NNT: 2016SACLS460 THÈSE DE DOCTORAT DE L’UNIVERSITÉ PARIS-SACLAY PRÉPARÉE À L’UNIVERSITÉ PARIS-SUD ÉCOLE DOCTORALE 575 EOBE Electrical, optical, bio-physics and engineering Spécialité de doctorat: Physique Par Mme. Mai Chung NGUYEN Etude théorique de nanodispositifs électroniques et thermoélectriques à base de jonctions contraintes de graphène Thèse présentée et soutenue à Orsay, le 2 Décembre 2016: Composition du Jury: M. Henri HAPPY Université de Lille 1, IEMN Président Mme. Mireille MOUIS CNRS, IMEP-LAHC, Grenoble Rapporteur M. Nicolas CAVASSILAS Université d'Aix-Marseille, IM2NP Rapporteur M. Viet Hung NGUYEN IMCN, Université Catholique de Louvain Examinateur M. Huy-Viet NGUYEN Institute of Physics Hanoi Co-directeur de thèse M. Philippe DOLLFUS CNRS, C2N, Orsay Directeur de thèse Acknowledgements Firstly, I would like to express my sincere gratitude to my supervisors, Dr. Philippe DOLLFUS and Dr. HuyViet NGUYEN for their support, patience, motivation, and immense knowledge in all the time of study and writing of this thesis. I also express my deepest thanks to Dr. Vi- etHung NGUYEN, who has ever kept a post-doc position in our group (COMICS), for his training, helping and sharing in research. To be hon- est, I am really very lucky to work with them. Without their guidance and encouragement, this work would have taken a longer time to complete. Besides my advisors, I would like to take this opportunity to thank all my colleagues in COMICS group (Prof. Arnaud Bournel, Dr. Jerome Saint-Martin, Dr. Damien Querlioz, Dr. Michele Amato, Mr. Christophe Chassat, Dr. Tran Van Truong, Dr. Jerome Larroque, Dr. Thanayut Kaewmaraya, Mr. Jean Choukroun, Mr. Brice Davier, Ms. Li Su and others) for stimulating discussions, a friendly atmosphere and all the fun (we have) in life as well as in work. My sincere thanks also goes to Universit Paris-Sud, Universit Paris-Saclay and CNRS (Centre National de la Recherche Scientifique), University Sci- ence and Technology of Hanoi (USTH) and Vietnamese government for the financial supports. Last but not least, I would like to thank to my parents, my husband, my sisters and brothers for supporting and encouraging me spiritually throughout writing this thesis, without that this work could not finished on time. Thank you very much, everyone! Orsay, 23th August, 2016 MaiChung NGUYEN Abstract Due to its outstanding physical properties, graphene is expected to be- come a new generation material, able to replace or complement traditional semiconductors in device technology. Hence, many studies have been led to explore the potential of this material immediately after the successful fabrication of a single layer of graphene in 2004. However, applications of graphene in electronic devices are still questionable due to the gapless character of this material. In particular, regarding electronic applications, the absence of energy bandgap in the band structure makes it difficult to switch off the current in graphene devices like transistors. Regarding thermoelectric properties, the gapless character is also a strong drawback since it prevents the separation of the opposite contributions of electrons and holes to the Seebeck coefficient. Thus, a sizable bandgap in graphene is a requirement to overcome the disadvantages of graphene and to fully benefit from its excellent conduction properties. It has been shown that many nanostructuring techniques can be used to open such a bandgap in graphene, e.g., graphene nanoribbons, graphene bilayer with a perpendic- ular electric field, graphene nanomesh lattices, channels based on vertical stack of graphene layers, mixed graphene/hexagonal boron nitride struc- tures, nitrogen doped graphene, and so on. However, each of these meth- ods has its own fabrication issues and/or need to be further confirmed by experiments. In this work, we focus on strain engineering, which offers a wide range of opportunities for modulating the electronic properties of graphene nanostructures. For this theoretical work, all calculations were performed using essentially two main methods, i.e., an atomistic tight binding Hamiltonian model to describe the electronic structure and the non-equilibrium Green's function approach of quantum transport. The main aim is to analyze in details the strain effects in graphene and to pro- vide strategies of strain engineering to improve the performance of both electronic (transistors and diodes) and thermoelectric devices. After introducing the general context if this work and the numerical tech- niques developed for this purpose, we first analyze the only effect of strain. Actually, if uniformly applied, a strain of large amplitude (> 23%) is re- quired to open a bandgap in the band structure of graphene. However, we show that with a strain of only a few percent, the strain-induced shift of