UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS FÍSICAS Departamento de Física de Materiales TESIS DOCTORAL Electronic and thermal properties of Graphene nanoestructures Propiedades electrónicas y térmicas de nanoestructuras de grafeno MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR Marta Saiz Bretín Directores Francisco Domínguez-Adame Acosta Andrey V. Malyshev Madrid Ed. electrónica 2019 © Marta Saiz Bretín, 2019 Electronic and Thermal Properties of Graphene Nanostructures Propiedades Electrónicas y Térmicas de Nanoestructuras de Grafeno By MARTA SAIZ BRETÍN Under the supervision of FRANCISCO DOMÍNGUEZ-ADAME ACOSTA and ANDREY V. MALYSHEV Departmento de Física de Materiales UNIVERSIDAD COMPLUTENSE DE MADRID A thesis submitted for the degree of DOCTOR OF PHILOSOPHY JUNE 2019 Table of Contents Page List of Publications v Abbreviations vii Abstract ix Resumen xiii 1 Introduction 1 1.1 Electronic Properties of Graphene . . . . . . . . . . . . . . . . . 2 1.2 Thermal Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Thermoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Objectives and Outline . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Conductance Quantization in Graphene Nanoconstrictions 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 System, Model and Methodology . . . . . . . . . . . . . . . . . . 15 2.3 Single Nanoconstriction . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Impact of Edge Orientation . . . . . . . . . . . . . . . . . 17 2.3.2 Extrapolation to Larger Systems . . . . . . . . . . . . . 19 2.3.3 Role of the Nanoconstriction Shape . . . . . . . . . . . . 20 2.4 Rough Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Extrapolation to Larger Systems . . . . . . . . . . . . . 25 2.5 Multiple Nanoconstrictions . . . . . . . . . . . . . . . . . . . . . 27 2.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 i TABLE OF CONTENTS 3 Twisted Graphene Nanoribbons as Non-Linear Electronic Devices 33 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 System, Model, and Methodology . . . . . . . . . . . . . . . . . . 35 3.3 Structural Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Electron Transmission . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.1 Impact of Structural Relaxation . . . . . . . . . . . . . . 41 3.4.2 Effect of a Transverse Electric Field . . . . . . . . . . . 43 3.5 Current-Voltage Characteristics . . . . . . . . . . . . . . . . . . 45 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Spin-Polarized Transport in Graphene Nanorings 49 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 System, Model and Methodology . . . . . . . . . . . . . . . . . . 50 4.3 Non-Ferromagnetic Rings . . . . . . . . . . . . . . . . . . . . . . 53 4.3.1 Simplified Model . . . . . . . . . . . . . . . . . . . . . . . 55 4.4 Ferromagnetic Rings . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4.1 Effect of the Ferromagnetic Layer . . . . . . . . . . . . . 56 4.4.2 Side-Gate Control . . . . . . . . . . . . . . . . . . . . . . 58 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5 Thermoelectricity in Graphene Nanorings: Electronic Contribution 63 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 System, Model and Methodology . . . . . . . . . . . . . . . . . . 65 5.3 Thermoelectricity in Graphene Rings . . . . . . . . . . . . . . . 67 5.4 Effect of a Side-Gate Voltage . . . . . . . . . . . . . . . . . . . . 70 5.5 Edge Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6 Molecular Dynamics Study of the Lattice Thermal Conductivity in Graphene Nanostructures 75 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2 System, Model and Methodology . . . . . . . . . . . . . . . . . . 77 6.3 Influence of the Contacts . . . . . . . . . . . . . . . . . . . . . . . 81 6.4 Thermal Conductivity in Graphene Nanorings . . . . . . . . . . 83 ii TABLE OF CONTENTS 6.4.1 Role of Dimensions . . . . . . . . . . . . . . . . . . . . . . 83 6.4.2 Effect of Temperature . . . . . . . . . . . . . . . . . . . . 85 6.4.3 Edge Disorder and Functionalization . . . . . . . . . . . 87 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7 Impact of the Device Geometry on Electron and Phonon Transport in Graphene Nanorings 91 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 System, Model and Methodology . . . . . . . . . . . . . . . . . . 93 7.3 Phonon Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.4 Electron Transport and Thermoelectric Response . . . . . . . . 100 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8 Conclusions 105 A Quantum Transmitting Boundary Method and Transfer Matrix Method 111 B Density-Functional Tight-Binding Method 117 C Molecular Dynamics 121 Bibliography 125 iii List of Publications Publications Related to This Thesis 1. Enhancing thermoelectric properties of graphene quantum rings M. Saiz-Bretín, A. V. Malyshev, P. A. Orellana, and F. Domínguez- Adame Physical Review B 91, 085431 (2015). 2. Control of spin-polarised currents in graphene nanorings M. Saiz-Bretín, J. Munárriz, A. V. Malyshev, and F. Domínguez-Adame Physics Letters A 379, 2102 (2015). 3. Lattice thermal conductivity of graphene nanostructures M. Saiz-Bretín, A. V. Malyshev, F. Domínguez-Adame, D. Quigley, and R. A. Roemer Carbon 127, 64 (2018). 4. Quantized electron transport through graphene nanoconstrictions V. Clericò, J. A. Delgado-Notario, M. Saiz-Bretín, C. Hernández Fuentevilla, A. V. Malyshev, J. D. Lejarreta, E. Diez, and F. Domínguez-Adame Physica Status Solidi A 215, 1701065 (2018). 5. Twisted graphene nanoribbons as nonlinear nanoelectronic devices M. Saiz-Bretín, F. Domínguez-Adame, and A. V. Malyshev Accepted for publication in Carbon 6. Impact of the device geometry on electron and phonon transport in graphene nanorings M. Saiz-Bretín, L. Medrano Sandonas, R. Gutierrez, G. Cuniberti, and F. Domínguez-Adame Accepted for publication in Physical Review B v LIST OF PUBLICATIONS 7. Quantum nanoconstrictions fabricated by cryo-etching in encapsulated graphene V. Clericò, J. A. Delgado-Notario, M. Saiz-Bretín, A. V. Malyshev, Y. M. Meziani, P. Hidalgo, B. Méndez, M. Amado, F. Domínguez-Adame, and E. Diez Under review 8. Valley-symmetry preserved transport in smooth graphene nanoconstrictions M. Saiz-Bretín In Preparation Other Publications 1. Interactions and thermoelectric effects in a parallel-coupled double quantum dot M. A. Sierra, M. Saiz-Bretín, F. Domínguez-Adame, and D. Sánchez Physical Review B 93, 235452 (2016). 2. Spin-dependent electronic lenses based on hybrid graphene nanostructures Y. Baba and M. Saiz-Bretín Under review 3. Tuning the thermoelectric response of silicene nanoribbons with monovacancies C. Núñez, M. Saiz-Bretín, P. A. Orellana, L. Rosales, and F. Domínguez- Adame Under review vi Abbreviations 2D Two-Dimensional 3D Three-Dimensional AB Aharonov-Bohm AIREBO Adaptive Intermolecular Reactive Empirical Bond-Order AFM Atomic Force Microscope aGNR Armchair Graphene Nanoribbon CNT Carbon Nanotube DFT Density Functional Theory DFTB Density-Functional Tight-Binding EIP Empiric Interatomic Potential FI Ferromagnetic Insulator GNC Graphene Nanoconstriction GNR Graphene Nanoribbon hBN Hexagonal Boron Nitride LAMMPS Large-scale Atomic/Molecular Massively Parallel Simulator MD Molecular Dynamics mfp Mean Free Path NEMD Non-Equilibrium Molecular Dynamics vii ABBREVIATIONS QPC Quantum Point Contact QTBM Quantum Transmitting Boundary Method REBO Reactive Empirical Bond-Order SEM Scanning Electron Microscope TB Tight Binding TMM Transfer Matrix Method zGNR Zigzag Graphene Nanoribbon viii Abstract Nanoscience is one of the most exciting research fields of modern science due to the novel and unexpected properties that materials exhibit at the scale of individual atoms. In order to exploit these properties in future applications, a deep understanding of the behavior of matter at the nanoscale is required. This thesis aims to present new findings related to the modeling of elec- tronic and thermal transport in several graphene nanostructures, including graphene nanoconstrictions, twisted graphene ribbons and graphene rings. Graphene Nanoconstrictions Among the exceptional electronic properties of graphene, the extremely long mean free path of charge carriers stands out, making this material an ideal system to study electronic phenomena in the coherent regime. Conductance quantization in integer multiples of the fundamental quantum 2e2/h is one of the hallmarks of ballistic quantum transport. In the case of graphene, the valley degree of freedom can also result in 4e2/h conductance steps. In this thesis, the influence of several geometric factors in the conductance quan- tization of graphene nanoconstrictions is thoroughly analyzed, finding the valley-symmetry is more easily preserved in the presence of zigzag edges, short constrictions and rounded-shaped profiles. Rough edges are found to deeply degrade electronic transport, although signs of quantization are still noticeable for low edge roughness. Additionally, an effective approach to ex- trapolate results from small systems to experimentally feasible sized systems is derived and tested by comparing with experimental results. These find- ings could result in the design of more optimized tailored structures able to preserve the size quantization and the valley symmetry. Twisted Graphene Nanoribbons The unique geometry of helical conformations can give rise to peculiar proper- ties, especially when these structures are subjected to an electric field. Due to ix ABSTRACT its remarkable mechanical properties, graphene nanoribbons can be shaped into different geometries, including twisted conformations. In this thesis we analyze the mechanical deformation of twisted graphene ribbons and the impact of the induced strain on their electronic properties, demonstrating its effect can be neglected in relevant cases. Additionally, when a transverse elec- tric field is applied, the twisted conformation induces a periodic potential for charge carriers along the nanoribbon, resulting in a superlattice-like energy band structure. We show that the current-voltage characteristics of this type of system can be controlled by the electric field, allowing the twisted ribbon to operate as a field-effect transistor or as a tunnel diode. Graphene Nanorings The long coherence length of charge carriers in graphene also allows to study electron quantum interference, being graphene nanorings one of the most suitable nanostructures to this end. Previous studies demonstrated that tuning the phase difference between electrons traveling along different arms of the ring with magnetic fields or electrostatic gates can result in destructive and constructive interference, and thus, in a way of controlling the electric current. In this thesis, we find that an asymmetric connection between the ring and the contacts, or alternatively the application of a side-gate voltage accross the arms of the ring, can give rise to sharp features and Fano resonances in the electron transmission spectra. We demonstrate these characteristics can result in a highly spin-polarized transmission when a ferromagnetic insulator is placed on top of the graphene nanoring. Also, Fano resonances are found to have a great impact on the thermoelectric properties of graphene rings. Although thermal phenomena are not directly responsible for the electrical functionality of devices, they can limit their performance, making essential the search for superior thermal conductors capable of heat removal. On the other hand, thermoelectric energy conversion requires materials with a strongly suppressed thermal conductivity. Therefore, a better understanding of heat transport at the nanoscale is crucial. In this regard, we analyze the thermal properties of graphene nanorings, both at high and low temperature. At high temperature the quantum effects can be neglected and molecular dynamics simulations are the most suitable tool. On the contrary, at low temperature and in the absence of phonon-phonon interactions, density-functional tight- binding simulations in combination with Green function’s techniques are employed. We obtain that above a threshold temperature, which depends on the specific geometry configuration, the presence of bends, corners, or a hole in the path from source to drain, strongly degrade the thermal con- ductance as compared to uniform ribbons. Below the threshold temperature, graphene nanorings can show a higher thermal conductance than ribbons of x ABSTRACT the same width. We suggest and provide some evidence of phonon interference to be responsible for this counter-intuitive behavior, opening new possibilities of engineering phonon transport in a similar way as electron transport in nanorings. In conclusion, we demonstrate that graphene nanorings are highly versa- tile structures which can be employed for spintronics, thermal managament and thermoelectric applications. Electron transport in such systems can be engineering by using quantum interference effects while phonon conductance can be greatly modified with an appropriate design. xi Resumen La Nanociencia es una de las disciplinas de mayor crecimiento y relevancia en la actualidad debido a las nuevas e inesperadas propiedades que exhiben los materiales al reducir sus dimensiones. Sin embargo, para poder explotar estas propiedades de forma óptima en posibles aplicaciones, aún se necesita una mayor comprensión del comportamiento de la materia en la nanoescala. El objetivo principal de esta tesis es estudiar el transporte electrónico y térmico en distintas nanoestructuras de grafeno, entre las que se incluyen nanoconstricciones, nanocintas de grafeno helicoidales y anillos de grafeno. Nanoconstricciones de Grafeno Entre las excepcionales propiedades electrónicas del grafeno destaca el gran recorrido libre medio de los electrones, haciendo de este material un sistema ideal para estudiar fenómenos electrónicos en el régimen balístico y coherente. La cuantización de la conductancia en múltiplos del cuanto fundamental 2e2/h es una de las señas de identidad del transporte cuántico balístico. En el caso del grafeno, además, debido al grado de libertad de valle, la cuantización puede resultar en algunos casos en múltiplos de 4e2/h. En esta tesis anali- zamos el papel que juegan distintos factores geométricos en la cuantización de la conductancia en nanoconstricciones de grafeno. Entre los principales resultados encontramos que la simetría de valle se conserva más fácilmente cuando consideramos bordes de tipo zigzag, en las constricciones más cortas y en los perfiles más redondeados. También encontramos que los bordes rugosos degradan el transporte electrónico, aunque aún se observan algunos signos de cuantización cuando la rugosidad es baja. Por último, desarrollamos un nuevo y efectivo método que nos permite extrapolar resultados de sistemas pequeños a sistemas con tamaños experimentalmente realistas. La validez del método es comprobada comparando con medidas experimentales. Estos hallazgos podrían resultar en el diseño y fabricación de estructuras más óptimas y capaces de preservar tanto la cuantización de la conductancia como la simetría de valle. xiii RESUMEN Nanocintas de Grafeno Helicoidales La geometría única de las estructuras helicoidales puede dar lugar a propieda- des muy peculiares, especialmente cuando estas configuraciones son sometidas a un campo eléctrico. Debido a las excepcionales propiedades mecánicas del grafeno, las nanocintas de este material pueden moldearse dando lugar a distintas geometrías, como por ejemplo a configuraciones helicoidales. En esta tesis analizamos la deformación mecánica de las cintas de grafeno helicoidales y su impacto sobre las propiedades electrónicas, demostrando que su efecto es despreciable en los casos más relevantes. Además, observamos que la apli- cación de un campo eléctrico transversal da lugar a una estructura de bandas similar a la de una superred. Finalmente, demostramos que este tipo de sis- temas pueden ser de gran utilidad para aplicaciones en nanoelectrónica, ya que el campo eléctrico nos permite obtener curvas corriente-voltaje similares a las de los transistores de efecto campo o los diodos túnel. Anillos de Grafeno La gran longitud de coherencia de los portadores de carga en grafeno también nos permite estudiar efectos de interferencia cuántica, siendo los anillos de grafeno una de las nanoestructuras más apropiadas para este fin. Estudios previos han demostrado que la diferencia de fases entre los electrones que viajan por distintos brazos del anillo puede ser controlada mediante campos magnéticos y electrostáticos, lo que resulta en interferencias constructivas y destructivas y, por tanto, en una forma de controlar la corriente eléctrica a través del sistema. En esta tesis hemos demostrado que una conexión asimétrica entre los contactos y el anillo, o alternativamente la aplicación de un potencial lateral entre ambos brazos, puede dar lugar a formas abruptas y resonancias Fano en el espectro de transmisión electrónica. Estas característi- cas, por su parte, pueden resultar en una transmisión altamente polarizada en espín cuando se deposita un material ferromagnético sobre el anillo de grafeno. También, encontramos que las resonancias Fano tienen un gran impacto sobre las propiedades termoeléctricas de estos sistemas. Aunque los fenómenos térmicos no son directamente responsables de la respuesta electrónica de un dispositivo, pueden limitar su funcionamiento, lo que hace esencial la búsqueda de sistemas con alta conductividad térmica. Por otro lado, la tecnología termoeléctrica requiere de materiales con una conductividad térmica muy reducida. Por tanto, una mejor comprensión del transporte térmico en la nanoescala es fundamental. En esta tesis analizamos las propiedades térmicas de los anillos de grafeno, tanto a alta como a baja temperatura. A alta temperatura los efectos cuánticos pueden ser despreciados y las simulaciones de dinámica molecular constituyen la herramienta más xiv RESUMEN adecuada. Por el contrario, a bajas temperaturas y en ausencia de interaccio- nes fonón-fonón, utilizamos simulaciones de DFTB (de la acepción inglesa density-functional tight-binding) junto con el formalismo de las funciones de Green. Los resultados que obtenemos indican que por encima de una tempe- ratura umbral, que va a depender de la geometría específica, la presencia de obstáculos geométricos (curvas, agujeros o esquinas) degrada fuertemente la conductancia térmica. Por debajo de la temperatura umbral, los anillos de grafeno presentan un comportamiento inesperado, siendo mejores conductores térmicos que las nanocintas de la misma anchura. Proponemos la interferencia de fonones como posible explicación de este fenómeno contraintuitivo. En conclusión, demostramos que los anillos de grafeno son unas estruc- turas muy versátiles con potencial para aplicaciones en espintrónica, gestión térmica o termoelectricidad. El transporte electrónico en estos sistemas puede explotar los efectos de interferencia mientras que el transporte térmico o de fonones puede ser modificado mediante un diseño adecuado. xv C H A P T E R 1 Introduction Nanoscience is one of the most exciting research fields of modern science. At the scale of individual atoms, materials and systems exhibit novel and unex- pected properties. Therefore, a deep understanding of the behavior of matter at the nanoscale is an essential step towards future applications of nanotech- nology. In this regard, it is widely believed that nanotechnology will lead to the next industrial revolution, comparable to that of semiconductor technol- ogy in the past century. Nanoscience and nanotechnology research promise breakthroughs in a wide range of areas, including materials, nanoelectronics, biotechnology, medicine, energy, or information technology [1]. Low dimensional materials such as graphene, which consists of a single- layer of carbon atoms discovered in 2004 [2], are expected to play a key role. Devices made of this material can be extremely compact. But most impor- tantly, the extraordinary properties of graphene allow for new devices and applications that could never have been anticipated. Electronic applications that could exploit its high charge-carrier mobility [3] were among the first to be envisaged [4, 5]. Flexible, wearable or transparent electronics that take advantage of graphene’s superior electronic properties, as well as its optical and mechanical features have also been envisioned as promising applica- 1 CHAPTER 1. INTRODUCTION tions [6, 7]. The potential of graphene photonics and optoelectronics have been demonstrated by advances ranging from solar cells and light-emitting devices to photodetectors and ultrafast lasers [6]. Additionally, this material is of interest in spintronics [8] because of the extraordinarily long spin-coherence length [9]. Lastly, its extremely high thermal conductivity [10, 11] and the possibility of modifying the thermal transport by different mechanisms, make graphene an ideal candidate for thermal management and thermoelectric applications [12]. Over the years, a variety of experimental techniques have been developed to synthesize, fabricate and characterize nanometer-sized objects. Manipulat- ing nanoscale matter is a challenging task though. Hence, theoretical models and numerical simulations are crucial to make progress in the fundamental knowledge of nanoscale physical properties, as well as to optimize the struc- tures and devices. It is the aim of this thesis to present new findings related to the modeling of electronic and thermal transport in graphene nanostructures, in which fascinating effects arise since its dimensions are smaller than the characteristic lengths associated with the physical phenomena. 