. The positive nucleus will be displaced in the direction of the field while the negative electrons will be displaced in the opposite direction, each atom thus becoming a small dipole and the material is said to be polarized. The dipole moment p, which is the physical value quantifying this polarization, is defined as the charge of the nucleus multiplied by the distance between the nucleus and the center of the electrons cloud and has the unitsC·m. At the macroscopic scale, polarization is better described by the dipole moment per unit volumePwith units C·m/m³=C/m², which is charge per unit surface. This is in fact a link to another way to represent the polarization: along the field lines, the positive and negative charges of all the aligned dipoles effectively cancel each other, except for the two charges at both extremities of the object which remain uncompensated. The surface density of these charges, calledbound chargesbecause they are linked to their

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respective atoms, has thus the same value and units as the polarization in the bulk of the material [84].

3.1.2 Spontaneous polarization

Figure 3.1:Hysteresis in an idealP–Eloop showing the switching of the polarization under the application of an electric field.

For many materials calledlinear dielectrics, the polarizationPis proportional to the electric field E, as long as the latter is not too strong. For ferroelectric materials, the relation betweenPandEis strongly non-linear as shown by the hysteresis loop in Fig.3.1. The microscopic origin of this behavior will be discussed in Section3.2. When starting from an initial unpolarized state, the polarization first rapidly increases and then gradually tends towards a saturation value as the electric field is ramped up. With the diminution of the field, the polarization decreases but remains at a finite value+Prcalledremanent polarizationatE =0. After an increase of the field in the opposite direction, the polarization reverses once the field reaches the intensity−Ec, called thecoercive field. As the field is brought back toE = 0, the polarization stays at a finite negative value−Pr. A positive coercive value+Ecis required to switch the polarization to positive values.Psis the spontaneous polarization and is usually defined as the extrapolation to zero field of the polarization values at high fields. In polycrystalline materials,Psis usually higher thanPr, but in single crystals, the two values can be almost identical [85]. In some materials, a built-in electric field shifts the hysteresis loop along the electric field axis, a phenomenon calledimprint. The behavior described in this paragraph is in fact the perfect example of the formal definition of a ferroelectric material: it must have two different, stable, non-zero polarization states in the absence of an electric field and an experimentally realizable external electric field has to be able to switch the polarization from one state to the other [86]. It is interesting to remark that the prefixferroin ferroelectricity has nothing to do with iron. It comes from

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3.1 Ferroelectricity

an analogy with ferromagnets, a class of materials presenting a similarly shaped hysteresis loop in the magnetization responsevs.applied magnetic field. As iron is the prototypical material for this class, it gave its name to the phenomenon.

3.1.3 Thermodynamic description

Figure 3.2:Spontaneous polarizationP_sas a function of temperatureTfor a system undergoing(a)a first-order phase transition and(b)a second-order phase transition at the Curie temperatureT_c.

The magnitude of the spontaneous polarizationPsdepends on the temperature.

In some ferroelectric materials, the polarization will drop abruptly at a critical tem- peratureTccalledCurie temperature, undergoing what is called afirst-order phase transition(see Fig.3.2(a)) while in others, the decrease will be progressive until the polarization reaches zero, a process known as asecond-order phase transition (see Fig.3.2(b)). Above the Curie temperature, the material is in the paraelectric phase while below it is in the ferroelectric phase. The Landau-Ginzburg-Devonshire formalism [87,88] is the usual approach to describe these phase transitions from a macroscopic perspective. In the vicinity ofTc, the free energyFof the system can be expanded as a polynomial function of the polarizationP:

F(P)=F0+αP²+γP⁴+δP⁶−EP (3.1) whereF0is the free energy of the unpolarized state,α,γ, andδare coefficients to be determined phenomenologically, andEis the electric field. In this formalism, close to a transition temperatureT0, it is assumed that the quadratic term depends linearly on the temperature withα=β(T −T0) and that the other terms are temperature independent. Moreover,βis positive for every known ferroelectrics [88] andδ has to be positive to ensure a positive energy when the polarization tends towards infinity. AtE=0, the qualitative behavior ofF(P) depends on the temperature and the sign ofγ.

