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Modeling graphene: magnetic, transport and optical properties
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Modeling graphene: magnetic, transport and optical properties
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MODELING GRAPHENE: MAGNETIC, TRANSPORT AND OPTICAL PROPERTIES by Yi Chen Chang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS AND ASTRONOMY) August 2012 Copyright 2012 Yi Chen Chang Dedication To my beloved parents for their over years breeding and education. To my husband, Chia-Chi, for his everlasting love and support. To my son, Cheng Hua, for the happiness he brings into my life. ii Acknowledgments This thesis would not have been possible without the support of many people. I would like to express my gratitude to my research advisor, Prof. Stephan Haas, for giving me opportunity to work in his group, for giving me freedom to choose my own research projects, and for offering invaluable support and guidance. I appreciate his assistance in writing papers and this thesis. Deepest gratitude are also due to Dr. Richard Thompson for offering many valuable courses and for stimulating my enthusiasm for theoretical physics. Many thanks go to Dr. Aiichiro Nakano, Dr. Werner D¨ appen, Dr. Nelson Bickers, and Dr. Chi Mak, for being my qualify and defense committees. I also want to thank Dr. Jia G Lu, for giving me opportunity to learn the low temperature measurement. Special thanks also go to all my group members for providing diversity academic environment. Many thanks to Rodrigo Muniz, whom I learned so much about the basic properties of plasmons and to Tameem Albash for numerous stimulating discussions in the strain induced pseudo magnetic field. I would also like to thank my friends, Hsiao- Hsuan Lin, Yaqi Tao, Yung-Hsu Lin, Ming-Chak Ho, Kok Wee Song, and Yung-Ching Liang, for sharing literature and invaluable assistance. Last, I would like to express my love and gratitude to my beloved parents; for their understanding and endless love, through the duration of my studies. I also want to thank my son, Cheng Hua, for bringing me happiness everyday. I must thank my husband and best friend, Chia Chi. I wouldn’t finish this thesis without his encouragement. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vi List of Figures vii Abstract xiv Chapter 1: Introduction 1 1.1 Electric Band Structure of Monolayer and Few Layer Graphene . . . 2 1.2 Zigzag Graphene Nanoribbons . . . . . . . . . . . . . . . . . . . . 7 1.3 Outline of This Work . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2: Electric-field Induced Semimetal-to-metal Transition in Few- layer Graphene 12 2.1 Experiment set up . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Simple Two Band(STB) Model . . . . . . . . . . . . . . . . 16 2.2.2 Puddle Models . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Chapter 3: Defect Induced Resonances and Magnetic Patterns in Graphene 29 3.1 Mean-field Hubbard Model . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.1 Density of States . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.2 Magnetic Pattern . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 4: The Magnetism of Graphene Under Strain 47 4.1 Uniaxial Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 iv 4.2 Non Uniform Strain Induced Pseudo Magnetic Field . . . . . . . . . 56 4.2.1 Conventional Quantum Hall Effect in Graphene . . . . . . . 56 4.2.2 Strain Induced Pseudo Magnetic Field . . . . . . . . . . . . 66 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Chapter 5: Polarization of Graphene 79 5.1 Self-consisted Field Approach . . . . . . . . . . . . . . . . . . . . 80 5.2 Optical Properties of Graphene . . . . . . . . . . . . . . . . . . . . 84 5.3 Plasmon in Graphene with Impurity . . . . . . . . . . . . . . . . . 88 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Chapter 6: Conclusions 98 Bibliography 101 v List of Tables Table 1.1 The values of the effective masses relative to the free electron massg 0 = 2m 0 =~ 2 and of the two-dimensional densities of states g relative to the value are shown for different numbers N of graphene layers. . . . . . . . . . . . . . . . . . . . . . 6 Table 2.1 The values of the effective massesm e =m 0 for electrons and m h =m 0 for holes relative to the free electron mass are shown for different numbers N of graphene layers. The normalized densities of states shown in the last column are the averages of the values shown in columns two and three. . . . . . . . . 25 vi List of Figures Figure 1.1 Illustration of Hybrid atomic orbitals sp2.[77] . . . . . . . . 1 Figure 1.2 Graphene honeycomb lattice and its Brillouin zone.[12] . . . 2 Figure 1.3 Electronic dispersion in the honeycomb lattice. Left: full energy spectrum. Right: zoom in of the energy bands close to one of the Dirac points[12]. . . . . . . . . . . . . . . . . 3 Figure 1.4 Lattice structure of bilayer graphene and its band structure.[12] 5 Figure 1.5 Geometry of zigzag graphene nanoribbons[12] . . . . . . . 7 Figure 1.6 Energy spectrum of zigzag graphene nanoribbon . . . . . . 8 Figure 1.7 (a) and (d) are band structures around the K and K’ points. (b) and (c) are the ground state as a function of position for A and B type sublattices at the K point. (e) and (f) are the ground states as function of position for A and B type sub- lattices at the K’ point. . . . . . . . . . . . . . . . . . . . . 