1.1 Electronic Properties of Graphene In order to understand the electronic properties of graphene, it is necessary to start by analyzing the atomic structure of this carbon-based material. The carbon atom has six electrons, two of them forming a closed 1s2 shell and four filling 2s and 2p states. In graphene, three of the outer electrons form hybrid sp2 covalent bonds separated by angles of 120◦. These are the σ bonds that lead to the honeycomb lattice of carbon atoms separated by a distance of 1.42Å. The remaining electron, corresponding to the non-hybridized p orbital perpendicular to the plane, forms a π bond and it is responsible for the electric conductivity in the system. The crystal structure comprises two non- equivalent sublattices (see Figure 1.1) and results in a unique band structure, in which the valence and conduction bands touch conically at the points K and K ′ at the corners of the graphene Brillouin zone (see Figure 1.2). These two non-equivalent points are called Dirac points and are the origin of the unique electronic properties of graphene [13, 14]. The low energy spectrum in the vicinity of the K and K ′ points exhibits a linear dispersion that can be 2 1.1. ELECTRONIC PROPERTIES OF GRAPHENE FIGURE 1.1. (a) The graphene honeycomb lattice. Sublattices A and B are shown in gray and white, respectively. (b) Brillouin zone of graphene. The Dirac points are located at K and K ′ points. FIGURE 1.2. Electronic dispersion in the honeycomb lattice. Zoom of the energy bands close to one of the Dirac points. well described by the massless Dirac equation [13, 14]. At higher energies or large momentum, however, the energy spectrum and the transport properties have to be calculated using the tight-binding (TB) approximation [15, 16] or, more accurately, ab initio methods. Comparing with the latter, the TB approximation allows to handle much larger systems and will be the tool of choice to model electronic properties throughout this thesis. The band structure and the electronic properties of this material are very sensitive to the specific geometry, size, and the nature of the edges, namely, zigzag or armchair. For example, within the single-orbital TB model, ribbons 3 CHAPTER 1. INTRODUCTION with zigzag edges are metallic, while those with armchair edges are either metallic or semiconducting, depending on the ribbon width [17, 18]. Also, atomic size fluctuation and especially rough edges, which are unavoidable in real samples yet, can have a great impact on the electronic properties [19, 20]. Its effect on the electrical conductance of graphene nanoconstrictions (GNCs) will be addressed in Chapter 2. Among the electronic properties of graphene, the extremely long mean free path and coherence length of charge carriers stand out [21]. Graphene nanoribbons (GNRs) behave as room-temperature ballistic electrical conduc- tors on a length scale greater than ten microns [22], what makes this material an ideal system to study electronic phenomena in the coherent regime. Con- ductance quantization in integer multiples of the fundamental quantum 2e2/h is one of the hallmarks of ballistic quantum transport and can be explained in terms of the Landauer formula [23] (1.1) G = 2e2 h ∑ n τn , where τn is the transmission coefficient for each transverse mode n that arises from the lateral confinement of the charge carriers. In graphene, the valley degree of freedom that comes from the two inequivalent K and K ′ valleys, can result in 4e2/h quantization steps in some cases [24–27]. The conductance quantization in GNCs will be analyzed in Chapter 2. The long coherence length of charge carriers in graphene also allows to study electron quantum interference. Graphene nanorings are one of the most suitable nanostructures to that end. It was demonstrated that tuning the phase difference between electrons traveling along different arms of the ring with magnetic fields [28–35] or electrostatic gates [36, 37] can result in destructive and constructive interference and thus in a way of controlling the electric current. Different aspects related to graphene nanorings will be thoroughly discussed from Chapter 4 to Chapter 7. Additionally, another way of current control by means of electric fields in twisted GNRs will be explored in Chapter 3. These kind of structures that proved to be experimentally feasible [38–41], take advantage of graphene superior electronic properties, as well as its high elasticity and remarkable mechanical properties [42]. The long spin-coherence length of graphene, up to several microns even at room temperature [9], has resulted in a considerable interest in the de- 4 1.2. THERMAL TRANSPORT velopment of devices in which the spin degree of freedom can be controlled. There are several ways to inject spins and induce magnetism in graphene. For instance, it can be done along the edges of nanoribbons [43], by intro- ducing defects [44, 45], depositing atoms or molecules [46, 47], or by growing graphene on magnetic substrates [48]. In particular, the use of ferromagnetic insulators (FIs) such as europium chalcogenides has received a great deal of attention [49–53]. This approach will be employed in Chapter 4 to design systems in which the electron transmission is highly spin-polarized. 1.2 Thermal Transport Ever increasing miniaturization of electronic devices has resulted in an un- precedented level of power dissipation, and therefore higher temperatures at the nanoscale level. Although thermal phenomena are not directly re- sponsible for the electrical functionality of devices, they can adversely affect their reliability and limit their performance, making essential the search for superior thermal conductors capable of heat removal [54–56]. On the other hand, thermoelectric energy conversion requires materials with a strongly suppressed thermal conductivity, as we will see in next section. Therefore, a better understanding of heat transport at the nanoscale is crucial for the future development of stable high-density integrated circuits as well as more efficient thermoelectric devices. The energy transferred from atoms and electrons of the hotter regions of a system to those in cooler regions is called heat. Conduction is the mode of heat transfer in which energy exchange takes place in solids due to the presence of a temperature gradient in the system. An empirical relationship between the heat flow and the temperature gradient was first formulated by Fourier in 1822. For a homogeneous and isotropic material, this relationship, known as Fourier’s law, can be expressed as [57] (1.2) J =−κ∇T . where ∇T is the temperature gradient, the heat flux vector J represents heat flow per unit time, per unit area, and κ is called the thermal conductivity of 5 CHAPTER 1. INTRODUCTION the material, which is a positive, scalar quantity 1. The sign in Equation (1.2) is negative, indicating that heat flows from high to low temperature. Although Fourier’s law accurately describes heat transport in macroscopic samples, there has been some controversy regarding its validity at nanometer length scales. Nevertheless, recent studies on thermal transport at the nanoscale concluded that Fourier’s law is essentially exact in the diffusive and ballistic limits [58]. In solid materials heat can be carried by lattice vibrations and electrons so that the thermal conductivity splits as κ= κlat+κel, where κlat and κel are the lattice2 and electron contributions, respectively. In metals, κel is dominant due to the large concentration of free carriers and its value can be estimated from the electrical conductivity σ via the Wiedemann-Franz law [59]. However, failure of this law has been predicted, for example in nanoscale systems with strong Coulomb interaction effects [60] or those showing Fano resonances in the transmission spectra [61]. On the contrary, in most semiconductors and insulators, heat conduction is dominated by lattice vibrations. This is also the case of carbon-based materials [12]. The basic energy quantum of lattice vibration are phonons, which can be treated as both waves and par- ticles. Thermal properties of nanoscopic systems can be very different from conventional bulk materials since size effects appear when the dimensions of the system are comparable to or smaller than the phonon characteristic lengths. The most important characteristic length of heat conduction is the phonon mean free path (mfp), which is the average distance that phonons travel before being scattered by other phonons, defects or other obstacles. Experimental works have found a phonon mfp of approximately 100 nm at room temperature for SiO2-supported graphene and 3 to 8 times higher in freely suspended graphene [11, 62]. Significant progress has been made in understanding heat transfer in nanostructures, such as nanowires [63], superlattices [64, 65] and nanocom- posites [66, 67]. In these nanostructures, interfaces and boundaries strongly degrade the thermal transport. For example, phonon-boundary scattering is 1In the most general case of an anisotropic material, the thermal conductivity would be a tensor. 2The terms lattice contribution κlat and phonon contribution κph will be equally used throughout this thesis. κlat (κph) will be preferably employed when using semi-classical (quantum) approaches. 6 1.2. THERMAL TRANSPORT thought to be responsible for the decrease of κ in silicon nanowires by more than two orders of magnitude with respect to the bulk value [63]. On the other hand, theoretical studies on low dimensional crystals have revealed exotic behavior that leads to infinitely large values of κ [68]. Indeed, an anomalous large κ has been experimentally observed in graphene [10, 11]. Graphene based materials, with the strong sp2 covalent bonding and the relatively low mass of carbon atoms, occupy a unique place in terms of their exceptional thermal properties [12, 69]. The first measurements on graphene revealed a value of κ in the range of 2000−5500 W/mK at room tempera- ture [10, 11], well above the bulk graphite limit and even higher than other carbon allotropes. In addition, many geometrical constraints and physical factors can be used to modify the thermal transport. For example, increas- ing the thickness as in few-layer graphene [70, 71], limiting the graphene lateral sizes as in GNRs or changing the termination from zigzag to arm- chair [62, 72, 73] can significantly lower κ. Some studies have demonstrated that κ in graphene flakes and GNRs increases with length up to micrometer sizes and then converges to the graphene value [74–76]. Rough edges [77–80], patterning [81–84], or the insertion of bends, corners and other obstacles in the phonon paths [85–87], are promising strategies to strongly reduce κ in GNRs, especially for narrow ribbon widths, indicating the relevance of boundary scattering of phonons. Defects [88–90], edge passivation [91, 92], and isotope effects [91, 93, 94] can also significantly degrade the thermal transport. Furthermore, mechanical strain [95–97] and the coupling with sub- strates [98] and other two dimensional materials [99, 100] arise as alternative approaches to alter κ. In this context, an enhancement of κ of up to 36% for strained GNRs was found [96]. Due to the simple crystal structure of graphene and the increased com- putational power, atomistic simulations emerge as the tool of choice for un- derstanding phonon transport in this material. These methods are based on modelling the lattice vibrations either in real space or reciprocal space. In the real space approach, we find molecular dynamics (MD) simulations, which use the classical trajectories of the atoms to obtain the thermal conductivity. An outline of the MD method, which will be used in Chapter 6, can be found in Ap- pendix C. The reciprocal space approach, based on a quantum mechanics point of view, consists of the decomposition of the lattice vibration in phonons, which 7 CHAPTER 1. INTRODUCTION are characterized by a certain frequency ω. Then, the thermal conductivity can be obtained using kinetic theory such as the Boltzmann transport equation. Green’s function techniques are also very useful at low temperatures, when elastic scattering is dominant and the phonon transport is ballistic or semibal- listic. This approach in combination with density-functional tight-binding (DFTB) simulations will be used in Chapter 7. 1.3 Thermoelectricity The conversion of heat into electric energy is expected to play a crucial role in the development of sustainable energy technologies capable of reducing greenhouse emissions and the dependence on fossil fuels [101, 102]. In this context, thermoelectric materials arise as the ideal candidates to perform this task. On one hand, these materials can convert heat emanating from a hot source such as the Sun, engines, boilers, or even the human body directly into electricity. On the other hand, electricity can also drive a thermoelectric device to work as a spot-size refrigerator without moving parts or greenhouse gases emissions. However, despite the plethora of applications with important impli- cations for the society that have been envisioned, the usage of thermoelectric technology is currently limited by the small efficiency [103]. Thermoelectric phenomena have been well known since the 19th century. It was in the early 1800s when Seebeck first observed that when two dissimilar materials are joined together with junctions at different temperatures T and T +∆T, a voltage difference ∆V proportional to the temperature difference is created. The ratio of the voltage developed to the temperature gradient is related to an intrinsic property of the materials called the Seebeck coefficient or thermopower S [104] (1.3) S = ∆V ∆T . A few years later, a related effect was discovered by Peltier. He found that when an electric current is passed through the junction, a small heating or cooling effect is produced, depending on the direction of the current. It was soon realized that in order for these processes to be efficient, the Seebeck coefficient and the electric conductivity σ would need to be high and, at the same time, the thermal conductivity κ would need to be reduced. Otherwise, 8 1.3. THERMOELECTRICITY electron scattering would generate heat along the system and the temperature difference that must be maintained between the cold and hot sides would produce a large heat backflow. All these requirements are reflected in the thermoelectric figure of merit ZT, which is the parameter generally used to assess the thermoelectric efficiency of a system [104] (1.4) ZT = σS2T κ , where σS2 is sometimes called power factor. Unfortunately, at that time, there were no materials in which the combination of properties was good enough to achieve reasonable efficiencies. The main problem lies in the fact that optimization of these three parameters is compromised because all of them are affected by the electronic properties of the material: electrons conduct heat as well as electric current, and the Seebeck effect decreases as the electrical conductivity increases. It was only in 1950s when the first applications were developed following the works of Goldsmid and Ioffe with the introduction of thermoelectric materials based on heavily doped semiconductors such as tellurides of antimony, bismuth and lead [105, 106]. These advances led to some applications such as solid-state cooling for infrared sensors or power generation for space applications. Nevertheless, with efficiencies too low for thermoelectric technology to be truly competitive and no better materials having emerged by the end of the 1960s, the belief that ZT might be limited to unity arose in the thermoelectric community. Despite the lack of a theoretical reason, this value was a practical upper limit for more than three decades. In the early 1990s Dresselhaus and Hicks suggested a new approach: to use quantum mechanics and low dimensionality to design high efficiency thermoelectric materials [107, 108]. These predictions were followed by sev- eral experiments that proved that thermoelectric properties are strongly enhanced at the nanoscale [109–113]. In particular, experiments performed on bismuth nanowire composites demonstrated that the thermopower was increased by a factor of 104 as compared with that of bulk bismuth due to size quantization effects [110]. Similarly, it was found ZT = 1.6 in a quantum dot superlattice [111] and ZT = 2.4 was reported in thin-film supperlattices at room temperature [109]. Low dimensionality and size quantization can improve the thermoelectric performance in two possible ways. On the one hand, reducing the lattice 9 CHAPTER 1. INTRODUCTION thermal conductivity κph by increasing phonon scattering is a promising strategy. This is especially relevant for materials with low electron density, such as semiconductors and insulators, since thermal transport is dominated by phonons. In the previous section, we already mentioned that interfaces and boundaries can strongly degrade thermal transport in nanostructures. Other examples of systems that show a strong phonon scattering are tai- lored crystal structures [114], quantum-dot superlattices [115] or nanopore arrays [116]. On the other hand, the electronic structure can be engineered so that high power factors are obtained. In this regard, Heremans et al. found an experimental increase of the thermoelectric efficiency in PbTe by a distortion of the electronic density of states [117]. Also, theoretical works found the thermopower could be improved by the increase in the density of states in molecular junctions [118], or by opening a gap in the quasiparticle spectrum of graphene [119]. This enhancement of the thermoelectric properties had been already predicted two decades ago by Mahan and Sofo [120], who pointed out that systems in which there is a local increase in the electronic density of states over a narrow energy range, i.e. systems whose energy-dependent electron transmission is a delta function, would maximize ZT. Although such an ideal system is not interesting for applications since the power output becomes exceedingly small and even minute perturbations would lead to a low ZT [121], these ideas pave the way to tailor the electron transmission in order to obtain sharper profiles that enhance the thermoelectric response of a system. In this context, a promising approach is to use quantum interference effects. These phenomena have already proved to play a role in the thermoelec- tric efficiency of single-molecule heterojunctions [122] and zero-dimensional systems [121]. In particular, quantum effects giving rise to sharp resonances such as Fano lineshapes were predicted to have an impact on the thermo- electric efficiency of quantum dot systems [61, 123–125], single-molecule devices [126], nanoscale junctions [127, 128], and DNA-like systems [129]. Their impact on graphene nanorings will be explored in Chapter 5. Graphene is a paradigmatic example of a two-dimensional material whose thermoelectric performance can be significantly improved after nanostruc- turing or functionalization. As we know from the previous section, its lattice thermal conductivity can be greatly reduced by several mechanisms, rang- ing from patterning and rough edges to hydrogen-passivation and isotope 10 1.4. OBJECTIVES AND OUTLINE engineering [77–94]. Additionally, the extremely long mean free path and coherence length of charge carriers in this material [21, 22], make graphene an ideal candidate to exploit quantum interference phenomena in order to en- hance the thermopower. The combination of these two approaches in graphene nanorings, namely, reducing κ by purely geometrical effects and, at the same time, enhancing the electronic transport by quantum interference, will be addressed in Chapter 7. 1.4 Objectives and Outline In this thesis, several graphene nanostructures with potential to be used as building blocks for nanodevices are analyzed. The focus is set on GNCs, twisted GNRs and nanorings. A TB approach has been employed to model electronic properties. Also, in Appendix A an outline of the numerical method based on the quantum transmitting boundary method (QTBM) and transfer matrix method (TMM) that is used to calculate the electron transmission coefficient through the structures is presented. In order to analyze thermal transport, two different approaches have been employed. At low temperatures the DFTB method in combination with Green’s functions techniques was used, whereas MD simulations were the tool of choice for higher temperatures. An outline of the DFTB and the MD methods are presented in Appendix B and Appendix C, respectively. A summary of the objectives for each chapter is listed below: • In Chapter 2, the impact of several factors on the electronic transport and the conductance quantization in GNCs are discussed. • In Chapter 3, the possibility of controlling electron transport and engi- neering the current-voltage characteristics in twisted GNRs by means of a transverse electric field is explored. • In Chapter 4, the spin polarization in devices based on symmetric and asymmetric graphene nanoring and a ferromagnetic insulator is studied. • In Chapter 5, the thermoelectric efficiency of symmetric and asymmetric graphene nanorings in the absence of phonon contribution is addressed. 