The case ofγ <0 is shown in Fig.3.3(a). For temperatures belowT₀, there are two local minima atP ,0, shown with green points. As the temperature is increased to values betweenT0 andT_c, a third local minima at P = 0 appears, but the two minima atP,0 stay thermodynamically favorable. AtT =Tc, the three minima are energetically degenerate and above the Curie temperatureTc, the

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paraelectric, unpolarized state at P = 0 becomes thermodynamically favorable, explaining the abrupt first-order transition in thePs(T) curve shown in Fig.3.2(a).

Note that during cooling the system can stay in the unpolarized, metastableP=0 state untilT =T0, unless an electric field or thermal fluctuations give it enough energy to reach one of the twoP,0 states. BelowT0,P=0 is no longer a local minimum and the system will become ferroelectric in any case. The difference betweenTcandT0explains the temperature hysteresis of the transition temperature which depends on whether the material is heated or cooled.

Figure 3.3:Non-constant free energyF−F₀as a function of the polarizationPusing the polynomial expression of Eq.3.1withE=0 for various temperatures. The green points show the local minima.

In(a)γis negative and the ferroelectric to paraelectric phase transition is first-order while in(b)γis positive and the transition second-order.

The case ofγ >0, shown in Fig.3.3(b), is easier. For all temperatures below T_c, there are only two local minima at P , 0 and the material is ferroelectric.

Their positions on the horizontal polarization axis continuously approachP=0 as the temperature is increased towardsT_c. At the Curie temperature, the curve is perfectly flat at P = 0 which becomes the only local minimum, leading to a smooth second-order phase transition towards the paraelectric state, as shown in Fig.3.2(b). For temperatures aboveTc,F(P) does not qualitatively change. In this case,T0 =Tcand no thermal hysteresis in the transition temperature is expected.

3.1.4 Crystallographic conditions

The existence of a well-defined polarization axis in the material in the absence of any external electric field imposes some conditions on the allowed symmetries of the unit cell. Indeed, the presence of an inversion center prevents any spontaneous polarization. From the 32 existing crystallographic point groups, 21 are non-centrosymmetric. Among these 21 point groups, all except the cubic group 432 arepiezoelectric, which means that the materials develop a surface charge in response to applied mechanical stress and undergo a deformation in response to an external electric field. The former phenomenon is called the piezoelectric

Probing nanoscale limits of polarization switching and controlling electronic properties in devices combining carbon nanotubes and ferroelectrics

Thesis

Reference

Probing nanoscale limits of polarization switching and controlling electronic properties in devices combining carbon nanotubes and

ferroelectrics

BLASER, Cédric

BLASER, Cédric. Probing nanoscale limits of polarization switching and controlling electronic properties in devices combining carbon nanotubes and ferroelectrics . Thèse de doctorat : Univ. Genève, 2014, no. Sc. 4711

URN : urn:nbn:ch:unige-418765

DOI : 10.13097/archive-ouverte/unige:41876

Probing nanoscale limits of

polarization switching and controlling electronic properties in devices combining carbon nanotubes and

ferroelectrics

THÈSE

Cédric Blaser

Contents

Résumé

CHAPTER 1

Introduction

1.1 General introduction

1.2 The present study

CHAPTER 2

Carbon nanotubes

2.1 Structural properties

2.2 History, synthesis, and imaging methods

2.2.1 Discovery of carbon nanotubes

2.2.2 Synthesis methods

2.2.3 Imaging methods

2.3 Electronic properties

2.4 Carbon nanotube-based field e ff ect transistors

2.5 Universal Conductance Fluctuations

CHAPTER 3

Ferroelectric materials

3.1 Ferroelectricity

3.1.1 Polarization

3.1.2 Spontaneous polarization

3.1.3 Thermodynamic description

3.1.4 Crystallographic conditions