the Dirac point in k-space may be enough to open a sizable conduc- tion gap (500 meV or more) in graphene heterojunctions made of un- strained/strained junctions, though the strained material remains gapless. After analyzing in details this property according to the amplitude and direction of strain and the direction of transport, we exploit this effect us- ing appropriate strain junctions to improve the behavior and performance of several types of devices. In particular, we show that with a strain of only 5%, it is possible to switch-off transistors efficiently, so that the ON/OFF current ratio can reach 105, which is a strong improvement with respect to pristine graphene transistors where this ratio cannot exceed 10. Then we show that by combining strain and doping engineering in such strain junctions the Seebeck coefficient can reach values higher than 1.4 mV/K, which is 17 times higher than in gapless pristine graphene. It can contribute to make graphene an excellent thermoelectric material. Finally, we study the effect of negative differential conductance (NDC) in graphene diodes made of either as single gate-induced strained barrier or a p-n junction. We show that appropriate strain engineering in these devices can lead to very strong NDC effects with peak-to-valley ratios of a few hundred at room temperature. Contents R´esum´e 1 Introduction 8 1 OVERVIEW 12 Part I Properties of graphene and strain effects 14 1.1 Graphene material . 14 1.2 Strain Engineering . 17 1.2.1 Uniform Strain . 18 1.2.2 Local Strain . 18 1.3 General theory of stress and strain . 19 1.3.1 Stress tensor . 20 1.3.2 Strain tensor . 21 1.3.3 The relationship between stress and strain . 23 1.4 Effect of strain on the electronic properties of materials . 25 1.5 Experimental techniques to generate strain in graphene . 27 1.6 Conclusions . 31 Part II Numerical techniques 33 1.7 Tight-binding method . 33 1.7.1 Secular equations . 34 1.7.2 Tight-binding calculations for carbon structures . 36 1.8 Non-equilibrium Green's function technique . 41 1.8.1 Basic equations . 41 1.8.1.1 Transmission . 42 i 1.8.1.2 Density of states (DOS) and local density of states (LDOS) . 43 1.8.1.3 Electric current . 44 1.8.1.4 Conductance . 45 1.8.2 Numerical techniques for Green's function calculation . 45 1.8.2.1 Calculation of the self energy . 45 1.8.2.2 Device Green's function calculation . 52 1.9 Conclusions . 53 2 ELECTRONIC PROPERTIES OF STRAINED GRAPHENE JUNC- TIONS 54 2.1 Introduction . 54 2.2 Electronic properties of graphene nanostructures . 55 2.2.1 Monolayer graphene . 55 2.2.2 Graphene nanoribbon structures . 59 2.2.3 Bilayer graphene . 62 2.2.4 Graphene nanomesh . 64 2.3 Electronic properties of strained graphene . 71 2.3.1 Basic equations . 72 2.3.2 Opening of bandgap under large strain . 77 2.3.3 The shift of Dirac point by a small strain of a few percent . 80 2.4 Opening of a conduction gap (transport gap) in strained graphene junc- tions . 81 2.4.1 The opening of conduction gap with strain of a few percent . 82 2.4.2 The dependence of conduction gap on strain amplitude (σ), direction of applied strain (θ) and transport direction (φ) in both of tensile and compressive cases . 85 2.4.2.1 Armchair transport direction (φ = 0) . 85 2.4.2.2 Zigzag transport direction (φ = 30◦) . 87 2.4.2.3 Any transport direction (0 < φ < 30◦) . 88 2.4.2.4 Compressive and tensile-strained graphene junctions 91 2.5 Conclusion . 94 ii 3 APPLICATION OF STRAINED JUNCTIONS IN GRAPHENE HET- EROCHANNELS 95 3.1 Introduction . 95 3.2 Improved performance of graphene transistors . 96 3.2.1 Field-effect transistor (FET) . 97 3.2.2 Different types of graphene transistors . 99 3.2.3 A high ON/OFF ratio in strained graphene transistor . 101 3.3 Enhanced thermoelectric effect . 104 3.3.1 What is the thermoelectric effect? . 105 3.3.2 The Seebeck effect in monolayer and bilayer graphene, graphene nanoribbons and graphene nanomesh . 107 3.3.3 The enhancement of Seebeck effect in graphene heterochannels by strain and doping engineering . 110 3.3.3.1 Strained device with uniform doping (∆U = 0) . 112 3.3.3.2 Strained device with both strain and doping engineer- ing (∆U 6= 0) . 113 3.3.3.3 Strained device with large doping . 116 3.4 Enhanced negative differential conductance (NDC) effect . 119 3.4.1 What is the NDC effect? . 119 3.4.2 Enhanced NDC effect in strained gaphene channels . 121 3.4.2.1 Enhanced NDC effect in single potential barrier struc- ture . 123 3.4.2.2 Enhanced NDC effect in PN junction . 128 3.5 Conclusion . 133 Summary and perspectives 134 A The relationship between the bond lengths in cases of (σ; θ) and (−σ; θ + 90◦) 137 B The equation of transmission 139 Bibliography 143 iii R´esum´e Le graph`ene,une couche unique d'atomes de carbone dispos´esen nid d'abeille poss`ede de nombreuses caract´eristiquesfascinantes.