11 CHAPTER 1. INTRODUCTION • In Chapter 6, the thermal conductivity of the aforementioned graphene rings is calculated in the high temperature regime. • In Chapter 7, thermal transport in the low temperature regime, where quantum effects come into play, is analyzed. Also, the complete thermo- electric response of graphene rings is obtained. Finally, in Chapter 8 the main results and conclusions are summarized. New research lines motivated by and derived from the present work will be commented as well. 12 C H A P T E R 2 Conductance Quantization in Graphene Nanoconstrictions 2.1 Introduction Graphene stands out as one of the most intriguing two-dimensional materials in which the possibility of achieving a fully-tunable electronic confinement was pursued from its discovery [130]. Furthermore, graphene has proved to be an ideal system to study electronic phenomena in the coherent regime due to the extremely long mean free path and coherence length of charge carriers in this material [21]. In particular, GNRs were found to behave as single-channel room-temperature ballistic electrical conductors on a length scale greater than ten microns [22]. Conductance quantization in integer multiples of the fundamental quan- tum 2e2/h is one of the hallmarks of ballistic quantum transport. This phe- nomenon can be explained in terms of the Landauer theory of transport [23] and it was first observed three decades ago with the discovery of a sequence of steps in the conductance of a semiconductor quantum point contact (QPC) [131, 132]. Semiconductor QPCs are basic elements of quantum transport circuitry 13 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... that allow for the control of the number of conducting modes. In these devices, the electron confinement is achieved by using electrostatic potentials that control the width of the constriction. In the case of graphene, there are two in- dependent ways of achieving quantized conductance, either using electrostatic potential gates placed on top (underneath) the flake to confine the electrons into a well defined area as in traditional QPCs [25, 27], or by physically tai- loring the flake into a narrow constriction through a mechanical or chemical etching definition [133–139]. The latter structures are commonly referred to as GNCs and are envisioned as central building blocks in graphene-based nanoelectronics [140]. These structures have recently attracted both theo- retical [141–147] and experimental [133–139] interest. However, they have turned into an arduous technological challenge because of the lack of control in the edge-definition process and the presence of rough edges. Both the gated graphene constrictions and the tailored GNCs, have yielded some signs of conductance quantization in multiples of 2e2/h, and even in multiples of 4e2/h in some cases [25–27, 133, 136, 138, 139]. Conductance quantization in steps of 4e2/h is a sign of valley-symmetry conservation in ad- dition to the spin degeneracy already included in the 2e2/h quantization [24]. The degeneracy of the valley degree of freedom has the potential to be exploited in the novel field of valleytronics [148, 149], in which the valley can be used in the same way as the electron spin is used in spintronics or quantum com- puting. Valley-symmetry conservation requires the suppression of intervalley scattering arising from the short-range scattering from disorder, atomically sharp defects or edge irregularities, among others [150, 151]. Experimental studies have reported conductance steps of 4e2/h in electrostatically induced QPCs, both in graphene [25] and bilayer graphene [27]. Also, 4e2/h quantiza- tion steps were recently reported in ultrashort physically etched constrictions in bilayer graphene [26]. Confining the carriers in bilayer graphene is easier due to the presence of an already existing gap. Also, shaping electrostratically a constriction allows to make it narrower, thus diminishing the number of channels responsible of electric conduction. However, tailored GNCs are more challenging to fabricate and not clear traces of 4e2/h quantization have been experimentally observed yet. A deeper understanding on how the exact geometry of the GNC affects the electron transport is needed to elucidate which conditions favour the appear- 14 2.2. SYSTEM, MODEL AND METHODOLOGY ance of conductance quantization and conservation of the valley-symmetry. For this reason, in this chapter we perform a detailed analysis on how differ- ent factors, such as the type of edge or the shape of the constriction, affect the conductance in GNCs. Because one of the main experimental problems is the presence of rough edges, we also address its effect on the electrical conductance. In addition, we derive some scaling laws that allow us to extrap- olate the results to much larger systems. Finally, our results are compared with an experimental 200 nm-wide GNC fabricated by the Grupo de Nanotec- nología at Universidad de Salamanca. Low edge roughness is achieved by applying a novel cryo-etching technique, and atomic force microscopy (AFM) measurements are employed to assess the actual profile of the sample edges, that are subsequently used as input for numerical calculations, resulting in an excellent agreement between the measured and calculated electrical conductance. 2.2 System, Model and Methodology A GNC consists of a GNR of width W and length L inserted between two wider semi-infinite GNRs of width WL. Connecting the narrow and the wide ribbons there is a tapered region at an angle α. The wider GNRs act also as device leads. A schematic diagram of a typical GNC is depicted in Figure 2.1(a). The electronic properties of graphene structures can be described by a simple TB model of π orbitals [15]. Using these orbitals as the basis set to represent the wave function, the Hamiltonian that governs the dynamics of the electron is given by (2.1) H =∑ i εi|i〉〈i|− ∑ 〈i, j〉 ti j|i〉〈 j| , where εi represents the onsite energy at the ith atom, |i〉 the atomic orbital in the same site, and ti j the hopping parameter between orbitals at sites i and j. The onsite energy εi can depend on the position of the ith atom due to different factors, such as the presence of a source-drain or a side-gate voltage, impurities or the proximity of another material. Some of these effects will be explored in the following chapters, but for now we will set εi = 0. In addition, the hopping parameter is considered to be uniform ti j ≡ t = 2.8 eV [13]. In the above equation only nearest-neighbors tunneling was considered. This 15 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... FIGURE 2.1. (a) Schematics of a GNC. (b) Dispersion relation for the different regions. Inside the wider leads, there is a higher density of transverse modes than in the middle region. approximation provides reliable results near the K and K ′ points of the Brillouin zone [16]. Assuming that electron-phonon and electron-electron interactions in our samples are negligible, electrons can be considered to be in the fully co- herent regime, travelling ballistically through the system. Combining the QTBM [152] and the TMM [153] adapted for graphene [33, 154] (see Ap- pendix A for further details), the wave function |ψn〉 in the whole sample and the transmission coefficient τn(E) for each mode n and for each energy E are calculated. The modes, also known as channels or subbands, arise from the lateral confinement at the leads [23]. At very low temperature, the electrical conductance G is directly proportional to the transmission coefficient (2.2) G = 2e2 h ∑ n τn(E) . 16 2.3. SINGLE NANOCONSTRICTION 2.3 Single Nanoconstriction In this section we perform a detailed analysis of the influence of the edge orientation and the nanoconstriction shape on the electrical conductance of a single GNC as the one depicted in Figure 2.1(a). In the following we assume W = L = 10 nm, WL = 10W , and α= 45◦, unless otherwise specified. 2.3.1 Impact of Edge Orientation In graphene, the lattice orientation can give rise to two types of edges, namely, zigzag and armchair. Within the single-orbital TB model, ideal GNRs with zigzag edges (zGNRs) are metallic, while those with armchair edges (aGNRs) are either metallic or semiconducting, depending on the ribbon width. The width can be specified by the number of dimer lines N across the ribbon. Families with N = 3p and N = 3p+1 (p being a non-negative integer) have a semiconductor-type spectra with wide gap (scaling inversely proportional to the nanoribbon width), while the family with N = 3p+2 is metallic [17, 18]. The electrical conductance curves for a zGNR and the three families of aGNRs, all of them with W ' 10 nm, are plotted in Figure 2.2(a) (dotted lines). As it is expected, the conductance is quantized in abrupt steps [24, 144]. This stepwise profile is associated with the increase in the number of bands or transverse modes crossing the Fermi energy [see Figure 2.1(b)]. By using the Dirac equation, Tworzydło et al. found that the conductance quantum in GNRs is 4e2/h, with the factor 4 accounting for both spin and valley degeneracy. The only exception would be the lowest mode, in which the valley degeneracy is broken [24]. However, more accurate numerical calculations using a TB model showed the valley degeneracy is only approximate, at least for armchair GNRs [24]. This is in agreement with the results shown in Figure 2.2(a), in which we observe the conductance is quantized in steps of 2e2/h for all the aGNRs, and in steps of 4e2/h for the zGNR. On the contrary, size quantization is smoothed out in the case of GNCs, as it is demonstrated in Figure 2.2(a) for GNCs (solid lines) with the same width W in the narrow region as the previous GNRs (dotted lines). We ob- serve the quantization plateaus are smeared out and evenly spaced modu- lations (kinks) appear. This can be explained in terms of reflection at the leads. For a fixed Fermi energy, the number of transverse modes in the leads 17 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... FIGURE 2.2. (a) Electrical conductance for GNRs (dotted lines) of width W and GNCs (solid lines) with zigzag and the three families of armchair edges. The dimensions are W = L ' 10 nm, WL = 10W and α= 45◦. The curves are shifted upwards by 2e2/h for better visualization. (b) Electrical conductance at low energies for the armchair GNC with 3p+1. The steps represent the conductance for ribbons of widths corresponding to the narrow (W) and wide (WL) ribbons that form the GNC. (c) Square modulus of the wave functions for the three subbands at the energy marked in (b). is larger than in the narrower GNR region [see Figure 2.1(b)]. The mismatch between the wave functions in these regions can lead to backscattering and a lower transmission. We find that the electron backscattering in the zigzag case is much weaker than in the armchair one, leading to more pronounced conductance plateaus, close to those of the ideal uniform zGNR. Some au- thors attribute this difference to the current densities being much lower near zigzag edges [144]. According to these findings, conductance quantiza- tion, and in particular 4e2/h steps and valley-symmetry conservation, would be more easily spotted in zigzag-edged GNCs. However, since the current etching techniques does not allow for a precise control on the edge type in fabricated graphene nanostructures, understanding which factors favour the 18 2.3. SINGLE NANOCONSTRICTION observation of conductance quantization also in less favourable edged GNCs is crucial. For this reason, in the following we will restrict our study to armchair GNCs. Notice also the appearance, in some cases, of a sharp peak at low energies. This kind of resonances are also found in graphene junctions and some authors associate them with localized states at the zigzag edges [155]. We observe they can even arise for energies lying within the band gap of the narrow ribbon. This is observed at E = 26 meV in the top curve of Figure 2.2(a), corresponding to the 3p+1 aGNC. In this case, the sharp resonance appears at an energy that lies in the band gap of the narrow ribbon. However, at E = 26 meV there are already three subbands corresponding to the wider ribbon [see Figure 2.2(b)]. To better understand this phenomenon, in Figure 2.2(c) we plot the square modulus of the wave function |Ψ|2 for each subband. We observe that for two of the subbands the wave function is reflected in the leads, whereas for the remaining one, the wave function is strongly localized in the constriction region, especially at the corners where both armchair and zigzag edges coexist. This ultimately results in resonant transmission. 2.3.2 Extrapolation to Larger Systems As the constriction becomes wider, the number of transverse modes for a given energy increases and so does the conductance. But interestingly, provided that we keep within the same armchair family, the features are essentially the same when G is plotted against the dimensionless magnitude WkF , where W is the width of the constriction and kF the Fermi wave number. This is shown in Figure 2.3(a), where we consider GNCs of several widths W while the ratio between the different geometric parameters (α= 45◦, L =W and WL = 10W) is kept fixed. For W = 1.5 nm, we observe the conductance clearly differs from the curves for larger systems. However, as the system becomes larger, the conductance curves are more similar. Even for W = 3 nm the resemblance with systems 10 times larger is significant at low energies. This finding has an important consequence: We can extrapolate results to much larger systems as the ones fabricated experimentally and otherwise computationally very expensive (or even impossible) to simulate. Looking to further optimize the simulations, in Figure 2.3(b) we see the width of the leads WL can be decreased 19 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... FIGURE 2.3. (a) Electrical conductance for GNCs with several widths W, while keeping WL = 10W, L = W and α = 45◦. (b) Electrical conductance for GNCs with several WL, while keeping W = L = 10 nm and α= 45◦. The curves are shifted upwards by 2e2/h for better visualization. (while keeping α= 45◦ and W = 10 nm) until approximately WL = 2W without significantly affecting the results. 2.3.3 Role of the Nanoconstriction Shape To better understand the electronic transport in these structures, now we analyze how the exact shape of the constriction affects the conductance. The focus is set on three different aspects, namely, the constriction length L, the angle α of the tapered region, and the roundness of the constriction corners. We start studying the effect of the length by increasing L from 0 to 10W [see Figure 2.4(a)]. We observe that the traces of conductance quantization in steps slightly lower than 4e2/h are clearly visible for L < W. As L becomes larger, the quantization steps are blurred because of the increasing number of kinks that emerge in the conductance. These peaks can be attributed to Fabry- Pérot-like oscillations due to multiple reflections at the leads [147]. Regarding the angle dependence, in Figure 2.4(b) we plot the conductance for different slopes of the constriction. Note that only for three cases (α= 30◦,60◦,90◦) the 20 2.3. SINGLE NANOCONSTRICTION FIGURE 2.4. Electrical conductance for GNCs with widths W = 10 nm and WL = 10W. (a) Results for α = 45◦ and varying lengths L specified in the plot. (b) Results for L = W and varying α. The gray steps represent the conductance for a ribbon of the same width (W = 10 nm). The curves are shifted upwards by 2e2/h for better visualization. (c) Schematic diagram of the edge orientation for each α considered in (b). edge orientation is purely zigzag or armchair in the tapered region, while any other case will mix both type of edges [see Figure 2.4(c)]. For α = 60◦ the armchair orientation is preserved along the whole constriction, including the ribbons and the tapered region. This results in a smaller band gap. This angle also divides two regimes, one with a narrower band gap for higher angles, and another with a wider band gap for lower angles. Additionally, limiting cases, α= 90◦ and α= 15◦, allow to better discern the quantization, although the peaks that partially mask the plateaus are still present. Besides 21 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... FIGURE 2.5. (a) Electrical conductance for a sharp GNC and some rounded GNCs. The degree of roundness increases downwards, being the sharp one at the top and the circular one at the bottom. The gray steps represent the conductance for a ribbon of the same width (W = 10 nm). The curves are shifted upwards by 2e2/h for better visualization. (b) Schematics for some of these GNCs. that, no remarkable features are observed, and we conclude that the angle of the constriction does not play a major role in the appearance of well-defined conductance quantization. So far we have considered constrictions with different sizes and angles, but in all cases the corners were sharp [see Figure 2.5(b)]. We now focus on how the electronic transport is influenced by increasingly rounded corners. Figure 2.5(a) shows the electrical conductance for GNCs ranging from sharp corners to a fully circular profile. We observe that when the corners are slightly rounded the conductance mostly resembles that of the sharp (not rounded) case. As the degree of roundness increases, however, the whole conductance shifts towards higher energies and the kinks begin to smooth out until they disappear for the circular constriction. This indicates the backscattering at the junctions between regions of different width is fully suppressed and a very clear 4e2/h quantization with very pronounced steps arises. 22 2.4. ROUGH EDGES Therefore, two main strategies can be used to obtain a 4e2/h conductance quantization, namely, reducing the length of the GNC in such a way that L 2.5W the conductance is greatly degraded and the slope is much lower than in the case of the single constriction studied in previous sections (take as a reference the gray steps corresponding to the conductance of an ideal GNR). Although some features of conductance quantization are still present, the overall conductance is quite noisy, resembling the conductance of highly disordered ribbons. A 27 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... FIGURE 2.8. Schematics of a double GNC (n = 2). FIGURE 2.9. (a) Electrical conductance for double GNCs with W = 10 nm, WL = 10W, L =W and varying D (and α). The gray steps represent the conductance for a ribbon of the same width. The curves are shifted upwards by 2e2/h for better visualiza- tion. (b) Electrical conductance for GNCs with varying number of constrictions (n). All the parameters as in (a) and, additionally, D = 1.2W (α= 82.5◦). dramatic change occurs when the distance D is sufficiently small (D < 2.5W). In this case, well defined peaks (resonances) appear in the conductance. As the distance D decreases this behavior is more pronounced. In Figure 2.9(b), we also analyze the conductance when D = 1.2W and the number of constrictions 28 2.6. EXPERIMENTAL RESULTS n is increased. Although adding more constrictions degrades the overall conductance, we observe the peaks or resonances are still clearly noticeable even for n = 10. 2.6 Experimental Results Finally, according to our proposal and design, graphene-based single nanocon- strictions were fabricated by Grupo de Nanotecnología at Universidad de Salamanca [157]. First, graphene was encapsulated between thin layers of hexagonal boron nitride (hBN). Then, the constrictions were defined on the hBN/graphene/hBN heterostructure by means of an innovative implemen- tation of the cryo-etching method, which allows for an unprecedented con- trol of the structure edges. Additionally, due to the use of lattice-matched hBN layers a high electron mobility was obtained in the GNCs, typically µ' 150000 cm2/Vs at room temperature. Figure 2.10(a) shows a tilted scanning electron microscope (SEM) image of the GNC, in which the sandwiched structure (hBN/graphene/hBN) is colored in sky blue. The lateral width is W ' 206 nm and the length is L ' 200 nm. In order to get a quantitative analysis of the edge roughness, AFM measurements were performed on the fabricated GNC by Grupo de Física de Nanomateriales Electrónicos at Universidad Complutense de Madrid. Figure 2.10(b) displays the AFM image of the whole GNC. The edge and the corresponding contour profile within the square marked in panel (a) are shown in Figure 2.10(c). The AFM images reveal an excellent definition and smooth edges. The average edge roughness can be estimated from the AFM image, obtaining a value around 2 nm. Also, the roughness parameter d as defined in Section 2.4 is found to be close to 4 nm [see Figure 2.10(c)]. These measures are the first quantitative estimation of the roughness in encapsulated GNCs. Thanks to the accurate profile obtained with AFM measurements, we are able to account for the real edge roughness in our simulations. Because the size of the experimental system is too large to be simulated, we use the extrapolation approach developed in Section 2.4. By using this approach and the edge roughness extracted from the AFM profile, first, we check if the results are independent of the width W. In Figure 2.11(a) we observe the conductance slope is practically the same for the W = 100 nm and W = 140 nm 29 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... FIGURE 2.10. (a) Tilted SEM micrograph of an encapsulated GNC with lateral width W ' 206 nm and length L ' 200 nm. The hBN/graphene/hBN heterostructure is colored in sky blue, the SiO2 substrate in violet and the contacts in yellow gold. (a) AFM image of the same GNC. (b) Contour plot taken from the square highlighted in panel (b), where the dashed lines indicate the values used to estimate the edge roughness. GNCs, indicating that the extrapolation is reliable for much larger systems. Indeed, Figure 2.11(b) shows an excellent agreement between the slope of the simulated conductance (dashed line) and the experimental one measured at T = 3.1 K (solid line). This result has important implications. First, it indicates that the transport in the experimental GNC is almost truly ballistic due to the high mobility and the smooth edges obtained by the novel use of the cryo-etching procedure on the hBN/graphene/hBN heterostructures. Additionally, it demonstrates the effectiveness of the extrapolation approach 30 2.7. CONCLUSIONS FIGURE 2.11. (a) Conductance for GNCs using the extrapolation ap- proach for AFM extracted edge roughness. Several widths W indicated in the plot are considered. (b) Comparison of simulated and experimental conductance of a GNC of width W = 206 nm measured at T = 3.1 K. The simulated curve has been shifted right by 5WkF to account for the residual doping of the exper- imental GNC. The inset shows the position of the plateaus of the experimental conductance separated by 2e2/h and matching integer values from G = 10e2/h onwards. developed in Section 2.4 for the case of rough edges. Furthermore, a closer inspection of the experimental conductance G [see the inset in Figure 2.11(b)] yields another remarkable finding: up to six plateaus are clearly observed in steps of 2e2/h. 2.7 Conclusions In summary, we have performed an extensive analysis of the influence of dif- ferent factors on the electronic transport and the conductance quantization in GNCs. We found that zGNRs present a conductance quantization in multiples of 4e2/h as a fingerprint of valley-symmetry conservation [24]. For this reason, the 4e2/h quantization should be more easily observed in zigzag GNCs. On the 31 CHAPTER 2. CONDUCTANCE QUANTIZATION IN GRAPHENE... contrary, in aGNRs the valley degeneracy is only approximate, hence 2e2/h quantization steps arise. However, we found that armchair GNCs can recover the valley-symmetry conservation in some specific cases, namely for very short constrictions lengths (L 5 can be obtained in some cases. Figures 5.4(a) and (b) show the transmission coefficients corresponding to some of the above-mentioned cases, specifically the N = 3p asymmetric and symmetric rings with VG = 46 mV and VG = 21 mV respectively. These transmission coefficients display new sharp Fano line-shapes with very abrupt slopes that arise due to VG. The occurrence of these sharp Fano resonances gives rise to the increase of both the thermopower S and the figure of merit ZTel , as can be seen in the middle and lower panels of Figure 5.4. The good thermoelectric performance, however, is degraded on increasing temperature. 70 5.4. EFFECT OF A SIDE-GATE VOLTAGE FIGURE 5.4. (a) Transmission coefficient, (c) Seebeck coefficient, and (e) figure of merit for a asymmetric ring with N = 3p and VG = 46 mV. (b) Transmission coefficient, (d) Seebeck coefficient, and (f) figure of merit for a symmetric ring with N = 3p and VG = 21 mV. The ribbon width in all cases is w = 12.1 nm. Figure 5.6 shows the dependence with temperature of the maximum ZTel for the two cases discussed above. We observe the maximum value is reached around 2−5 K, depending on the specific case, and then it rapidly decays. However, at a temperature of 20 K, ZTel is still well above 1. Higher tempera- tures were not considered since the phonon contribution is not included in our calculations. 71 CHAPTER 5. THERMOELECTRICITY IN GRAPHENE NANORINGS... FIGURE 5.5. Figure of merit ZTel as a function of the chemical po- tential µ and the side-gate voltage VG for the asymmetric (upper panels) and symmetric rings (lower panels). The ribbon width for left (right) panels corresponds to w = 12.1 nm (w = 12.5 nm). FIGURE 5.6. Maximum value of the figure of merit as a function of temperature for the rings analyzed in Figure 5.5, i.e. the asym- metric and symmetric rings with VG = 46 mV and VG = 21 mV, respectively. 72 5.5. EDGE DISORDER 5.5 Edge Disorder Finally, because atomic size imperfections, especially at the edges, are un- avoidable in real samples, we address the effects of edge disorder on the thermoelectric properties of the graphene rings. To do so, we randomly re- move carbon atoms from the zigzag edges with some given probability p. In the case of armchair edges, pairs of neighbor atoms are removed with the same probability. By removing pairs rather than single atoms we ensure that there are no dangling atoms in the sample, so we do not have to deal with complicated edge reconstruction effects. The transmission coefficient and the corresponding Seebeck coefficient and figure of merit calculated for typical realizations of disorder for the asymmetric ring with N = 3p+1 are shown in Figure 5.7. In contrast to GNRs [116], we have found that the electronic properties of graphene quantum rings are not markedly deteriorated by edge disorder. We observe that when the edge disorder is small (p ≤ 0.05) the trans- FIGURE 5.7. (a) Schematic diagram of a graphene nanoring displaying edge disorder. (b) Transmission coefficient, (c) Seebeck coefficient, and (d) figure of merit for an asymmetric ring with N = 3p+1 (w = 12.5 nm). 73 CHAPTER 5. THERMOELECTRICITY IN GRAPHENE NANORINGS... mission coefficient is slightly distorted but the features that give rise to a good thermoelectric performance are not destroyed. For this reason the Seebeck coefficient and the figure of merit are generally degraded but still remains high. In addition, new resonances can appear in the transmission coefficient, leading to new peaks of the figure of merit. This is due to the fact that we are removing atoms from the edges, making the nanoribbons effectively narrower. Because GNRs have a width-dependent gap, these narrower areas will act as barriers, causing new resonances to appear. 5.6 Conclusions In summary, we have studied the thermoelectric properties of symmetrically and asymmetrically connected quantum rings based on graphene. The trans- mission coefficient of the rings manifests Breit-Wigner line-shapes in the former case and can show Fano line-shapes in the latter (depending on the width of the nanoribbons comprising the rings). While Breit-Wigner line- shapes lead to a moderate thermoelectric response, the occurrence of Fano line-shapes causes a dramatic enhancement of the thermoelectric efficiency of these nanodevices. Therefore the non-gated asymmetric rings seem to be more promising from the point of view of applications. However, even if a non-gated ring does not support Fano anti-resonances (e.g., symmetrically connected rings), the application of a side-gate voltage can introduce asymmetry in the system and induce such features in the transmission spectrum, which consequently leads to an enhancement of the thermoelectric response and values of ZTel as high as 8. We note that the contribution of phonons to the thermal conductance has been neglected in our calculations. This contribution would lead to a smaller figure of merit because it increases the denominator of ZT. In the following chapters we will address this issue and analyze if a careful engineering of the edges can reduce the effect of phonons, as suggested by some authors [83, 86]. Finally, we have also shown that the predicted effects are robust under moderate edge disorder. Precise positions of the Fano features can be very highly dependent on the system geometry and fabrication imperfections (disorder). Therefore the possibility to induce and control such features by electrostatic means seems very promising from the point of view of optimization of thermoelectric properties. 74 C H A P T E R 6 Molecular Dynamics Study of the Lattice Thermal Conductivity in Graphene Nanostructures 6.1 Introduction Advances in nanotechnology demand a better understanding of heat transport in nanoscale systems. The increased levels of dissipated power in ever smaller devices make essential the search for high thermal conductors [54–56]. On the other hand, thermoelectric energy conversion requires materials with a strongly suppressed thermal conductivity κ, but still a high electric con- ductivity σ [101, 102]. Both electrons and lattice vibrations contribute to the heat current and consequently the thermal conductivity splits as κ= κel +κlat. Unfortunately, parameters κ and σ cannot be adjusted independently in the vast majority bulk materials. For instance, the ratio σ/κel in metals is de- termined from the Wiedemann-Franz law [59]. Hence reducing the lattice thermal conductivity κlat by increasing phonon scattering is one of the most promising routes to improve thermoelectric materials. 75 CHAPTER 6. MOLECULAR DYNAMICS STUDY OF THE LATTICE... Graphene occupies a unique place in terms of its thermal properties because it possesses one of the highest known lattice thermal conductivities. Although a high value of κlat is undesirable for thermoelectric applications, thermal transport in graphene can be significantly degraded by different mechanisms. For example κlat can be greatly reduced in GNRs by rough edges [79], hydrogen-passivation [91] and patterning [81–84, 86]. In this context, Mazzamuto et al. found that a structure combining zigzag and arm- chair edges and different widths presents a low thermal conductivity [83]. Since graphene is envisioned as a material of choice for a variety of appli- cations in future electronics, understanding how heat is carried in different graphene nanostructures is crucial. One of these structures are graphene nanorings, which arise as ideal candidates for exploiting quantum interfer- ence phenomena [32, 36, 37, 51, 210, 211]. These effects were used in Chap- ter 4 for designing a new spintronic device and also in Chapter 5, where we demonstrated that graphene nanorings may be useful as thermoelectric de- vices too [212]. We found that quantum interference effects can lead to a large Seebeck coefficient and hence a high figure of merit ZT when the heat current is solely due to electrons. Yet, lattice heat conduction, which is expected to be the most important contribution to heat transport in carbon materials due to the strong covalent sp2 bonding, had not been studied in graphene nanorings. In this work, we address the contribution of the atomic lattice to heat transport in armchair GNRs and nanorings by using non-equilibrium molecu- lar dynamics (NEMD) simulations as implemented in the LAMMPS Molecular Dynamics Simulator [213]. NEMD simulations provide a direct method to calculate the lattice thermal conductivity. To this end, a heat flow establishes a temperature gradient across the system and Fourier’s law brings about an estimate of the lattice thermal conductivity. First, we analyze the impact of the size of the contacts on the results. Then, we compare the thermal conduc- tivities of GNRs and rectangular graphene nanorings of widths up to 6 nm over a wide range of temperature. Our study proves that the lattice thermal conductivity κlat is greatly reduced in nanorings as compared to GNRs due to higher scattering of lattice vibrations at bends. Finally, we also demonstrate that edge disorder has a weaker impact on the heat current in nanorings as compared to GNRs. 76 6.2. SYSTEM, MODEL AND METHODOLOGY 6.2 System, Model and Methodology Our study is focused on three different types of graphene nanostructures connected to leads. The first system under consideration is a uniform rect- angular aGNR of width W, as seen in the top panel of Figure 6.1. We only consider aGNRs since many studies have demonstrated they have a lower thermal conductivity than zGNRs [72, 78, 214], thus being more suitable for FIGURE 6.1. Schematic view of the analysed structures. The top panel shows a GNR of width W. The middle (bottom) panel shows a square asymmetric (symmetric) graphene nanoring. The red (blue) area represents the hot (cold) contact where an amount of energy ∆ε is introduced (removed) in every time step of the NEMD simulation. 77 CHAPTER 6. MOLECULAR DYNAMICS STUDY OF THE LATTICE... thermoelectric applications. The second (third) kind of nanostructures are asymmetric (symmetric) graphene nanorings similar to the ones studied in previous chapters (see Figure 6.1 for schematics). For definiteness, the rings will be square, all nanoribbons forming a given structure will have the same width W and the size of the inner hole will be W ×W . In this work we use NEMD simulations [213, 215] in which the classical trajectories of Lagrangian particles (in our case, carbon atoms) are obtained by numerically solving Newton’s equation of motion (see Appendix C for further details). Atoms with initial positions and velocities are exposed to collisions governed by an empirical interatomic potential. At each time step, the force acting on each atom is obtained, and then, positions and velocities are updated. This is an excellent approximation for a wide range of materials; only when we consider light atoms or vibrational motion with a frequency ω such that ~ω> kBT should we worry about quantum effects [216]. As a general rule, the classical treatment will be valid if the interparticle distance is much larger than the thermal de Broglie wavelength Λth = h/ √ 2πmkBT. This is indeed the scenario in our simulations since the distance between C atoms is about 0.142 nm while Λth = 0.030 nm at room temperature and Λth = 0.043 nm at T = 100 K. NEMD simulations provide a direct method to calculate lattice thermal conductivity by applying a perturbation to the system and measuring the response. Most suitable choices of perturbations would be either imposing a thermal gradient ∇T across the system or introducing a heat flow J. Through- out this work we consider the latter case, that is, a heat flow is introduced and the subsequent thermal gradient is calculated [92, 217]. Then, Fourier’s law is applied to obtain the lattice thermal conductivity κ (because we only address the lattice contribution to the thermal conductivity we omit the subscript hereafter, unless otherwise stated) (6.1) J =−κ∇T . The small size of the systems under study poses a question about the va- lidity of Fourier’s law at the nanoscale but a comparison with more elaborated approaches, such as the phonon Boltzmann transport equation, is beyond the scope of this work. Nevertheless, previous studies on steady-state thermal transport in nanostructures concluded that Fourier’s law is essentially exact 78 6.2. SYSTEM, MODEL AND METHODOLOGY in the diffusive and ballistic limits (see Ref. [58] and references therein for further details). In this work, a time step of 0.5 fs is used in the simulations and carbon- carbon interactions are described by the adaptive intermolecular reactive bond order (AIREBO) potential [218], which depends not only on the distance between atoms but also on the local atomic environment, and therefore im- plicitly contains many-body information (see Appendix C for further details). This potential has already been successfully implemented to study thermal and mechanical properties of graphene [52, 89, 92]. The method we use is depicted in Figure 6.2 and has the following steps: 1. The initial configuration is first equilibrated at a temperature T (typi- cally the mean target temperature) during 2×106 time steps by keeping the two outermost rows of atoms at each end fixed while applying a Nosé-Hoover thermostat to the rest of the atoms which are free to move in three dimensions. A Nosé-Hoover thermostat is a constant temper- ature method in which some variables are added to the equations of motion [219]. 2. When the configuration is equilibrated, we introduce a heat flow by adding at each time step a small amount of energy (+∆ε) into the hot contact and removing the same amount of energy (−∆ε) from the cold contact. This energy addition (subtraction) is done by rescaling the ve- locity vectors at both contacts. In order to avoid non-linear temperature profiles when ∆ε is too large or temperature fluctuations when ∆ε is too small, we adjust ∆ε so that ∆T = Tmax −Tmin ' 0.2T. Since the value of ∆ε is unknown beforehand it is found by performing iterative simula- tions and adjusting ∆ε at each iteration step to obtain the target ∆T/T ratio. The whole system is then switched to the constant volume and constant energy ensemble and we run at least 107 time steps to allow the system to attain steady state. 3. Once steady state is reached, the three components of velocity of each atom are averaged over 107 time steps. Then, the system is divided into slices and the temperature within each slice is obtained from the 79 CHAPTER 6. MOLECULAR DYNAMICS STUDY OF THE LATTICE... FIGURE 6.2. Step diagram of the method usded to obtain the thermal conductivity. 80 6.3. INFLUENCE OF THE CONTACTS equipartition theorem as (6.2) T = M 3NkB ∑ i ( 〈vi〉2x +〈vi〉2y +〈vi〉2z ) , where M is the mass of the carbon atom, N is the number of atoms in each slab, kB denotes the Boltzmann’s constant and 〈vi〉µ stands for the time averaged µ-component of velocity of atom i. 4. The temperature gradient ∂T/∂x is determined from a linear fit. 5. The heat current, which can be defined as the amount of energy trans- ferred per unit time and cross sectional area, is calculated as (6.3) J = 1 Wd ∆ε ∆t , where ∆t is the time step and d the graphene thickness, taken approx- imately as the diameter of a carbon atom d = 0.144 nm [72]. Finally, the lattice thermal conductivity is obtained using Fourier’s law [equa- tion (6.1)]. We have also tried an alternative method of the conductivity calculation, consisting in maintaining a fixed temperature difference between the con- tacts and calculating the energy flux. In this case temperature profiles can be highly nonlinear, characterized by abrupt temperature changes at the contacts (resulting from a mismatch between the dispersion relation of the fixed temperature parts and the rest of the system [220]). To avoid the non- physical temperature kinks, larger contacts have to be considered increasing the simulation time considerably, for which reason the latter approach was abandoned. 6.3 Influence of the Contacts In this section we analyze how the size of the contacts can influence the results. To do so, we consider an aGNR with fixed length LS and width W and vary the contact length LC (see Figure 6.1 for a schematic view). We first plot the temperature profile across the system for different values of LC. In Figure 6.3 one can observe that it is linear and smooth in the interface between the system and the contacts in all cases. However the slope of the 81 CHAPTER 6. MOLECULAR DYNAMICS STUDY OF THE LATTICE... FIGURE 6.3. Temperature profile and linear fit for aGNRs of width W = 1.2 nm, LS = 8.4 nm and three different values of LC indi- cated in the panels. The gray areas represent the contact regions of length LC. linear fit to the points is dependent on the contact size. As a consequence, κ will be also influenced by the specific choice of contacts, as it can be seen in Figure 6.4, where we plot κ as a function of the contact length LC. We observe that κ increases with LC and although it seems that it tends to saturate as LC becomes larger, a constant value is not reached for LC = 3LS yet, which is the larger contact size consider in this work. To understand this behavior we have to look again at Figure 6.3 and note the thermal gradient spans not only across the "device" part of the system (the central part of length LS) but also across part of the contacts. As the contacts grows larger, the expansion of the thermal gradient across them is also larger. Then, the effective length of the system is larger than LS and increases with LC. Therefore, the thermal conductivity is also contact size dependent because κ is a length-dependent magnitude in nanometer-sized GNRs [72, 75]. This occurs because the contacts (heat 82 6.4. THERMAL CONDUCTIVITY IN GRAPHENE NANORINGS FIGURE 6.4. Lattice thermal conductivity κ as a function of the ratio LC/LS for aGNRs of width W = 1.2 nm and LS = 8.4 nm. source and heat sink) cannot be considered as isothermal classical boundary conditions. Although the average temperature remains constant, there is a temperature gradient within contacts. They are part of the system, so that vibrational modes are characterized by the whole dimension and not only by the size of the intermediate zone [221]. In order to avoid any dependence on the length dimension, the length of the system and the contacts (cold and hot regions) will be fixed throughout this work so that we take LC = 12.6 nm and LS = 25.4 nm. 6.4 Thermal Conductivity in Graphene Nanorings 6.4.1 Role of Dimensions First, we focus on how the thermal conductivity is affected by the width of the nanoribbons. To do so we fix the temperature at T = 300 K. Figure 6.5(a) shows the thermal conductivity for GNRs and the two types of nanorings for widths up to W = 6 nm. We observe that κ monotonously increases with the width W , both for GNRs and nanorings. This is in agreement with previous results [72, 78] where the same trend was found for aGNRs at room temperature. It can be understood as follows. Wider GNRs have a larger number of vibrational modes while the number of edge localized modes does not change. Thus, the 83 CHAPTER 6. MOLECULAR DYNAMICS STUDY OF THE LATTICE... FIGURE 6.5. (a) Lattice thermal conductivity as a function of width W for aGNRs and rings. (b) Ratio κring/κribbon as a function of the ribbon width W . edge effect decreases and κ increases with W. At a threshold width, larger than the ones considered in this work, κ will reach the value of graphene (2000−5500 W/mK at room temperature [10, 11]) and then stay constant due to intermode scattering arising in the anharmonic lattice. Although κ also increases with W for nanorings, it remains lower than for GNRs at all W considered by us. We interpret this as the effect of mixing different edges, both armchair and zig-zag, that leads to a mismatch of the vibrational modes [83] and by scattering at the bends [52, 86]. Although absolute values of κ obtained by NEMD simulations depend on the choice of interatomic potential, boundary conditions, simulated system di- mensions and chosen method of imposing heat flux and temperature gradient, our results remain relevant because we are addressing the relative reduction of the thermal conductivity due to the nanostructuring. In the lower panel 84 6.4. THERMAL CONDUCTIVITY IN GRAPHENE NANORINGS of Figure 6.5 the ratio κring/κribbon is plotted, where κribbon is the thermal conductivity of GNRs and κring indicates the lattice thermal conductivity of the corresponding symmetric/asymmetric ring of the same ribbon width. Our results show that symmetric nanorings reduce the thermal conductivity more efficiently, reaching only 60% of that of the GNR with the same width up to 3.5 nm. This can be explained by the existence of more bends in symmetric rings, leading to a stronger scattering and suppression of the thermal con- ductivity. As the ribbon width decreases, the mismatch of vibration modes at different regions become more important and the suppression of κ is stronger. 6.4.2 Effect of Temperature In this section we analyze the temperature dependence of κ. We fix the ribbon width at W = 2.5 nm and vary the temperature from 100 K up to 1000 K. Classical NEMD simulations are considered valid near and above Debye’s temperature (TD ≈ 322 K for GNRs [214]), where all vibrational modes are fully excited. At lower temperatures, quantum effects cannot be neglected. To mitigate this limitation, a quantum correction was developed using the following expression [214, 222] (6.4) TMD = TD 3 + 2T3 Q T2 D ∫ TD/TQ 0 x2 ex −1 dx , where TMD, TD and TQ are simulation temperature, Debye’s temperature and quantum corrected temperature, respectively. The corrected thermal conductivity, (6.5) κD = κdTMD dTQ , is obtained by equating the heat fluxes obtained from Fourier’s law in the classical (non-corrected) and quantum systems. This correction has been imple- mented in many MD studies, including those analyzing graphene [214, 222]. However, some authors have questioned the validity of these quantum correc- tions [223]. For this reason, in Figure 6.6 we plot both the quantum corrected and uncorrected values of the thermal conductivity. There is no quantum correction available at low temperatures (shadowed areas in Figure 6.6) [222]. Our results show that for the three considered graphene structures, κ first 85 CHAPTER 6. MOLECULAR DYNAMICS STUDY OF THE LATTICE... FIGURE 6.6. (a) Lattice thermal conductivity as a function of temper- ature for aGNRs and nanorings with W = 2.5 nm. Dotted lines represent the quantum corrected value of the conductivity for each structure. There is no quantum correction available for tem- peratures within the shadowed area. (b) Ratio κring/κribbon as a function of temperature T. increases very quickly with T until it reaches a maximum value and then it slowly decreases. When no quantum correction is considered, the maximum κ value is reached at lower temperatures. We also find that low temperatures favor a reduction of the thermal conductivity in rings, both in symmetric and asymmetric configurations (see lower panel of Figure 6.6). As before, the reduction is stronger for symmetric rings. For these rings, κ is only about 40% of the conductance of the corresponding GNR for temperatures near 100 K, and even for temperatures as high as 1000 K symmetric rings cause a significant decrease of the thermal conductivity (κring < 0.8κribbon). 86 6.4. THERMAL CONDUCTIVITY IN GRAPHENE NANORINGS 6.4.3 Edge Disorder and Functionalization Thus far, we have considered ideal nanostructures with perfect edges. However, real samples would present imperfections or disorder, which can affect the thermal response of the system. There exist different sources of disorder, such as charged impurities in the substrate, native defects and imperfections of FIGURE 6.7. (a) Schematic representation of an asymmetric graphene ring with edge disorder. (b) Lattice thermal conductivity as a func- tion of the probability of removal p for W = 2.5 nm and T = 300 K. All thermal conductivities are normalized by the thermal conduc- tivity of the corresponding perfect ribbon or ring, κ0. (c) Ratio κring/κribbon as a function of the probability of removal p. 87 CHAPTER 6. MOLECULAR DYNAMICS STUDY OF THE LATTICE... the edges. The impact of the latter on the lattice thermal conductivity will probably be rather significant, especially in nanoscale systems. In order to estimate a possible impact of the edge disorder (also known as rough edges) on the heat current, we consider disordered samples in which we delete carbon atoms from the edges. As we did in the previous chapter, carbon atoms are randomly removed from the zigzag edges with some given probability p. To avoid dangling atoms in the armchair edges, pairs of neighbor atoms are removed with the same probability. We find that the thermal conductivity is almost independent of the exact position of the removed atoms as long as the probability p remains the same. Even so, results are averaged over five realizations of the disordered sample. Figure 6.7(b) shows the thermal conductivity of the GNRs and nanorings as a function of the probability p when W = 2.50 nm and T = 300 K. All thermal conductivities are normalized to the thermal conductivity of the corresponding perfect nanostructure, denoted by κ0. Our results show that rough edges decrease κ in the three studied nanostructures. This is in good agreement with the results reported in References [78, 79], where it was found that rough edges can cause a suppression of the thermal conductivity in GNRs due to the scattering of vibrational modes. Furthermore, Figure 6.7(b) shows GNRs would be the most affected, while symmetric rings would be the least. It is worth mentioning that κring(p)< κribbon(p) continues to be true in spite of the already marked decrease observed for κribbon(p) when increasing p. Figure 6.7(c) demonstrates that the thermal conductivity in symmetric rings is about 70% of the corresponding ribbon conductivity for p = 20%, which is the largest probability of removal considered in this work. One major issue regarding heat transport in narrow GNRs is to elucidate up to what extent would chemical functionalization affect the results. Among various functional groups, hydrogen has attracted considerable interest in recent years (see Reference [92] and references therein). In order to answer this question, we have conducted NEMD simulations of GNRs and rings with edges saturated by hydrogen atoms. We found that the thermal conductivity slightly decreases as compared to open GNRs and rings of the same size. This finding seems consistent with NEMD simulations carried out in hydrogenated GNRs [92] and GNRs with hydrogen termination [78]. The decrease of the thermal conductivity is more pronounced in rings than in GNRs because the 88 6.5. CONCLUSIONS larger edge length of the former. However, in both cases the reduction of the thermal conductivity is not significant and the effects of hydrogen-saturated edges can be safely neglected. 6.5 Conclusions In this chapter, we have studied the lattice thermal conductivity of graphene nanorings and GNRs by means of NEMD simulations. We found a significant reduction of the thermal conductivity κlat in symmetric rings, especially for narrow ribbon widths and low temperatures, as compared to uniform GNRs. Even at temperatures as high as 1000 K, our results show a substantial de- crease of κlat. The impact of rough edges on heat transport was also addressed and we concluded that it is higher in GNRs. Nevertheless, the thermal con- ductivity of disordered nanorings is considerably smaller than that of GNRs at the same magnitude of disorder. Therefore, at this point we know nanorings present two main advantages for exploiting their thermoelectric properties as compared to GNRs. First, as we demonstrated in Chapter 5, quantum interference effects enhance the Seebeck coefficient and the electronic contribution to the thermoelectric figure of merit. Second, the scattering of vibrational modes at bends yields a strong reduction of the lattice thermal conductivity κlat. We notice, however, the quantum interference effects that enhance the electronic properties take place at temperatures below 10 K, while MD results are not reliable at such low temperatures. This issue will be addressed in the next chapter by using a combination of DFTB calculations and Green’s function techniques. 89 C H A P T E R 7 Impact of the Device Geometry on Electron and Phonon Transport in Graphene Nanorings 7.1 Introduction Exploration of new materials and novel physical mechanisms for heat-to- electricity conversion lies at the heart of thermoelectric research. We learned in previous chapters that the thermoelectric efficiency of materials and devices is usually quantified by the dimensionless figure of merit ZT =σS2T/κ [104]. Here, S stands for the Seebeck coefficient, and σ and κ are the electrical and thermal conductances at temperature T, respectively. Because both electrons and phonons can contribute to the heat current, the thermal conductance is split as κ = κel +κph. Although the figure of merit has formally no upper bound, values of ZT around 3−4 are needed for thermolectrics to compete with other energy-generation technologies [102]. Bulk materials are, however, usually well below this threshold because of the unfavorable interdependence of S, σ, and κ in 3D systems. 91 CHAPTER 7. IMPACT OF THE DEVICE GEOMETRY ON ELECTRON... Miniaturization of materials was predicted long ago to improve the ther- moelectric efficiency [107, 108]. Phonon scattering in nanostructures is a promising route to reduce the phonon thermal conductivity while maintaining good electric properties. Additionally, quantum effects allow these systems to overcome classical limitations as the Wiedemann-Franz law. For instance, sharp resonances in the electron transmission (such as Fano lineshapes) make the ratio σ/κel increases above this limit [61, 128, 129]. The combina- tion of these two factors, namely reduced κph and enhanced σ/κel, explains why nanometer-sized objects can exhibit values of ZT unachievable by bulk materials [109–115]. Graphene nanorings are ideal examples of nanometer-sized systems that can exploit both approaches to improve the thermoelectric efficiency. On one side, thermal transport in graphene is strongly affected by different mech- anisms, such as rough edges [79], hydrogen-passivation, [91] and pattern- ing [81–84]. For instance, arrays of nanopores [87] and double-bends [86] drastically reduce the phonon contribution in GNRs. Also, using NEMD sim- ulations, we demonstrated in Chapter 6 that the thermal conductivity in graphene rings can be greatly reduced as compared with GNRs of the same width [224]. It should be noticed, however, that NEMD simulations exclude quantum effects and, consequently, they cannot be regarded as a valid tool for studying graphene nanostructures below 107 K. It is precisely at low temperatures where graphene rings can make use of quantum interference phenomena. In Chapter 5, we proved that these effects can give rise to Fano resonances that strongly enhance the Seebeck coefficient S and the electronic contribution to the figure of merit ZTel of graphene rings [212]. In this chapter, we analyze both electron and phonon transport, as well as the full thermoelectric response of graphene rings. Because MD simulations are not valid at the low temperatures required for the quantum interference effects to be noticeable, we first explore the phonon transport properties in the low temperature regime by using the DFTB method [191] combined with Green’s function (GF) techniques. We study three different configurations to elucidate the optimal geometry yielding the lower κph value. Once this magnitude is determined, we analyze the electron transport as we did in Chapter 5, that is, using the QTBM [152] combined with the TMM [153] adapted for graphene [33, 154]. Finally, the corresponding ZT is obtained 92 7.2. SYSTEM, MODEL AND METHODOLOGY FIGURE 7.1. Schematic diagram of the three configurations of graphene nanorings connected to two ideal leads, represented by shadowed areas. (a) Symmetric, (b) asymmetric and (c) orthogo- nal configurations. Throughout this work we take w0 = 1.23 nm, w1 ≈ w and l0 = 1.7 nm. over a wide range of parameters, finding a remarkable enhancement as compared to GNRs. Additionally, we observe the phonon contribution does not significantly degrade the thermoelectric response of graphene rings in the vicinity of Fano resonances, which are the features that yield the largest values of ZT. 7.2 System, Model and Methodology The system under study is a graphene rectangular ring with arms of width w attached to two semi-infinite aGNRs of width w0, which act as contacts. Depending on where these contacts are placed, we obtain different configura- 93 CHAPTER 7. IMPACT OF THE DEVICE GEOMETRY ON ELECTRON... tions, which we call symmetric, asymmetric, and orthogonal (see Figure 7.1 for schematics). Throughout this work we take w0 = 1.23 nm, l0 = 1.7 nm and we also consider the ring to be square and uniform (l = 3w0, w1 ≈ w) unless otherwise stated. Additionally, all edges are H-passivated. Structural relaxation of the graphene nanostructures was carried out by means of the DFTB method (see Appendix B) as implemented in the DFTB+ software package [191]. This method is based on the density-functional theory, but also uses ideas of the semi-empirical TB approach, hence the combination of accuracy and numerical efficiency. In particular, it allows dealing with systems up to 2000 atoms in a quantum simulation, and has demonstrated to provide reliable results for graphene-based systems [225, 226]. We have used the Slater-Koster parameters obtained by Niehhaus et al. [192] for C and H atoms. The geometry optimization was performed by using the conjugate gradient method, until the absolute value of the interatomic forces was below 10−5 atomic units. Interestingly, most of the Γ (including all the acoustic phonons as well) and K phonon modes have been shown to be uncoupled to electrons [227]. Therefore, as a reasonable approximation, electron-phonon interaction is not considered in our calculations. Consequently, phonon and electron transport can be treated independently within the Landauer approach [23] in terms of the corresponding electron τel(E) and phonon τph(ω) transmission coefficients. For the low temperatures mostly considered here, phonon-phonon interactions can also be safely neglected. Phonon transport properties have been computed by combining the DFTB method with GF techniques [228], as implemented in the in-house version of DFTB+ code developed in TU Dresden [226, 229, 230]. Based on this, we divide the whole system into three regions (see Figure 7.1), namely left and right contacts, and a device (scattering) region. Then, we calculate the phonon transmission coefficient τph(ω) as: (7.1) τph(ω)=Tr (GrΓLGaΓR) , where the broadening functions are ΓL/R = i(Σr L/R −Σa L/R) with ΣL/R being the contact self-energies. The retarded Green’s function is Gr = (ω21−K −Σr L − Σr R)−1, where K is the dynamical matrix, ω is the phonon frequency, and 1 is the unit matrix. K is obtained by numerically differentiating the forces calculated using the DFTB method. Using this formalism, the heat flux can 94 7.2. SYSTEM, MODEL AND METHODOLOGY be written in the Landauer form (7.2) J = ∫ ~ω 2π τph(ω)[N1(ω,T1)−N2(ω,T2)]dω with kB and ~ being the Boltzmann and Planck constants, respectively, and Ni(ω,Ti) = {exp[~ω/kBTi]−1}−1 the Bose-Einstein distribution of contact i at the temperature Ti. Because we restrict ourselves to the linear regime, the temperature difference ∆T between contacts is assumed to be small and a linear expansion in ∆T of the quantity N(ω,T +∆T)−N(ω,T) can be performed. Finally, the thermal conductivity is obtained as (7.3) κph = J ∆T = ~2 2πkBT2 ∫ ∞ 0 ω2e~ω/kBT (e~ω/kBT −1)2 τph(ω)dω . To calculate the electron transport properties, we have employed the same method that we used in Chapter 5. The nanostructures are described by a nearest-neighbor TB Hamiltonian (7.4) H =∑ i εi|i〉〈i|− t ∑ 〈i, j〉 |i〉〈 j| , where |i〉 is the π-orbital of the i-th carbon atom, and the hopping parameter and site energy are set to t = 2.8 eV and εi = 0, respectively. Assuming that electrons are in the fully coherent regime, we use the QTBM [152] combined with the TMM [153] adapted for graphene [33, 154] to calculate wave functions and the electron transmission coefficient τel(E) (see Appendix A for further details). The transmission coefficient τel(E) has also been calculated using the GF-DFTB approach for testing purposes, obtaining an excellent agreement between both methods. Therefore, in the following we use the above-described approach since it demands less computational resources. The Seebeck coefficient S, the electrical conductance σ, and the electronic thermal conductance κel can be calculated from the following expressions (see Section 5.2 in Chapter 5 for further details) S =− 1 eT K1 K0 ,(7.5a) σ= e2K0 ,(7.5b) κel = 1 T ( K2 − K2 1 K0 ) ,(7.5c) 95 CHAPTER 7. IMPACT OF THE DEVICE GEOMETRY ON ELECTRON... where the functions Kn are given in terms of the moments of the electronic transmission function (7.5d) Kn = 2 h ∫ ∞ −∞ ( − ∂ f ∂E ) (E−µ)nτel(E)dE . It is worth mentioning that these expressions are valid in the linear response regime. In this equation µ is the chemical potential of the graphene contacts, and f (E)= {exp[(E−µ)/kBT]+1}−1 is the Fermi distribution function. Finally, including the phonon thermal conductance κph previously calculated, the thermoelectric figure of merit ZT can be straightforwardly obtained. 7.3 Phonon Transport We first address the study of the phonon transport properties of the different configurations described in the previous section, namely symmetric, asym- metric and orthogonal. To do so, we keep w0 = 1.23 nm and consider three different arm widths: w = 0.4w0, w = w0, and w = 1.5w0. Note the latter case does not present a hole. The phonon transmission coefficient τph(ω) and the phonon thermal con- ductance κph for a GNR of width w0 (as reference system) and all the above- mentioned rings are presented in Figure 7.2. We observe that there are two clearly distinguishable regimes in the thermal conductance for the three con- figurations. In the case of symmetric rings, the critical temperature separating both regimes is around 30−50 K [see Figure 7.2(b)]. Above this temperature, the GNR has the highest thermal conductance and for the rings, it decreases as the arms becomes narrower. As we demonstrated in Chapter 6, wider GNRs present a higher thermal conductivity because the number of vibra- tional modes is larger and the edge effect less pronounced. However, below the critical temperature, the behavior is radically the opposite and all the rings show a larger conductance than the GNR. In Figure 7.2(a), it can also be seen that the phonon transmission τph decreases for narrower w with the exception of the lowest frequencies, where a peak arises for the case w = 0.4w0. Precisely at low temperatures the phonon distribution is shifted to the long-wavelength (or low frequency) portion of the spectrum. Further below, it will be discussed whether or not phonon interference is responsible for this counter-intuitive phenomenon. 96 7.3. PHONON TRANSPORT FIGURE 7.2. Phonon transmission coefficient τph as a function of the phonon frequency (panels a, c and e) and phonon thermal conductance κph as a function of temperature (panels b, d and f) for symmetric (top), asymmetric (middle) and orthogonal (bottom) configurations. For all the rings w0 = 1.23 nm and w is indicated in the legend. Grey shadowed curves represent the results for a GNR of width w0. Both the asymmetric and orthogonal configurations present a similar behavior to the symmetric one. However, while the asymmetric configuration also shows a critical temperature around 30−50 K, the orthogonal one has a lower one around 10 K. Again, below the critical temperature the thermal conductance is higher for the rings when comparing to the GNR geometry and the effect of the ratio w/w0 is, a priori, unpredictable. At higher temperatures, the main difference with the symmetric rings is that now the systems with w = w0 and w = 1.5w0 display almost the same conductance. An explanation 97 CHAPTER 7. IMPACT OF THE DEVICE GEOMETRY ON ELECTRON... FIGURE 7.3. (a) Schematic diagram of a symmetric ring with w = 0.4w0 and varying length l. (b) Phonon transmission τph as func- tion of the phonon frequency and (c) phonon thermal conductance κph as a function of temperature for the above depicted rings. For all the rings w0 = 1.23 nm, w = 0.4w0, and l is indicated in the plot. Grey shadowed curves represent the results for a GNR of width w0. of this phenomenon relies on the fact that phonon transport from source to drain is not affected by the hole placed sideway. Once the hole is large enough to alter the transport path, the thermal conductance is degraded. To assess if the unexpected behavior at low temperatures is due to phonon interference effects, we study the symmetric ring with w = 0.4w0 and vary the length of the arms l (see Figure 7.3). The transmission for the case l = 3w0 was the one that presented a very pronounced peak at low frequencies. How- ever, when we vary l, both the peak and the conductance decrease. This could be caused by a phonon interference mechanism. To further understand it, in 98 7.3. PHONON TRANSPORT FIGURE 7.4. (a) Local phonon density of states at ω= 18 cm−1 for the three rings considered in Figure 7.3. (b) Phonon thermal conduc- tance κph as a function of the length l for several temperatures given in the plot. Figure 7.4, we plot the local phonon density of states (LDOS) at ω= 18 cm−1, which is the frequency at which the peak in the transmission spectrum oc- curs. The LDOS per atomic site is defined as ηi(ω) = −(2ω/π) (Im Gr[ω])ii. Accordingly, we observe that for l = 3w0 the density is localized in the whole system as it occurs in resonant transmission. When the length is decreased to l = 2.66w0 the density is localized at the edges and corners of the ring. However, when l = 3.35w0, which was the ring with the lower conductance at low temperature in Figure 7.3(c), the phonon density is only localized at 99 CHAPTER 7. IMPACT OF THE DEVICE GEOMETRY ON ELECTRON... the corners. Additionally, in Figure 7.4(b) we observe the dependence of the thermal conductance on the length l is a non-monotonic function, especially at low temperatures. Similar results were obtained for smaller rings (not shown here). These results reinforced the idea that phonon interference could be responsible for the behavior of graphene rings at low temperatures. We also notice that phonon–phonon interactions should be taken into ac- count at high temperatures. Even though, the results are in good qualitative agreement with our MD study from the previous chapter, which automatically include anharmonicities to all orders, where we analyze the thermal conduc- tance of graphene rings with w = w0 [224]. In brief, by using both methods, MD and DFTB in combination with GF technique, we see that the thermal conductance of symmetric graphene rings is markedly reduced compared to other configurations. The only exception occurs at very low temperatures, when quantum effects play a significant role and classical simulations are no longer valid. 7.4 Electron Transport and Thermoelectric Response Next, we focus on the electron transport through the rings. We set the temper- ature at T = 4 K so that quantum interference effects are not smoothed out. The electron transmission τel for all the rings studied in the previous section is plotted in Figure 7.5. As expected, we observe the symmetric rings present the softest spectrum with smooth resonances, while both the asymmetric and orthogonal show more abrupt features, including Fano resonances (asymmet- ric lineshapes). Both configurations, asymmetric and orthogonal, introduce geometrical asymmetries in the system which induce Fano resonances in the electron transmission spectrum, as we showed in Chapters 4 and 5 [201, 212]. The electric thermal conductance κel is plotted in Figure 7.6 along with the electron contribution to the thermoelectric figure of merit ZTel = S2σT/κel and the full figure of merit including both electron and phonon contributions, namely ZT = ZTelκel/(κel +κph). In most cases we observe that the order of magnitude of κel is about 10−12 W/K, which is similar to the order of magnitude of the phonon thermal conductance at 4 K. The thermoelectric 100 7.4. ELECTRON TRANSPORT AND THERMOELECTRIC RESPONSE FIGURE 7.5. Electron transmission τel as a function of the incoming electron energy for symmetric, asymmetric, and orthogonal con- figurations. The curves are shifted upwards by a factor of 1 for clarity. FIGURE 7.6. Thermoelectric figure of merit ZTel (electron contribu- tion only) and full ZT (left axis), and electric thermal conductance κel (right axis) for symmetric, asymmetric, and orthogonal config- urations. 101 CHAPTER 7. IMPACT OF THE DEVICE GEOMETRY ON ELECTRON... figure of merit reaches the highest values at energies in the vicinity of Fano resonances and pronounced slopes of the transmission coefficient. In the case of the symmetric configuration, values of ZT ≈ 1 are only obtained when there is no hole in the system (w = 1.5w0) since geometries with w < 1.5w0 always yield smooth transmission profiles. For the asymmetric and orthogonal rings, values of ZT around unity or even higher are obtained in all cases, indicating that these configurations would be more suitable for thermoelectric applications. Additionally, as we demonstrated in Chapter 5, the application of a side-gate voltage between the two arms of the rings would allow to obtain much higher values of ZTel for both symmetric and asymmetric configurations. In some cases values as high as 8 were found. The main limitation of that previous study was the exclusion of the phonon contribution to the thermal conductance that ultimately degrades the figure of merit. However, as it can be observed in Figure 7.6, the addition of the phonon conductance in the calculation of ZT does not significantly degrade the efficiency in the vicinity of abrupt slopes and Fano resonances, which are the features that yield the largest values of ZT. 7.5 Conclusions In conclusion, we have theoretically studied phonon and electron transport in several ring-shaped graphene nanostructures, to ultimately assess their thermoelectric performance. Regarding the lattice thermal transport, we found that there are two clearly distinguishable regimes in the thermal conductance. For temperatures above the threshold temperature (which lies in the range 10−50 K, depending on the specific configuration) the GNR presents the highest thermal conductance and, for the rings, it decreases as the arms of the ring become narrower. Also, the ring hole does not affect thermal transport as long as it does not alter the path from source to drain. These results qualitatively agree with the results we obtained in Chapter 6. On the contrary, at low temperatures, when quantum effects come into play, the behavior is the opposite since nanorings present higher thermal conductance than the GNRs of the same width in most cases. We suggest and provide some evidence of phonon interference to be responsible for this counter-intuitive behavior, opening new possibilities of engineering phonon transport in a 102 7.5. CONCLUSIONS similar way as electron transport in nanorings. As for electron transport, we found that the asymmetric and the orthogonal configurations show sharper electron transmission profiles than the symmet- ric ones, including Fano resonances. These features give rise to higher values of the thermoelectric figure of merit ZT. Additionally, although we observe that electrons and phonons contribute equally to the thermal conductance at T = 4 K, the inclusion of the phonon contribution does not significantly degrade ZT in the vicinity of high slopes and Fano resonances. The low ther- mal conductance together with the use of quantum interference effects to obtain Fano resonances to enhance the power factor, make graphene nanorings promising candidates for low temperature thermoelectric applications. 103 C H A P T E R 8 Conclusions In this chapter the main results are summarized and a critical analysis is made, exploring the limitations of our approaches and commenting the possi- ble prospects as well. Several graphene nanostructures have been extensively studied throughout this thesis, hence we separate the conclusions according to each one of these systems, namely, graphene nanoconstrictions (GNCs), twisted graphene nanoribbons (GNRs) and graphene nanorings. Graphene Nanoconstrictions The impact of several factors on the electronic transport and the conductance quantization in GNCs was thoroughly analyzed in Chapter 2. Our main findings are listed below: • Zigzag GNRs present a 4e2/h stepwise electrical conductance and thus preserve the valley-symmetry. On the contrary, in armchair GNRs the valley degeneracy is broken and the electrical conductance shows 2e2/h steps. 105 CHAPTER 8. CONCLUSIONS • Regardless of the type of edges, when GNRs are tailored into GNCs, size quantization is smoothed out due to backscattering. • The 4e2/h quantization can be more easily observed in zigzag GNCs. However, armchair GNCs can recover the valley-symmetry conservation in some specific cases, namely for very short constrictions lengths and for rounded-shaped profiles. The latter approach proved to yield more pronounced 4e2/h plateaus. • For ribbons comprising multiple constrictions, we found well defined peaks (resonances) when the separation among constrictions is suffi- ciently small. • Rough edges were found to highly degrade the conductance, although signs of quantization were still noticeable for low edge roughness. • We propose an effective scaling method which allows to extrapolate results from small systems to experimentally feasible sized systems. Its validity was tested by comparing with experimental high quality GNCs. The profile of the sample edges obtained by AFM measurements was employed to simulate the actual GNC, finding an excellent agreement between the measured and calculated electrical conductance. In conclusion, our findings provide a deeper insight into the role of geo- metric factors in the electronic transport in GNCs, which could result in the fabrication of more optimized tailored structures able to preserve the size quantization and the valley symmetry. The high control in the edge-definition process required to this end could already be feasible, as recent experiments have demonstrated (see Section 2.6 and Reference [157]). Therefore, the possi- bility of using GNCs for valleytronics applications could be a reality in the near future. Twisted Graphene Nanoribbons The possibility of controlling electron transport and engineering the current- voltage characteristics in twisted zigzag graphene nanoribbons (zGNRs) by means of a transverse electric field was explored in Chapter 3. The following results were found: 106 • Below the threshold value of the torsion parameter (W/λ< 0.25), elec- tron transport in zGNRs is barely affected by the mechanical deforma- tion induced by the twisted conformation. • The transmission coefficient for twisted zGNRs is highly sensitive to the transverse electric field and, thus, the source-drain electric current can be effectively controlled, allowing the system to operate as an efficient field-effect transistor with the on/off ratio on the order of 1000. • If the working point is chosen appropriately, that is, the chemical po- tential of the leads and the transverse electric field are set to suitable values, twisted zGNRs can manifest negative differential resistance (NDR) and an N-type current–voltage characteristics similar to those of tunnel diodes. • In some cases, current-voltage curves can display multiple-peak NDR, opening a possibility for applications in multiple-valued logic circuits. In conclusion, our findings suggest that the current–voltage characteristics of twisted zGNRs can be effectively engineered to suit different nanoelectronic applications. Graphene Nanorings Several configurations of graphene nanorings were analyzed throughout Chap- ters 4, 5, 6 and 7. Regarding the electron transport in these systems, the following results were found: • An asymmetric connection between the ring and the contacts can give rise to sharp features in the electron transmission spectra, including Fano resonances. • The application of a side-gate voltage accross the arms of the ring can lead to the appearance of new Fano resonances, regardless of the symmetry of the system configuration. • In Chapter 4, it was found that these features can be used to obtain a highly spin-polarized transmission when a ferromagnetic insulator is placed on top of the graphene nanoring. 107 CHAPTER 8. CONCLUSIONS • In Chapter 5, Fano resonances were proved to have a great impact on the thermoelectric properties of the graphene rings. At low temperature and in the absence of phonon contribution, a figure of merit as high as ZT ' 8 was obtained for certain values of the side-gate voltage. Regarding the thermal properties, two different methods were used, namely, MD simulations (Chapter 6) and DFTB simulations in combination with GF techniques (Chapter 7). The former approach is valid if the quantum effects can be neglected, that is, for a high enough temperature (approximately T > 107 K in the case of graphene), while the latter is suitable in the absence of phonon–phonon interactions, that is, in the low temperature regime. The main results are summarized below: • For a temperature above the threshold one (which lies in the range 10−50 K, depending on the specific geometry configuration) the pres- ence of bends, corners, or a hole in the path from source to drain, strongly degrade the thermal conductance κlat as compared to uniform GNRs. The decrease is especially remarkable for symmetric rings and narrow widths. These results were obtained by using both methods. • As the temperature increases, κlat in rings is less reduced with respect to GNRs. However, even at 1000 K, our MD simulations show that κlat in a 2.5-nm-wide symmetric ring is only 80% of the value of κlat in the corresponding 2.5-nm-wide uniform GNR. • Below the threshold temperature, graphene nanorings present a higher κlat than ribbons of the same width in the vast majority of cases. We suggest and provide some evidence of phonon interference to be respon- sible for this counter-intuitive behavior, opening new possibilities of engineering phonon transport in a similar way as electron transport in nanorings. • Even though the phonon (or lattice) contribution is thought to be the most important contribution to heat transport in carbon-based materials due to the strong covalent sp2 bonding, we found that the electron and phonon contributions are of the same order of magnitude (κph ∼ κel) for graphene rings at low temperature. 108 • Regarding the thermoelectric response of the rings, the inclusion of the phonon contribution does not significantly degrade ZT in the vicin- ity of high slopes and Fano resonances, which are the features that were showed to enhance the electronic contribution to ZT. Therefore, graphene nanorings arise as promising candidates for low temperature thermoelectric applications. In conclusion, we have demonstrated that graphene nanorings are highly versatile structures which can be employed for spintronics, thermal managa- ment and thermoelectric applications. Electron transport in such systems can be engineering by using quantum interference effects while phonon conduc- tance can be greatly modified with an appropriate design. Prospective Research Throughout this thesis several approximations were made. Particularly, since charge carriers were treated as non-interacting particles, a possible extension of these works could be the inclusion of the electron–electron and electron– phonon interactions, as well as effects related to substrates or enviroment. Additionally, the following topics could also be envisioned as an extension of the present thesis: • A more extensive analysis on GNRs with several nanoconstrictions. • The effect of the transverse electric field on twisted armchair GNRs, edge-disordered GNRs, or more realistic twisted conformations obtained, for example, from MD simulations. As we demonstrated in Chapter 3, the use of higher accuracy methods would be required to calculate the electron properties in armchair ribbons. • The effect of a ferromagnetic insulator put in contact with the ring (as we did in Chapter 4) on the thermoelectric properties could be of interest in spin caloritronics. This type of device could show a spin- dependent Seebeck effect and may have a potential for spin caloritronic devices, which are multifunctional devices that combine spintronics with caloritronics and could play a significant role in low-power-consumption systems. 109 CHAPTER 8. CONCLUSIONS • A more extensive analysis of the use of the phonon interference effects found in Chapter 7 for an effectively engineering of thermal transport in graphene rings. 110 A P P E N D I X A Quantum Transmitting Boundary Method and Transfer Matrix Method Quantum Transmitting Boundary Method For structures in which electrons are in the ballistic and coherent regime, the electronic transport can be described by the effective-mass Schrödinger equation [231] (A.1) H |ψ〉 = E|ψ〉 . The quantum transmitting boundary method (QTBM) [152] is a technique for solving this equation for current-carrying states based on the finite-difference approximation. It enables to obtain the wave function in the whole system |ψ〉, and then, using this wave function, it is possible to calculate the transmission and reflection coefficients. In order to use this method, appropriate boundary conditions need to be formulated. The region of interest is divided into a device or scattering region D, and several lead regions, Li, i = 1,2, ..., N. Throughout this thesis, we set N = 2 111 APPENDIX A. QTBM AND TMM FIGURE A.1. Scheme of a system composed of a scattering region D and N = 2 semi-infinite leads, labelled Li. Each lead is split into identical cells C i, j with spatial period ∆i. because there are only a source and a drain leads in the studied structures (see Figure A.1 for schematics). Each lead Li consists on a quasi-one dimensional system with spatial period ∆i, made up of identical cells C i, j, j = 0,1..., each one containing Mi atoms. For each lead i at a given energy E, there are Mi ingoing eigenmodes |vin i,n〉 and Mi outgoing eigenmodes |vout i,n 〉, which are obtained from the TMM, which will be outline in next section. The wave function at the cell C i,0 can be written as |C i,0〉 = Ni∑ n=1 ai,n|vin i,n〉+ Ni∑ n=1 bi,n|vout i,n 〉 ≡ V̂ in i .ai + V̂ out i .bi , (A.2) where ai,n/bi,n are the amplitudes of the incoming/outgoing modes. Only the former are known quantities in our problem. Notice also the introduction of the matrix notation in the above equation. The wave function at the adjacent cell C i,1 is then obtained as |C i,1〉 = Ni∑ n=1 ai,n|vin i,n〉eikin i,n∆i + Ni∑ n=1 bi,n|vout i,n 〉eikout i,n∆i ≡ Ŵ in i .ai +Ŵout i .bi , (A.3) where kin/out i,n is the wavenumber for the corresponding ingoing/outgoing eigen- mode. Using Equations (A.2) and (A.3), the vector composed of unknown amplitudes bi, can be removed. Then, the following expression is obtained (A.4) |C i,1〉−Ŵout i ( V̂ out i )−1 |C i,0〉 = [ Ŵ in i −Ŵout i ( V̂ out i )−1 V̂ in i ] .ai , forming a set of ∑N i=1 Mi equations that can be written in a more compact way. Starting with (A.5) (H −E)|ψ〉 ≡ M̂0|ψ〉 = 0 , 112 the rows of M̂0 which correspond to atoms belonging to the C i,n cells are replaced by the left side of the corresponding Equation A.4. Similarly, the elements of the vector 0 in the same row are replace by the independent term (right side) of the corresponding Equation A.4. Finally, the wave function of the whole system |ψ〉 is obtained by solving a linear system (A.6) M̂|ψ〉 = b with both M̂ and b being sparse arrays. Transfer Matrix Method For a given energy E, the eigenmodes in the leads can be described by means of the transfer matrix method (TMM) [153]. Writing the Schrödinger equation for the the lead with hamiltonian Ĥ in the basis formed by the atoms belonging to the lth cell, we obtain (A.7) Ĥl,l−1ψl−1 + (Ĥl,l −E)ψl + Ĥl,l+1ψl+1 = 0 , where ψl is the set of coefficient of the wave function in that basis. It is common to write this expression in the transfer-matrix form (A.8)( ψl+1 ψl ) = ( −Ĥ−1 l,l+1(Ĥl,l −E) −Ĥ−1 l,l+1Ĥl,l−1 1 0 )( ψl ψl−1 ) = T ( ψl ψl−1 ) where T is the transfer matrix. The eigenvectors of T provide us the eigen- modes in the leads. Because this method requires a matrix inversion, when there are atoms in cell l that are not coupled to the atoms of the adjacent cell l+1, like in the case of graphene, singular matrix can be obtained. In this cases, a generalization of the TMM in which the unit cell is divided into two subcells, labelled L and R, can be used [33]. The sites in both of them must be connected to sites belonging to the same subcell or to its nearest neighbours. Furthermore, all sites belonging to L must be connected to both neighboring R subcells. Figure A.2 shows a valid two subcell division for an armchair and a zigzag GNR. 113 APPENDIX A. QTBM AND TMM FIGURE A.2. Scheme of an infinite GNR with (a) armchair, and (b) zigzag edges. A valid two subcell division is plotted using light/dark gray for L/R subcells Within this approach, the Schrödinger equation reads (A.9)  . . . ... ... ... ... ... · · · hL sL,R 0 0 · · · · · · s∗L,R hR dR,L 0 · · · · · · 0 d∗ R,L hL sL,R · · · · · · 0 0 s∗L,R hR · · · ... ... ... ... ... . . .   ... ψl,L ψl,R ψl+1,L ψl+1,R ...  = E  ... ψl,L ψl,R ψl+1,L ψl+1,R ...  , where hL and hR are square matrices, while sL,R and dR,L are in general rectangular. From this equation we can obtain s∗L,Rψl,L +hRψl,R +dR,Lψl+1,L = Eψl,R , d∗ L,Rψl,R +hLψl+1,L + sR,Lψl+1,R = Eψl+1,L . (A.10) Combining both equations we can get one containing only L subcells sL,R(E−hR)−1dR,Lψl+2,L = = (E−d∗ R,L(E−hR)−1dR,L − sL,R(E−hR)−1s∗L,R −hL)ψl+1,L −d∗ R,L(E−hR)−1s∗L,Rψl,L , (A.11) 114 which can be rewritten as (A.12) ψl+2,L = T̂1,1ψl+1,L + T̂1,2ψl,L , and, therefore, in a transfer matrix form (A.13) ( ψl+2,L ψl+1,L ) = ( T̂1,1 T̂1,2 1 0 ) ( ψl+1,L ψl,L ) . Further details on this method and its application to graphene structures can be found in Reference [154]. 115 A P P E N D I X B Density-Functional Tight-Binding Method The density-functional tight-binding (DFTB) method is based on density functional theory (DFT) as formulated by Hohenberg, Kohn and Sham [232, 233], and it also uses ideas of the semiempirical tight-binding (TB) method [234]. In DFT the energy is a functional of the electron density n(r) and for a system of electrons can be written as (B.1) E[n(r)]= Ts +Eext +EH +Exc +EII , where Ts is the non-interacting kinetic energy, Eext the external interactions (including electron-ion interactions), EH the Hartree energy, and EII ion-ion interactions. The term Exc = (T −Ts)+ (Eee −EH), where T is the kinetic energy of the interacting system and Eee the electron-electron interaction, is the exchange-correlation (xc) energy. This essential term includes and hides all the difficult many-body effects. DFTB can be derived from a Taylor series expansion of the density func- tional total energy around a properly chosen reference density n0(r) ≡ n0, i.e. the superposition of the atomic densities as if atoms in the system 117 APPENDIX B. DENSITY-FUNCTIONAL TIGHT-BINDING METHOD were free and neutral (see [235, 236] and references therein). This density does not minimize the energy, but neighbors the true minimization density nmin(r)= n0(r)+δn0(r), where δn0(r)≡ δn0 is supposed to be small. Expand- ing E[n] at n0 to second order, the energy reads (B.2) E[δn]≈ EBS[δn]+ECoul[δn]+Erep. The first term is the band-structure energy EBS[δn]=∑ a fa〈ψa|− 1 2 ∇2 +Vext +VH[n0]+Vxc[n0]|ψa〉 =∑ a fa〈ψa|H0|ψa〉 , (B.3) where fa is the occupation of a single-particle state ψa. The second term corresponds to the energy from charge fluctuations ECoul[δn]= 1 2 ∫ ∫ (δ2Eexc[n0] δnδn′ + 1 |r− r′| ) δnδn′d3rd3r′ = 1 2 ∑ i j γi j(Ri j)∆qi∆q j , (B.4) where ∆qi is the atomic population of atom i and γi j an analytical function. The third term is called the repulsive energy Erep =−1 2 ∫ VH[n0]n0d3r+Exc[n0]+EII − ∫ Vxc[n0]n0d3r = ∑ i< j V i j rep(Ri j) , (B.5) and can be approximated as a sum of pair-wise repulsive functions which are obtained by fitting to high-level theoretical calculations. Since tight-binding assumes tightly bound electrons, the KS orbitals ψi are expanded to a set of localized atomic orbitals φi (B.6) ψi = ∑ µ cµiφµ , which are obtained from DFT calculations of the corresponding atoms. The band structure energy becomes (B.7) EBS =∑ a fa ∑ µν ca∗ µ ca ν〈φµ|H0|φν〉 = ∑ a fa ∑ µν ca∗ µ ca νH0 µν . 118 The tight-binding formalism enters by accepting the matrix elements H0 µν as the principal parameters of the method. With all the terms alredy defined in the energy expression, the minimum is found by variation of δ(E −∑ a εa〈ψa|ψa〉), where εa are undetermined Lagrange multipliers, and obtain (B.8) ∑ ν ca ν(Hµν−εaSµν)= 0 , with (B.9) Hµν = H0 µν+ 1 2 Sµν ∑ k (γik +γ jk)∆qk . Because the Hamiltonian H0 µν and overlap Sµν matrix elements contain one- and two-centre contributions only, they can be calculated and tabulated in advance as functions of the distance between atomic pairs. Finally, as in Kohn-Sham equations in DFT, Equations (B.8) and (B.9) have to be solved self-consistently. From an initial guess for {∆qi} one obtains Hµν, then by solving Equation (B.8) one obtains new {ca µ} and finally new {∆qi}, iterating until self-consistency is achieved. 119 A P P E N D I X C Molecular Dynamics Molecular dynamics (MD) is a technique for computing both equilibrium and non-equilibrium properties of classical many-body systems. It is considered a classical method because the motion of the constituent particles, i.e. atoms or molecules, obeys the laws of classical mechanics [216, 237]. MD simulations are based on the numerical integration of Newton’s equation of motion for a set of N interacting particles (C.1) mi d2r i dt2 = j=N∑ j=1, j 6=i Fi j i = 1,2, ...N , where mi and r i are, respectively, the mass and the position of the particle i, and Fi j denotes the force exerted on particle i by particle j. This force Fi j corresponds to the negative derivative of the interatomic potential V between those particles (C.2) Fi j =− ∂V ∂r i j . To start with an MD simulation one needs to know the initial coordinates r i and velocities vi. In crystalline solids, the starting coordinates are defined by the crystal symmetry and the positions of the atoms within the unit cell. 121 APPENDIX C. MOLECULAR DYNAMICS The starting velocities can be chosen randomly from a Maxwell-Boltzmann or Gaussian distribution at a given temperature T. Repeating the integration with the updated configuration allows following the dynamics and obtaining the trajectory of the system under study. Then, after an equilibration time, physical quantities of interest can be obtained by the time average along the trajectory. Several algorithms have been developed to integrate the equations of motion, all of them assuming positions and velocities can be approximated by a Taylor series expansion. Because of its simplicity and stability, the velocity- Verlet algorithm [238], based on the original Verlet method [239], is one of the most frequently used. By using this approach, the positions are obtained as (C.3) r(t+∆t)= r(t)+v(t)∆t+ F(t) 2m ∆t2 , and the velocities by (C.4) v(t+∆t)= v(t)+ F(t+∆t)+F(t) 2m ∆t , where ∆t is the time step in the MD simulation. An appropriate value of ∆t needs to be chosen, since too large time steps can lead to algorithm errors and too small values result in much longer simulation time. Interatomic Potentials The accuracy and predictive power of MD simulations strongly depends on the interatomic potential used to describe the interactions between the con- stituent atoms of the system. These atomic interactions can be described at different levels of sophistication, being first-principle calculations the most accurate approach. Although the power of current supercomputers have made ab initio MD simulations feasible [240], they are extremely time consuming and unpractical when dealing with large systems or long time scales. In contrast, empirical interatomic potentials (EIPs), which include several pa- rameters that are fitted to reproduce experimental results, or results obtained by higher accuracy methods, can deal with systems up to millions of atoms and times of the order of nanoseconds [241]. One successful EIP for treating covalent bonding interactions is the Tersoff-type potential [242, 243]. Based on this model, several EIPs have 122 been developed for carbon, being the most widely used the hydrocarbon po- tential by Brenner [244], and its extensions [218, 245, 246]. Among these, the AIREBO potential [218] was the one employed in Chapter 5. By using the AIREBO potential, the entire system energy can be obtained as a sum of three different terms (C.5) E = 1 2 ∑ i ∑ j 6=i EREBO i j +ELJ i j + ∑ k 6=i j ∑ l 6=i jk ETORSION i jkl . The reactive empirical bond-order (REBO) term, EREBO i j , has the same func- tional form as the hydrocarbon REBO potential developed by Brenner [245]. This term is exclusively short-ranged, i.e. two atoms interact only if they are separated by a distance less than a covalent-bonding cutoff of rmax i j (2 Å for C-C bonds). Then, the interaction is described as (C.6) EREBO i j =V R i j (r i j)+bi jV A i j (r i j) , where V R i j and V A i j are repulsive and attractive pairwise potentials that de- pend only on the atom type (carbon or hydrogen) and the distance between the atoms r i j. However, the value of EREBO i j is also dependent on the local environment close to the i− j bond through the many-body term bi j. This term is characteristic of Tersoff-type potentials, and it accounts for the bond angles, coordination numbers, and other chemical effects that affect the strength of a covalent bonding. Additionally, the Lennard-Jones term, ELJ i j , adds longer- ranged interactions using a form similar to the standard Lennard-Jones potential, and the torsional term, ETORSION i jkl , is a 4-body potential dependent on the dihedral angles. Detailed formulas for these terms can be found in Reference [218]. 123 Bibliography [1] B. Bhushan (ed.) “Springer handbook of nanotechnology”. Springer (2003). [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov “Electric field effect in atomically thin carbon films”. Science 306, 666 (2004). [3] S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak, and A. K. Geim “Giant intrinsic carrier mobilities in graphene and its bilayer”. Phys. Rev. Lett. 100, 016602 (2008). [4] Y. Wu, D. B. Farmer, F. Xia, and P. Avouris “Graphene electronics: Materials, devices, and circuits”. Proc. IEEE 101, 1620 (2013). [5] G. Fiori, F. Bonaccorso, G. Iannaccone, T. Palacios, D. Neumaier, A. Seabaugh, S. K. Banerjee, and L. Colombo “Electronics based on two- dimensional materials”. Nat. Nanotechnol. 9, 768 (2014). [6] F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari “Graphene photonics and optoelectronics”. Nat. Photonics 4, 611 (2010). [7] J.-H. Ahn and B. H. Hong “Graphene for displays that bend”. Nat. Nanotechnol. 9, 737 (2014). [8] W. Han, R. K. Kawakami, M. Gmitra, and J. Fabian “Graphene spin- tronics”. Nat. Nanotechnol. 9, 794 (2014). [9] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees “Electronic spin transport and spin precession in single graphene layers at room temperature”. Nature 448, 571 (2007). 125 BIBLIOGRAPHY [10] A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau “Superior thermal conductivity of single-layer graphene”. Nano Lett. 8, 902 (2008). [11] S. Ghosh, I. Calizo, D. Teweldebrhan, E. P. Pokatilov, D. L. Nika, A. A. Balandin, W. Bao, F. Miao, and C. N. Lau “Extremely high thermal con- ductivity of graphene: Prospects for thermal management applications in nanoelectronic circuits”. Appl. Phys. Lett. 92, 151911 (2008). [12] A. A. Balandin “Thermal properties of graphene and nanostructured carbon materials”. Nat. Mater. 10, 569 (2011). [13] A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim “The electronic properties of graphene”. Rev. Mod. Phys. 81, 109 (2009). [14] M. Katnelson “Graphene: Carbon in two dimensions”. Cambridge Uni- versity Press (2012). [15] P. R. Wallace “The band theory of graphite”. Phys. Rev. 71, 622 (1947). [16] S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejón “Tight-binding description of graphene”. Phys. Rev. B 66, 035412 (2002). [17] K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus “Edge state in graphene ribbons: Nanometer size effect and edge shape depen- dence”. Phys. Rev. B 54, 17954 (1996). [18] L. Yang, C.-H. Park, Y.-W. Son, M. L. Cohen, and S. G. Louie “Quasipar- ticle energies and band gaps in graphene nanoribbons”. Phys. Rev. Lett. 99, 186801 (2007). [19] E. R. Mucciolo, A. H. Castro Neto, and C. H. Lewenkopf “Conductance quantization and transport gaps in disordered graphene nanoribbons”. Phys. Rev. B 79, 075407 (2009). [20] I. Martin and Y. M. Blanter “Transport in disordered graphene nanorib- bons”. Phys. Rev. B 79, 235132 (2009). 126 BIBLIOGRAPHY [21] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and de W. A. Heer “Electronic confinement and coherence in patterned epitaxial graphene”. Science 312, 1191 (2006). [22] J. Baringhaus, M. Ruan, F. Edler, A. Tejeda, M. Sicot, A. Taleb-Ibrahimi, A.-P. Li, Z. Jiang, E. H. Conrad, C. Berger, C. Tegenkamp, and W. A. de Heer “Exceptional ballistic transport in epitaxial graphene nanorib- bon”. Nature 506, 349 (2014). [23] S. Datta “Electronic transport in mesoscopic systems”. Cambridge University Press (1995). [24] J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J. Beenakker “Sub-poissonian shot noise in graphene”. Phys. Rev. Lett. 96, 246802 (2006). [25] M. Kim, J.-H. Choi, S.-H. Lee, K. Watanabe, T. Taniguchi, S.-H. Jhi, and H.-J. Lee “Valley-symmetry-preserved transport in ballistic graphene with gate-defined carrier guiding”. Nat. Phys. 12, 1022 (2016). [26] H. Lee, G.-H. Park, J. Park, G.-H. Lee, K. Watanabe, T. Taniguchi, and H.-J. Lee “Edge-limited valley-preserved transport in quasi-1D constriction in bilayer graphene”. Nano Lett. 18, 5961 (2018). [27] H. Overweg, H. Eggimann, X. Chen, S. Slizovskiy, M. Eich, R. Pisoni, Y. Lee, P. Rickhaus, K. Watanabe, T. Taniguchi, V. Fal’ko, T. Ihn, and K. Ensslin “Electrostatically induced quantum point contacts in bilayer graphene”. Nano Lett. 18, 553 (2018). [28] S. Russo, J. B. Oostinga, D. Wehenkel, H. B. Heersche, S. S. Sobhani, L. M. K. Vandersypen, and A. F. Morpurgo “Observation of Aharonov- Bohm conductance oscillations in a graphene ring”. Phys. Rev. B 77, 085413 (2008). [29] M. Huefner, F. Molitor, A. Jacobsen, A. Pioda, C. Stampfer, K. Ensslin, and T. Ihn “The Aharonov-Bohm effect in a side-gated graphene ring”. New J. Phys 12, 043054 (2010). 127 BIBLIOGRAPHY [30] J. Wurm, M. Wimmer, H. U. Baranger, and K. Richter “Graphene rings in magnetic fields: Aharonov–Bohm effect and valley splitting”. Semi- cond. Sci. Technol. 25, 034003 (2010). [31] M. Zarenia, J. M. Pereira, A. Chaves, F. M. Peeters, and G. A. Farias “Simplified model for the energy levels of quantum rings in single layer and bilayer graphene”. Phys. Rev. B 81, 045431 (2010). [32] Z. Wu, Z. Z. Zhang, K. Chang, and F. M. Peeters “Quantum tunneling through graphene nanorings”. Nanotechnology 21, 185201 (2010). [33] J. Schelter, D. Bohr, and B. Trauzettel “Interplay of the Aharonov-Bohm effect and Klein tunneling in graphene”. Phys. Rev. B 81, 195441 (2010). [34] V. H. Nguyen, Y. M. Niquet, and P. Dollfus “Aharonov-Bohm effect and giant magnetoresistance in graphene nanoribbon rings”. Phys. Rev. B 88, 035408 (2013). [35] D. Faria, R. Carrillo-Bastos, N. Sandler, and A. Latgé “Fano resonances in hexagonal zigzag graphene rings under external magnetic flux”. J. Phys. Condens. Matter 27, 175301 (2015). [36] J. Munárriz, F. Domínguez-Adame, and A. V. Malyshev “Towards graphene-based quantum interference devices”. Nanotechnology 22, 365201 (2011). [37] D. J. P. de Sousa, A. Chaves, J. M. Pereira, and G. A. Farias “Interferom- etry of Klein tunnelling electrons in graphene quantum rings”. J. Appl. Phys. 121, 024302 (2017). [38] T. W. Chamberlain, J. Biskupek, G. A. Rance, A. Chuvilin, T. J. Alexan- der, E. Bichoutskaia, U. Kaiser, and A. Khlobystov “Size, structure, and helical twist of graphene nanoribbons controlled by confinement in carbon nanotubes”. ACS Nano 6, 3943 (2012). [39] A. L. Elías, A. R. Botello-Méndez, D. Meneses-Rodríguez, V. J. González, D. Ramírez-González, L. Ci, E. Muñoz-Sandoval, P. M. Ajayan, H. Ter- rones, and M. Terrones “Longitudinal cutting of pure and doped car- bon nanotubes to form graphitic nanoribbons using metal clusters as nanoscalpels”. Nano Lett. 10, 366 (2010). 128 BIBLIOGRAPHY [40] A. N. Khlobystov “Carbon nanotubes: from nano test tube to nano- reactor”. ACS Nano 12, 9306 (2011). [41] L. Zhang and X. Wang “Atomistic insights into the nanohelix of hydro- genated graphene: Formation, characterization and application”. Phys. Chem. Chem. Phys. 16, 2981 (2014). [42] D. G. Papageorgiou, I. A. Kinloch, and R. J. Young “Mechanical prop- erties of graphene and graphene-based nanocomposites”. Prog. Mater. Sci. 90, 75 (2017). [43] Y.-W. Son, M. L. Cohen, and S. G. Louie “Half-metallic graphene nanorib- bons”. Nature 444, 347 (2006). [44] O. V. Yazyev “Magnetism in disordered graphene and irradiated graphite”. Phys. Rev. Lett. 101, 037203 (2008). [45] H.-X. Yang, M. Chshiev, D. W. Boukhvalov, X. Waintal, and S. Roche “Inducing and optimizing magnetism in graphene nanomeshes”. Phys. Rev. B 84, 214404 (2011). [46] D. Soriano, N. Leconte, P. Ordejón, J.-C. Charlier, J.-J. Palacios, and S. Roche “Magnetoresistance and magnetic ordering fingerprints in hydrogenated graphene”. Phys. Rev. Lett. 107, 016602 (2011). [47] K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, and R. K. Kawakami “Magnetic moment formation in graphene detected by scattering of pure spin currents”. Phys. Rev. Lett. 109, 186604 (2012). [48] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi “Proximity-induced ferromagnetism in graphene revealed by the anomalous Hall effect”. Phys. Rev. Lett. 114, 016603 (2015). [49] H. Haugen, D. Huertas-Hernando, and A. Brataas “Spin transport in proximity-induced ferromagnetic graphene”. Phys. Rev. B 77, 115406 (2008). [50] A. G. Swartz, P. M. Odenthal, Y. Hao, R. S. Ruoff, and R. K. Kawakami “Integration of the ferromagnetic insulator EuO onto graphene”. ACS Nano 6, 10063 (2012). 129 BIBLIOGRAPHY [51] J. Munárriz, A. V. Malyshev, P. A. Orellana, and F. Domínguez-Adame “Graphene nanoring as a tunable source of polarized electrons”. Nan- otechnology 23, 205202 (2012). [52] H. X. Yang, A. Hallal, D. Terrade, X. Waintal, S. Roche, and M. Chshiev “Proximity effects induced in graphene by magnetic insulators: First- principles calculations on spin filtering and exchange-splitting gaps”. Phys. Rev. Lett. 110, 046603 (2013). [53] A. Hallal, F. Ibrahim, H. Yang, S. Roche, and M. Chshiev “Tailoring magnetic insulator proximity effects in graphene: first-principles calcu- lations”. 2D Mater. 4, 025074 (2017). [54] D. Vasileska, K. Raleva, and S. M. Goodnick “Modeling heating effects in nanoscale devices: the present and the future”. J. Comput. Electron. 7, 66 (2008). [55] E. Pop, S. Sinha, and K. E. Goodson “Heat generation and transport in nanometer-scale transistors”. Proc. IEEE 94, 1587 (2006). [56] E. Pop “Energy dissipation and transport in nanoscale devices”. Nano Res. 3, 147 (2010). [57] J. H. Lienhard IV and J. H. Lienhard V “A heat transfer textbook”. Phlogiston Press (2003). [58] J. Kaiser, T. Feng, J. Maassen, X. Wang, X. Ruan, and M. Lundstrom “Thermal transport at the nanoscale: A Fourier’s law vs. phonon Boltz- mann equation study”. J. Appl. Phys. 121, 044302 (2017). [59] J. M. Ziman “Principles of the theory of solids”. Cambridge University Press (1972). [60] B. Kubala, J. König, and J. Pekola “Violation of the Wiedemann-Franz law in a single-electron transistor”. Phys. Rev. Lett. 100, 066801 (2008). [61] G. Gómez-Silva, O. Ávalos-Ovando, M. L. Ladrón de Guevara, and P. A. Orellana “Enhancement of thermoelectric efficiency and violation of the Wiedemann-Franz law due to Fano effect”. J. Appl. Phys. 111, 053704 (2012). 130 BIBLIOGRAPHY [62] M.-H. Bae, Z. Li, Z. Aksamija, P. N. Martin, F. Xiong, Z.-Y. Ong, I. Knezevic, and E. Pop “Ballistic to diffusive crossover of heat flow in graphene ribbons”. Nat. Commun. 4, 1734 (2013). [63] D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar “Thermal conductivity of individual silicon nanowires”. Appl. Phys. Lett. 83, 2934 (2003). [64] S.-M. Lee, D. G. Cahill, and R. Venkatasubramanian “Thermal conduc- tivity of Si–Ge superlattices”. Appl. Phys. Lett. 70, 2957 (1997). [65] T. Borca-Tasciuc, D. Achimov, W. L. Liu, G. Chen, H.-W. Ren, C.-H. Lin, and S. S. Pei “Thermal conductivity of InAs/AlSb superlattices”. Microscale Thermophys. Eng. 5, 225 (2001). [66] R. Yang and G. Chen “Thermal conductivity modeling of periodic two- dimensional nanocomposites”. Phys. Rev. B 69, 195316 (2004). [67] M.-S. Jeng, R. Yang, D. Song, and G. Chen “Modeling the thermal conductivity and phonon transport in nanoparticle composites using Monte Carlo simulation”. J. Heat Transfer 130, 042410 (2008). [68] G. Basile, C. Bernardin, and S. Olla “Momentum conserving model with anomalous thermal conductivity in low dimensional systems”. Phys. Rev. Lett. 96, 204303 (2006). [69] D. L. Nika and A. A. Balandin “Two-dimensional phonon transport in graphene”. J. Phys. Condens. Matter 24, 233203 (2012). [70] S. Ghosh, W. Bao, D. L. Nika, S. Subrina, E. P. Pokatilov, C. N. Lau, and A. A. Balandin “Dimensional crossover of thermal transport in few-layer graphene”. Nat. Mater. 9, 555 (2010). [71] W.-R. Zhong, M.-P. Zhang, B.-Q. Ai, and D.-Q. Zheng “Chirality and thickness-dependent thermal conductivity of few-layer graphene: A molecular dynamics study”. Appl. Phys. Lett. 98, 113107 (2011). [72] Z. Guo, D. Zhang, and X.-G. Gong “Thermal conductivity of graphene nanoribbons”. Appl. Phys. Lett. 95, 163103 (2009). 131 BIBLIOGRAPHY [73] Y. Xu, X. Chen, B.-L. Gu, and W. Duan “Intrinsic anisotropy of thermal conductance in graphene nanoribbons”. Appl. Phys. Lett. 95, 233116 (2009). [74] D. L. Nika, A. S. Askerov, and A. A. Balandin “Anomalous size depen- dence of the thermal conductivity of graphene ribbons”. Nano Lett. 12, 3238 (2012). [75] M. Park, S.-C. Lee, and Y.-S. Kim “Length-dependent lattice thermal conductivity of graphene and its macroscopic limit”. J. Appl. Phys. 114, 053506 (2013). [76] X. Xu, L. F. C. Pereira, Y. Wang, J. Wu, K. Zhang, X. Zhao, S. Bae, C. T. Bui, R. Xie, J. T. L. Thong, B. H. Hong, K. P. Loh, D. Donadio, B. Li, and B. Özyilmaz “Length-dependent thermal conductivity in suspended single-layer graphene”. Nat. Commun. 5, 3689 (2014). [77] H. Sevinçli and G. Cuniberti “Enhanced thermoelectric figure of merit in edge-disordered zigzag graphene nanoribbons”. Phys. Rev. B 81, 113401 (2010). [78] W. J. Evans, L. Hu, and P. Keblinski “Thermal conductivity of graphene ribbons from equilibrium molecular dynamics: Effect of ribbon width, edge roughness, and hydrogen termination”. Appl. Phys. Lett. 96, 203112 (2010). [79] A. V. Savin, Y. S. Kivshar, and B. Hu “Suppression of thermal conduc- tivity in graphene nanoribbons with rough edges”. Phys. Rev. B 82, 195422 (2010). [80] Z. Aksamija and I. Knezevic “Lattice thermal conductivity of graphene nanoribbons: Anisotropy and edge roughness scattering”. Appl. Phys. Lett. 98, 141919 (2011). [81] Y. Chen, T. Jayasekera, A. Calzolari, K. W. Kim, and M. Buon- giorno Nardelli “Thermoelectric properties of graphene nanoribbons, junctions and superlattices”. J. Phys. Condens. Matter 22, 372202 (2010). 132 BIBLIOGRAPHY [82] Y. Xu, X. Chen, J.-S. Wang, B.-L. Gu, and W. Duan “Thermal transport in graphene junctions and quantum dots”. Phys. Rev. B 81, 195425 (2010). [83] F. Mazzamuto, V. Hung Nguyen, Y. Apertet, C. Caër, C. Chassat, J. Saint-Martin, and P. Dollfus “Enhanced thermoelectric properties in graphene nanoribbons by resonant tunneling of electrons”. Phys. Rev. B 83, 235426 (2011). [84] H.-S. Zhang, Z.-X. Guo, X.-G. Gong, and J.-X. Cao “Thermal conductivity of sawtooth-like graphene ribbons: A molecular dynamics study”. J. Appl. Phys. 112, 123508 (2012). [85] P. Yang, Y. Tang, H. Yang, J. Gong, Y. Liu, Y. Zhao, and X. Yu “Thermal management performance of bent graphene nanoribbons”. RSC Adv. 3, 17349 (2013). [86] K.-M. Li, Z.-X. Xie, K.-L. Su, W.-H. Luo, and Y. Zhang “Ballistic thermo- electric properties in double-bend graphene nanoribbons”. Phys. Lett. A 378, 1383 (2014). [87] M. Sharafat Hossain, F. Al-Dirini, F. M. Hossain, and E. Skafidas “High performance graphene nano-ribbon thermoelectric devices by incorporation and dimensional tuning of nanopores”. Sci. Rep. 5, 11297 (2015). [88] J. Haskins, A. Kınacı, C. Sevik, H. Sevinçli, G. Cuniberti, and T. Çağın “Control of thermal and electronic transport in defect-engineered graphene nanoribbons”. ACS Nano 5, 3779 (2011). [89] T. Y. Ng, J. J. Yeo, and Z. S. Liu “A molecular dynamics study of the thermal conductivity of graphene nanoribbons containing dispersed Stone-Thrower-Wales defects”. Carbon 50, 4887 (2012). [90] D. Liu, P. Yang, X. Yuan, J. Guo, and N. Liao “The defect location effect on thermal conductivity of graphene nanoribbons based on molecular dynamics”. Phys. Lett. A 379, 810 (2015). 133 BIBLIOGRAPHY [91] J. Hu, S. Schiffli, A. Vallabhaneni, X. Ruan, and Y. P. Chen “Tuning the thermal conductivity of graphene nanoribbons by edge passivation and isotope engineering: A molecular dynamics study”. Appl. Phys. Lett. 97, 133107 (2010). [92] Q.-X. Pei, Z.-D. Sha, and Y.-W. Zhang “A theoretical analysis of the thermal conductivity of hydrogenated graphene”. Carbon 49, 4752 (2011). [93] S. Chen, Q. Wu, C. Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff “Thermal conductivity of isotopically modified graphene”. Nat. Mater. 11, 203 (2012). [94] X. Li, J. Chen, C. Yu, and G. Zhang “Comparison of isotope effects on thermal conductivity of graphene nanoribbons and carbon nanotubes”. Appl. Phys. Lett. 103, 013111 (2013). [95] X. Zhai and G. Jin “Strechting-enhanced ballistic thermal conductance in graphene nanoribbons”. Europhys. Lett. 96, 16002 (2011). [96] P. S. E. Yeo, K. P. L. Loh, and C. K. Gan “Strain dependence of the heat transport properties of graphene nanoribbons”. Nanotechnology 23, 495702 (2012). [97] N. Bonini, J. Garg, and N. Marzari “Acoustic phonon lifetimes and thermal transport in free-standing and strained graphene”. Nano Lett. 12, 2673 (2012). [98] J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T. Pettes, X. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R. S. Ruoff, and L. Shi “Two-dimensional phonon transport in supported graphene”. Science 328, 213 (2010). [99] S. Srinivasan and G. Balasubramanian “Reduced thermal transport in the graphene/MoS2/graphene heterostructure: A comparison with freestanding monolayers”. Langmuir 34, 3326 (2018). [100] K.-J. Tielrooij, N. C. H. Hesp, A. Principi, M. B. Lundeberg, E. A. A. Pogna, L. Banszerus, Z. Mics, M. Massicotte, P. Schmidt, D. Davy- 134 BIBLIOGRAPHY dovskaya, D. G. Purdie, I. Goykhman, G. Soavi, A. Lombardo, K. Watan- abe, T. Taniguchi, M. Bonn, D. Turchinovich, C. Stampfer, A. C. Ferrari, G. Cerullo, M. Polini, and F. H. L. Koppens “Out-of-plane heat transfer in van der Waals stacks through electron–hyperbolic phonon coupling”. Nat. Nanotechnol. 13, 41 (2018). [101] L. E. Bell “Cooling, heating, generating power and recovering waste heat with thermoelectric systems”. Science 321, 1457 (2008). [102] J. He and T. M. Tritt “Advances in thermoelectric materials research: Looking back and moving forward”. Science 357, 1369 (2017). [103] C. B. Vining “An inconvenient truth about thermoelectrics”. Nat. Mater. 8, 83 (2009). [104] H. J. Goldsmid “Introduction to thermoelectricity”. Springer (2010). [105] H. J. Goldsmid and R. W. Douglas “The use of semiconductors in ther- moelectric refrigeration”. Br. J. Appl. Phys. 5, 386 (1954). [106] A. F. Ioffe “Semiconductor thermoelements and thermoelectric cooling”. Infosearch (1957). [107] L. D. Hicks and M. S. Dresselhaus “Effect of quantum-well structures on the thermoelectric figure of merit”. Phys. Rev. B 47, 12727 (1993). [108] L. D. Hicks and M. S. Dresselhaus “Thermoelectric figure of merit of a one-dimensional conductor”. Phys. Rev. B 47, 16631 (1993). [109] R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn “Thin- film thermoelectric devices with high room-temperature figures of merit”. Nature 413, 597 (2001). [110] J. P. Heremans, C. M. Thrush, D. T. Morelli, and M.-C. Wu “Thermo- electric power of bismuth nanocomposites”. Phys. Rev. Lett. 88, 216801 (2002). [111] T. C. Harman, P. J. Taylor, M. P. Walsh, and B. E. LaForge “Quantum dot superlattice thermoelectric materials and devices”. Science 297, 2229 (2002). 135 BIBLIOGRAPHY [112] A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard, and J. R. Heath “Silicon nanowires as efficient thermoelectric materials”. Nature 451, 168 (2008). [113] A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Na- jarian, A. Majumdar, and P. Yan “Enhanced thermoelectric performance of rough silicon nanowires”. Nature 451, 163 (2008). [114] A. A. Olvera, N. A. Moroz, P. Sahoo, P. Ren, T. P. Bailey, A. A. Page, C. Uher, and P. F. P. Poudeu “Partial indium solubility induces chemical stability and colossal thermoelectric figure of merit in Cu2Se”. Energy Environ. Sci. 10, 1668 (2017). [115] A. Khitun, A. Balandin, J. L. Liu, and K. L. Wang “In-plane lattice thermal conductivity of a quantum-dot superlattice”. J. Appl. Phys. 88, 696 (2000). [116] P.-H. Chang and B. K. Nikolić “Edge currents and nanopore arrays in zigzag and chiral graphene nanoribbons as a route toward high-ZT thermoelectrics”. Phys. Rev. B 86, 041406 (2012). [117] J. P. Heremans, V. Jovovic, E. S. Toberer, A. Saramat, K. Kurosaki, A. Charoenphakdee, S. Yamanaka, and G. J. Snyder “Enhancement of thermoelectric efficiency in PbTe by distortion of the electronic density of states”. Science 321, 554 (2008). [118] P. Murphy, S. Mukerjee, and J. Moore “Optimal thermoelectric figure of merit of a molecular junction”. Phys. Rev. B 78, 161406 (2008). [119] S. G. Sharapov and A. A. Varlamov “Anomalous growth of thermoelectric power in gapped graphene”. Phys. Rev. B 86, 035430 (2012). [120] G. D. Mahan and J. O. Sofo “The best thermoelectric”. Proc. Natl. Acad. Sci. U.S.A. 93, 7436 (1996). [121] O. Karlström, H. Linke, G. Karlström, and A. Wacker “Increasing ther- moelectric performance using coherent transport”. Phys. Rev. B 84, 113415 (2011). 136 BIBLIOGRAPHY [122] J. P. Bergfield and C. A. Stafford “Thermoelectric signatures of coher- ent transport in single-molecule heterojunctions”. Nano Lett. 9, 3072 (2009). [123] Y. S. Liu and X. F. Fang “Enhancement of thermoelectric efficiency in a double-quantum-dot molecular junction”. J. Appl. Phys. 108, 023710 (2010). [124] P. Trocha and J. Barnaś “Large enhancement of thermoelectric effects in a double quantum dot system due to interference and Coulomb correlation phenomena”. Phys. Rev. B 85, 085408 (2012). [125] M. A. Sierra, M. Saiz-Bretín, F. Domínguez-Adame, and D. Sánchez “Interactions and thermoelectric effects in a parallel-coupled double quantum dot”. Phys. Rev. B 93, 235452 (2016). [126] C. M. Finch, V. M. García-Suárez, and C. J. Lambert “Giant ther- mopower and figure of merit in single-molecule devices”. Phys. Rev. B 79, 033405 (2009). [127] V. M. García-Suárez, R. Ferradás, and J. Ferrer “Impact of Fano and Breit-Wigner resonances in the thermoelectric properties of nanoscale junctions”. Phys. Rev. B 88, 235417 (2013). [128] R.-N. Wang, G.-Y. Dong, S.-F. Wang, G.-S. Fu, and J.-L. Wang “Impact of contact couplings on thermoelectric properties of anti, Fano, and Breit-Wigner resonant junctions”. J. Appl. Phys. 120, 184303 (2016). [129] H.-H. Fu, L. Gu, D.-D. Wu, and Z.-Q. Zhang “Enhancement of the thermoelectric figure of merit in DNA-like systems induced by Fano and Dicke effects”. Phys. Chem. Chem. Phys. 17, 11077 (2015). [130] M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim “Energy band-gap engi- neering of graphene nanoribbons”. Phys. Rev. Lett. 98, 206805 (2007). [131] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon “Quantized conductance of point contacts in a two-dimensional electron gas”. Phys. Rev. Lett. 60, 848 (1988). 137 BIBLIOGRAPHY [132] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones “One- dimensional transport and the quantisation of the ballistic resistance”. J. Phys. C: Solid State Phys. 21, 209 (1988). [133] N. Tombros, A. Veligura, J. Junesch, M. H. D. Guimarães, I. J. Vera- Marun, H. T. Jonkman, and B. J. van Wees “Quantized conductance of a suspended graphene nanoconstriction”. Nat. Phys. 7, 697 (2011). [134] B. Terrés, J. Dauber, S. Trellenkamp, U. Wichmann, and C. Stampfer “Disorder induced Coulomb gaps in graphene constrictions with different aspect ratios”. Appl. Phys. Lett. 98, 032109 (2011). [135] J. Baringhaus, M. Settnes, J. Aprojanz, S. R. Power, A.-P. Jauho, and C. Tegenkamp “Electron interference in ballistic graphene nanoconstric- tions”. Phys. Rev. Lett. 116, 186602 (2016). [136] B. Terrés, L. A. Chizhova, F. Libisch, J. Peiro, D. Jörger, S. Engels, A. Girschik, K. Watanabe, T. Taniguchi, S. V. Rotkin, J. Burgdörfer, and C. Stampfer “Size quantization of Dirac fermions in graphene constrictions”. Nat. Commun. 7, 11528 (2016). [137] R. K. Kumar, D. A. Bandurin, F. M. D. Pellegrino, Y. Cao, A. Principi, H. Guo, G. H. Auton, M. B. Shalom, L. A. Ponomarenko, G. Falkovich, K. Watanabe, T. Taniguchi, I. V. Grigoreva, L. S. Levitov, M. Polini, and A. K. Geim “Superballistic flow of viscous electron fluid through graphene constrictions”. Nat. Phys. 13, 1182 (2017). [138] J. M. Caridad, S. P. Power, M. R. Lotz, A. A. Shylau, J. D. Thomsen, L. Gammelgaard, T. J. Booth, A.-P. Jauho, and P. Bøggild “Conductance quantization suppression in the quantum Hall regime”. Nat. Commun. 9, 659 (2018). [139] V. Clericò, J. A. Delgado-Notario, M. Saiz-Bretín, H. Fuentevilla, A. V. Malyshev, J. D. Lejarreta, E. Diez, and F. Domínguez-Adame “Quantized electron transport through graphene nanoconstrictions”. Phys. Status Solidi A 215, 1701065 (2018). 138 BIBLIOGRAPHY [140] D. A. Areshkin and C. T. White “Building blocks for integrated graphene circuits”. Nano Lett. 7, 3253 (2007). [141] F. Muñoz-Rojas, D. Jacob, J. Fernández-Rossier, and J. J. Palacios “Co- herent transport in graphene nanoconstrictions”. Phys. Rev. B 74, 195417 (2006). [142] F. Muñoz-Rojas, J. Fernández-Rossier, L. Brey, and J. J. Palacios “Per- formance limits of graphene-ribbon field effect transistors”. Phys. Rev. B 77, 045301 (2008). [143] S. Ihnatsenka and G. Kirczenow “Effect of electron-electron interactions on the electronic structure and conductance of graphene nanoconstric- tions”. Phys. Rev. B 86, 075448 (2012). [144] S. Ihnatsenka and G. Kirczenow “Conductance quantization in graphene nanoconstrictions with mesoscopically smooth but atomically stepped boundaries”. Phys. Rev. B 85, 121407(R) (2012). [145] T. Gunst, T. Markussen, K. Stokbro, and M. Brandgybe “Inelastic vibra- tional signals in electron transport across graphene nanoconstrictions”. Phys. Rev. B 93, 245415 (2016). [146] A. Mreńca, K. Kolasińki, and B. Szafran “Conductance response of graphene nanoribbons and quantum point contacts in scanning gate measurements”. Semicond. Sci. Technol. 30, 085003 (2015). [147] M. H. D. Guimarães, O. Shevtsov, X. Waintal, and B. J. van Wees “From quantum confinement to quantum Hall effect in graphene nanostruc- tures”. Phys. Rev. B 85, 075424 (2012). [148] A. Rycerz, J. Tworzyd, and C. W. J. Beenakker “Valley filter and valley valve in graphene”. Nat. Phys. 3, 172 (2007). [149] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu “Valleytronics in 2D materials”. Nat. Rev. Mater. 1, 16055 (2016). 139 BIBLIOGRAPHY [150] D.-K. Ki, D. Jeong, J.-H. Choi, H.-J. Lee, and K.-S. Park “Inelastic scattering in a monolayer graphene sheet: A weak-localization study”. Phys. Rev. B 78, 125409 (2008). [151] F. V. Tikhonenko, A. A. Kozikov, A. K. Savchenko, and R. V. Gor- bachev “Transition between electron localization and antilocalization in graphene”. Phys. Rev. Lett. 103, 226801 (2009). [152] C. S. Lent and D. J. Kirkner “The quantum transmitting boundary method”. J. Appl. Phys. 67, 6353 (1990). [153] D. Z.-Y. Ting, E. T. Yu, and T. C. McGill “Multiband treatment of quan- tum transport in interband tunnel devices”. Phys. Rev. B 45, 3583 (1992). [154] J. Munárriz “Modelling of plasmonic and graphene nanodevices”. Springer (2014). [155] X.-L. Kong and Y.-J. Xiong “Resonance transport of graphene nanorib- bon T-shaped junctions”. Chin. Phys. Lett. 27, 047202 (2010). [156] P. T. Araujo, B. Terrones, and M. S. Dresselhaus “Defects and impurities in graphene-like materials”. Mater. Today 15, 98 (2012). [157] V. Clericò, J. A. Delgado-Notario, M. Saiz-Bretín, A. V. Malyshev, Y. M. Meziani, P. Hidalgo, B. Méndez, F. Amado, M. Domínguez-Adame, and E. Diez “Quantum nanoconstrictions fabricated by cryo-etching in en- capsulated graphene”, arxiv:1902.07459 (2019). [158] Z. Ren and P. X. Gao “A review of helical nanostructures: Growth theories, synthesis strategies and properties”. Nanoscale 6, 9366 (2014). [159] A. Shaikjee and N. J. Coville “The synthesis, properties and uses of carbon materials with helical morphology”. J. Adv. Res. 3, 195 (2012). [160] D. Klotsa, R. A. Romer, and M. S. Turner “Electronic transport in DNA”. Biophy. J. 89, 2187 (2005). [161] A. V. Malyshev “DNA double helices for single molecule electronics”. Phys. Rev. Lett. 98, 096801 (2007). 140 BIBLIOGRAPHY [162] R. Abdeddaïm, G. Guida, A. Priou, B. Gallas, and J. Rivory “Negative permittivity and permeability of gold square nanospirals”. Appl. Phys. Lett. 94, 081907 (2009). [163] J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, and G. von Freymann “Gold helix photonic metamaterial as broadband circular polarizer”. Science 325, 1513 (2009). [164] E. J. Smith, D. Makarov, S. Sanchez, V. M. Fomin, and O. G. Schmidt “Magnetic microhelix coil structures”. Phys. Rev. Lett. 107, 097204 (2011). [165] P. X. Gao, W. Mai, and Z. L. Wang “Superelasticity and nanofracture mechanics of ZnO nanohelices”. Nano Lett. 6, 2536 (2006). [166] F. Xu, W. Lu, and Y. Zhu “Controlled 3D buckling of silicon nanowires for stretchable electronics”. ACS Nano 5, 672 (2011). [167] S. Hwang, H. Kwon, S. Chhajed, J. W. Byon, J. M. Baik, J. Im, S. H. Oh, H. W. Jang, S. J. Yoon, and J. K. Kim “A near single crystalline TiO2 nanohelix array: enhanced gas sensing performance and its application as a monolithically integrated electronic nose”. Analyst 138, 443 (2013). [168] B. Göhler, V. Hamelbeck, T. Z. Markus, M. Kettner, G. F. Hanne, Z. Vager, R. Naaman, and H. Zacharias “Spin selectivity in electron transmission through self-assembled monolayers of double-stranded DNA”. Science 331, 894 (2011). [169] Z. Xie, T. Z. Markus, S. R. Cohen, Z. Vager, R. Gutiérrez, and R. Naaman “Spin specific electron conduction through DNA oligomers”. Nano Lett. 11, 4652 (2011). [170] R. Gutierrez, E. Díaz, C. Gaul, T. Brumme, F. Domínguez-Adame, and G. Cuniberti “Modeling spin transport in helical fields: Derivation of an effective low-dimensional hamiltonian”. J. Phys. Chem. C 117, 22276 (2013). [171] E. Díaz, R. Gutiérrez, C. Gaul, G. Cuniberti, and F. Domínguez-Adame “Coherent spin dynamics in a helical arrangement of molecular dipoles”. AIMS Mater. Sci. 4, 1052 (2017). 141 BIBLIOGRAPHY [172] O. V. Kibis, S. V. Malevannyy, L. Huggett, D. G. W. Parfitt, and M. E. Portnoi “Superlattice properties of helical nanostructures in a trans- verse electric field”. Electromagnetics 25, 425 (2005). [173] C. A. Downing, M. G. Robinson, and M. E. Portnoi “Nanohelices as superlattices: Bloch oscillations and electric dipole transitions”. Phys. Rev. B 94, 155306 (2016). [174] K. V. Bets and B. I. Yakobson “Spontaneous twist and intrinsic instabil- ities of pristine graphene nanoribbons”. Nano Res. 2, 161 (2009). [175] V. B. Shenoy, C. D. Reddy, A. Ramasubramaniam, and Y. W. Zhang “Edge-stress-induced warping of graphene sheets and nanoribbons”. Phys. Rev. Lett. 101, 245501 (2008). [176] H. Wang and M. Upmanyu “Saddle, twists, and curls: Shape transitions in freestanding nanoribbons”. Nanoscale 4, 3620 (2012). [177] D. Gunlycke, J. Li, J. W. Mintmire, and C. T. White “Edges bring new dimensions to graphene nanoribbons”. Nano Lett. 10, 3638 (2010). [178] I. Nikiforov, B. Hourahine, T. Frauenheim, and T. Dumitrica “Formation of helices in graphene nanoribbons under torsion”. J. Phys. Chem. Lett. 5, 4083 (2014). [179] X. Liu, F. Wang, and H. Wu “Anomalous twisting strength of tilt grain boundaries in armcahair graphene nanoribbons”. Phys. Chem. Chem. Phys. 17, 31911 (2015). [180] G. P. Tang, J. C. Zhou, Z. H. Zhang, X. Q. Deng, and Z. Q. Fan “Altering regularities of electronic transport in twisted graphene nanoribbons”. Appl. Phys. Lett. 101, 023104 (2012). [181] A. Sadrzadeh, M. Hua, and B. I. Yakobson “Electronic properties of twisted armchair graphene nanoribbons”. Appl. Phys. Lett. 99, 013102 (2011). [182] N. Xu, B. Huang, J. Li, and B. Wang “Semiconductor-metal and metal- semiconductor transitions in twisting graphene nanoribbons”. Solid State Commun. 202, 39 (2015). 142 BIBLIOGRAPHY [183] V. Atanasov and A. Saxena “Helicoidal graphene nanoribbons: chiral- tronics”. Phys. Rev. B 92, 035440 (2015). [184] N. Al-Aqtash, H. Li, L. Wang, W.-N. Mei, and R. F. Sabirianov “Elec- tromechanical switching in graphene nanoribbons”. Carbon 51, 102 (2013). [185] J. Jia, D. Shi, X. Feng, and G. Chen “Electromechanical properties of armchair graphene nanoribbons under local torsion”. Carbon 76, 54 (2014). [186] S. Cranford and M. J. Buehler “Twisted and coiled ultralong multilayer graphene ribbons”. Modelling Simul. Mater. Sci. Eng. 19, 054003 (2011). [187] S.-Y. Yue, Q.-B. Yan, Z.-G. Zhu, H. J. Cui, Q.-R. Zheng, and G. Su “First-principles study on electronic and magnetic properties of twisted graphene nanoribbon and Möbius strips”. Carbon 71, 150 (2014). [188] X. Wei, G. Guo, T. Ouyang, and H. Xiao “Tuning thermal conductance in the twisted graphene and gamma graphyne nanoribbons”. J. Appl. Phys. 115, 154313 (2014). [189] A. Antidormi, M. Royo, and R. Rurali “Electron and phonon transport in twisted graphene nanoribbons”. J. Phys. D 50, 234005 (2017). [190] W. Liu, S. Cai, and M. Deng “Thermoelectric performance of twisted graphene nanoribbons”. Adv. Mat. Res. 1070, 594 (2014). [191] B. Aradi, B. Hourahine, and T. Frauenheim “DFTB+, a sparse matrix- based implementation of the DFTB method”. J. Phys. Chem. A 111, 5678 (2007). [192] T. Niehaus, M. Elstner, T. Frauenheim, and S. Suhai “Application of an approximate density-functional method to sulfur containing com- pounds”. J. Mol. Struc. 541, 185 (2001). [193] R. M. Ribeiro, V. M. Pereira, N. M. R. Peres, P. R. Briddon, and A. H. Castro Neto “Strained graphene: tight-binding and density functional calculations”. New J. Phys. 11, 115002 (2009). 143 BIBLIOGRAPHY [194] R. Alaei and M. H. Sheikhi “Optical absorption of graphene nanorib- bon in transverse and modulated longitudinal electric field”. Fuller. Nanotub. Car. N. 21, 183 (2013). [195] K.-J. Gan, J.-J. Lu, W.-K. Yeh, Y.-H. Chen, and Y.-W. Chen “Multiple- valued logic design based on the multiple-peak BiCMOS-NDR circuits”. Eng. Sci. Technol. Int J. 19, 888 (2016). [196] P. M. Tedrow, J. E. Tkaczyk, and A. Kumar “Spin-polarized electron tunneling study of an artificially layered superconductor with internal magnetic field: EuO-Al”. Phys. Rev. Lett. 56, 1746 (1986). [197] T. Tokuyasu, J. A. Sauls, and D. Rainer “Proximity effect of a ferromag- netic insulator in contact with a superconductor”. Phys. Rev. B 38, 8823 (1988). [198] J. Zou, G. Jin, and Y.-Q. Ma “Negative tunnel magnetoresistance and spin transport in ferromagnetic graphene junctions”. J. Phys. Condens. Matter 21, 126001 (2009). [199] Y. Gu, Y. H. Yang, J. Wang, and K. S. Chan “Equilibrium spin current in ferromagnetic graphene junction”. J. Appl. Phys. 105, 103711 (2009). [200] S. Bandopadhyay, B. Dutta-Roy, and H. S. Mani “Understanding the Fano resonance through toy models”. Am. J. Phys. 72, 1501 (2004). [201] M. Saiz-Bretín, J. Munárriz, A. V. Malyshev, and F. Domínguez-Adame “Control of spin-polarised currents in graphene nanorings”. Phys. Lett. A 379, 2102 (2015). [202] K. Zollner, M. Gmitra, T. Frank, and J. Fabian “Theory of proximity- induced exchange coupling in graphene on hBN/(Co, Ni)”. Phys. Rev. B 94, 155441 (2016). [203] A. A. Balandin and O. L. Lazarenkova “Mechanism for thermoelectric figure-of-merit enhancement in regimented quantum dot superlattices”. Appl. Phys. Lett. 82, 415 (2003). [204] J. P. Heremans, M. S. Dresselhaus, L. E. Bell, and D. T. Morelli “When thermoelectris reach the nanoscale”. Nat. Nanotechnol. 8, 471 (2013). 144 BIBLIOGRAPHY [205] Y. Ouyang and J. Guo “A theoretical study on thermoelectric properties of graphene nanoribbons”. Appl. Phys. Lett. 94, 263107 (2009). [206] P. Divari and G. Kliros “Modeling the thermopower of ballistic graphene ribbons”. Physica E 42, 2431 (2010). [207] G. S. Kliros and P. C. Divari “Thermoelectric properties of gated graphene ribbons in the ballistic regime”. Superlattices Microstruct. 52, 221 (2012). [208] L. Rosales, C. D. Nuñez, M. Pacheco, A. Latgé, and P. A. Orellana “Graphene nanoribbon thermopower as a tool for molecular spec- troscopy”. J. Appl. Phys. 114, 153711 (2013). [209] A. S. Nissimagoudar and N. S. Sankeshwar “Electronic thermal conduc- tivity and thermopower of armchair graphene nanoribbons”. Carbon 52, 201 (2013). [210] R. Farghadan, A. Saffarzadeh, and E. Heidari Semiromi “Magnetic edge states in Aharonov-Bohm graphene quantum rings”. J. Appl. Phys. 114, 214314 (2013). [211] A. Mreńca-Kolasińska and B. Szafran “Lorentz force effects for graphene Aharonov-Bohm interferometers”. Phys. Rev. B 94, 195315 (2016). [212] M. Saiz-Bretín, A. V. Malyshev, P. A. Orellana, and F. Domínguez- Adame “Enhancing thermoelectric properties of graphene quantum rings”. Phys. Rev. B 91, 085431 (2015). [213] http://lammps.sandia.gov [214] J. Hu, X. Ruan, Z. Jiang, and Y. P. Chen “Molecular dynamics calculation of thermal conductivity of graphene nanoribbons”. AIP Conference Proceedings 1173, 135 (2009). [215] S. Plimpton “Fast parallel algorithms for short-range molecular dynam- ics”. J. Comput. Phys. 117, 1 (1995). 145 http://lammps.sandia.gov BIBLIOGRAPHY [216] D. Frenkel and B. Smit “Understanding molecular simulation: From algorithms to applications”. Academic Press (2002). [217] J. R. Lukes, D. Y. Li, X.-G. Liang, and C.-L. Tien “Molecular dynamics study of solid thin-film thermal conductivity”. J. Heat Transf. 122, 536 (2000). [218] S. J. Stuart, A. B. Tutein, and J. A. Harrison “A reactive potential for hydrocarbons with intermolecular interactions”. J. Chem. Phys. 112, 6472 (2000). [219] S. Nosé “A unified formulation of the constant temperature molecular dynamics methods”. J. Chem. Phys. 81, 1 (1984). [220] J. Shiomi “Nonequilibrium molecular dynamics methods for lattice heat conduction calculations”. Annu. Rev. Heat Transf. 17, 177 (2014). [221] P. Chantrenne and J.-L. Barrat “Finite size effects in determination of thermal conductivities: Comparing molecular dynamics results with simple models”. J. Heat Transf. 126, 577 (2004). [222] J. Hu, R. X., and Y.-P. Chen “Thermal conductivity and thermal rectifi- cation in graphene nanoribbons: A molecular dynamics study”. Nano Lett. 9, 2730 (2009). [223] J. E. Turney, J. H. McGaughey, and C. H. Amon “Assessing the ap- plicability of quantum corrections to classical thermal conductivity predictions”. Phys. Rev. B 79, 224305 (2009). [224] M. Saiz-Bretín, A. V. Malyshev, F. Domínguez-Adame, D. Quigley, and R. A. Römer “Lattice thermal conductivity of graphene nanostructures”. Carbon 127, 64 (2018). [225] H. Sevinçli, C. Sevik, T. Cagin, and G. Cuniberti “A bottom-up route to enhance thermoelectric figures of merit in graphene nanoribbons”. Sci. Rep. 3, 1228 (2013). [226] L. Medrano Sandonas, H. Sevinçli, R. Gutierrez, and G. Cuniberti “First- principle-based phonon transport properties of nanoscale graphene grain boundaries”. Adv. Sci. 5, 1700365 (2018). 146 BIBLIOGRAPHY [227] I. Milos̆ević, N. Kepc̆ija, E. Dobardz̆ić, M. Damnjanović, M. Mohr, J. Maultzsch, and C. Thomsen “Electron-phonon coupling in graphene”. Int. J. Mod. Phys. B 24, 655 (2010). [228] W. Zhang, T. S. Fisher, and N. Mingo “The atomistic Green’s function method: An efficient simulation approach for nanoscale phonon trans- port”. Numer. Heat Tr. B-Fund. 51, 333 (2007). [229] L. Medrano Sandonas, D. Teich, R. Gutierrez, T. Lorenz, A. Pecchia, G. Seifert, and G. Cuniberti “Anisotropic thermoelectric response in two-dimensional puckered structures”. J. Phys. Chem. C 120, 18841 (2016). [230] L. Medrano Sandonas, R. Gutierrez, A. Pecchia, G. Seifert, and G. Cuniberti “Tuning quantum electron and phonon transport in two- dimensional materials by strain engineering: A Green’s function based study”. Phys. Chem. Chem. Phys. 19, 1487 (2017). [231] J. H. Davies “The physics of low-dimensional semiconductors”. Cam- bridge University Press (1997). [232] P. Hohenberg and W. Kohn “Inhomogeneous electron gas”. Phys. Rev. 136, B864 (1964). [233] W. Kohn and L. J. Sham “Self-consistent equations including exchange and correlation effects”. Phys. Rev. 140, A1133 (1965). [234] C. M. Goringe, D. R. Bowler, and E. Hernández “Tight-binding modelling of materials”. Rep. Prog. Phys. 60, 1447 (1997). [235] P. Koskinen and V. Mäkinen “Density-functional tight-binding for be- ginners”. Comput. Mater. Sci. 47, 237 (2009). [236] M. Elstner and G. Seifert “Density functional tight binding”. Phil. Trans. R. Soc. A 372, 20120483 (2014). [237] K. Varga and J. Driscoll “Computational nanoscience: Applications for molecules, clusters, and solids”. Cambridge University Press (2011). 147 BIBLIOGRAPHY [238] W. C. Swope, H. C. Andersen, P. H. Berens, and K. R. Wilson “A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters”. J. Chem. Phys. 76, 637 (1982). [239] L. Verlet “Computer ‘experiments’ on classical fluids. I. Thermodynami- cal properties of Lennard-Jones molecules”. Phys. Rev. 159, 98 (1967). [240] F. Zuo, P. Panda, M. Kotiuga, J. Li, M. Kang, C. Mazzoli, H. Zhou, A. Barbour, S. Wilkins, B. Narayanan, M. Cherukara, Z. Zhang, S. K. R. S. Sankaranarayanan, R. Comin, K. M. Rabe, K. Roy, and S. Ramanathan “Habituation based synaptic plasticity and organismic learning in a quantum perovskite”. Nat. Commun. 8, 240 (2017). [241] Y. Shibuta, K. Oguchi, T. Takaki, and M. Ohno “Homogeneous nucle- ation and microstructure evolution in million-atom molecular dynamics simulation”. Sci. Rep. 5, 13534 (2015). [242] J. Tersoff “New empirical approach for the structure and energy of covalent systems”. Phys. Rev. B 37, 6991 (1988). [243] J. Tersoff “Modeling solid-state chemistry: Interatomic potentials for multicomponent systems”. Phys. Rev. B 39, 5566 (1989). [244] D. W. Brenner “Empirical potential for hydrocarbons for use in simulat- ing the chemical vapor deposition of diamond films”. Phys. Rev. B 42, 9458 (1990). [245] D. W. Brenner, O. A. Shenderova, J. A. Harrison, S. J. Stuart, B. Ni, and S. Sinnott “A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons”. J. Phys. Condens. Matter 14, 783 (2002). [246] L. Lindsay and D. A. Broido “Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene”. Phys. Rev. B 81, 205441 (2010). 148 Tesis Marta Saiz Bretín Portada Table of Contents List of Publications Abbreviations Abstract Resumen Introduction Electronic Properties of Graphene Thermal Transport Thermoelectricity Objectives and Outline Conductance Quantization in Graphene Nanoconstrictions Introduction System, Model and Methodology Single Nanoconstriction Impact of Edge Orientation Extrapolation to Larger Systems Role of the Nanoconstriction Shape Rough Edges Extrapolation to Larger Systems Multiple Nanoconstrictions Experimental Results Conclusions Twisted Graphene Nanoribbons as Non-Linear Electronic Devices Introduction System, Model, and Methodology Structural Relaxation Electron Transmission Impact of Structural Relaxation Effect of a Transverse Electric Field Current-Voltage Characteristics Conclusions Spin-Polarized Transport in Graphene Nanorings Introduction System, Model and Methodology Non-Ferromagnetic Rings Simplified Model Ferromagnetic Rings Effect of the Ferromagnetic Layer Side-Gate Control Conclusions Thermoelectricity in Graphene Nanorings: Electronic Contribution Introduction System, Model and Methodology Thermoelectricity in Graphene Rings Effect of a Side-Gate Voltage Edge Disorder Conclusions Molecular Dynamics Study of the Lattice Thermal Conductivity in Graphene Nanostructures Introduction System, Model and Methodology Influence of the Contacts Thermal Conductivity in Graphene Nanorings Role of Dimensions Effect of Temperature Edge Disorder and Functionalization Conclusions Impact of the Device Geometry on Electron and Phonon Transport in Graphene Nanorings Introduction System, Model and Methodology Phonon Transport Electron Transport and Thermoelectric Response Conclusions Conclusions Quantum Transmitting Boundary Method and Transfer Matrix Method Density-Functional Tight-Binding Method Molecular Dynamics Bibliography