11 vii Figure 2.1 Measured resistance of trilayer graphene as a function of the gate voltage at different temperatures (4.5, 10, 30, 50, 70, 90, 100, 124, 155, 170, 220, and 240 K). The upper left inset shows an optical image of the device. The current is injected between probes 1 and 4, and the voltage is measured between probes 2 and 3. The distance between probes 2 and 3 is L = 4:3m. The upper right insert shows the band structure for the simple two-band model with a band overlap E 0 of width shaded. . . . . . . . . . . . . . . . . . . . . . . 14 Figure 2.2 Temperature dependence of the resistance at several gate voltages ranging from -40V to 0 V . The points are data. The solid lines are fits to the overlapping-band theory, while the dotted lines are fits to the continuous-puddle model. . . . . . 16 Figure 2.3 The carrier concentration of electron(blue), holes(red) and total carrier with temperature (a) 4.5K, (b)70K, (c)170K, and (d) 240K as functions of gate voltage. . . . . . . . . . . 19 Figure 2.4 The mobilities e for electrons and h for holes as functions of temperature. . . . . . . . . . . . . . . . . . . . . . . . . 20 Figure 2.5 (Conductivity(black dot) and STB fitting(red dash) versus gate voltage at (a)T = 4:5K and (b)T = 240K. . . . . . . . 22 Figure 2.6 Illustration of electrostatic potential fluctuation due to impurities[90]. 23 Figure 2.7 (Conductivity versus gate voltage atT = 4:5K. The lowest curve shows the fit to the two-puddle theories. The mid- dle curve shows the fit to the three-puddle theory displaced up by 2 mS. The top curve shows the fit to the continuous- puddle theory displaced up by 4 mS. The same data points are shown three times with the corresponding upward dis- placements. . . . . . . . . . . . . . . . . . . . . . . . . . . 26 viii Figure 2.8 Conductivity versus gate voltage at . The bottom curve shows the fit to the overlapping-band theory. The top curve shows the fit to the continuous-puddle theory displaced up by 2 mS. The same data points are shown twice, the higher values being displaced up by 2 mS for comparison with theory. . . . 27 Figure 3.1 Schematic of the single orbit tight binding model.(a) a sin- gle atom with electrons in different orbits. (b) when atoms are brought together to form a solid. The outermost orbit becomes global, while the second outermost orbit is still localized. (c) The second outermost orbit is the most impor- tant feature when we are only interesting in the low energy physics. (d) The simplified model which describes the elec- trons live on the lattices site and hop between sites.[76] . . . 31 Figure 3.2 The total magnetization of the A and B sublattices as a func- tion of the on-site Coulomb repulsion. . . . . . . . . . . . . 32 Figure 3.3 (a) Zero-energy local density of states in a graphene sheet with a single vacancy. (b) Local density of states at a lat- tice site next to the vacancy (red dotted line) and at a site far away from it (black solid line). (c) Spatial dependence of the intensity of the low-energy peak in the local den- sity of states, corresponding to an impurity induced bound state. The solid red curve is a fit to ar 2 decay. (d) Reso- nance energy at impurity site as a function of the scattering strength U d . The red solid curve is a fit to the where W is 5.6eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Figure 3.4 Lattice structure with respect to the defect site. [63] . . . . . 36 Figure 3.5 (a) Local density of states at various distances from a zigzag line defect in a graphene sheet. The upper inset shows the intensity of the zero energy local density of states as a func- tion of the position along the direction perpendicular to the defect, indicated by the red line in the lower inset. (b) Same as part (a), but along the direction parallel to the defect, as shown in the lower inset. . . . . . . . . . . . . . . . . . . . 39 ix Figure 3.6 The local density of states on the site next to the vacancy. The impurity cluster contains a single vacancy (bottom) up to 13 vacancies (up). . . . . . . . . . . . . . . . . . . . . . 40 Figure 3.7 (a)The gloabal density of state with differnt defect concen- tration.(b) The intensity of resonance peak located at 0.25eV as a function of defect concentration. . . . . . . . . . . . . . 41 Figure 3.8 Magnetic patterns in a graphene sheet induced by (a) single defect placed at the center of the sheet, (b) zigzag line defect, and (c) armchair line defect. The positions of the impurity atoms are denoted by white symbols. The color scale ranges from dark blue (negative magnetization) to yellow (positive magnetization). . . . . . . . . . . . . . . . . . . . . . . . . 43 Figure 3.9 Magnetic patterns in a graphene sheet induced by a zigzag line defect placed at the center of the sheet. The positions of the impurity atoms are denoted by white symbols. Here, we consider various impurity scattering strengths (a)U d =t = 1, (b)U d =t = 3, and (c)U d =t = 5, while keeping U/t=1.2. The color scale ranges from green (negative magnetization) to yellow (positive magnetization). . . . . . . . . . . . . . . . 45 Figure 4.1 Magnetic pattern of (a) a relaxed graphene dot, (b) in the presence of a Zigzag strain, and (c) in the presence of an armchair strain. . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 4.2 (a) and (b) are the local density of states at the edge with uni- axial strain along the zigzag and armchair directions, respec- tively. The insets shows the local magnetic moments as a function of strain. . . . . . . . . . . . . . . . . . . . . . . 54 Figure 4.3 Variation of the total magnetic moment of the A and B sub- lattices as a function of the Hubbard interactionU=t. . . . . 55 Figure 4.4 Cyclotron motion of electrons in the presence of a uniform magnetic was applied . . . . . . . . . . . . . . . . . . . . . 57 x Figure 4.5 The Hall conductance as a function of magnetic filed in con- ventional 2DEG [18]. . . . . . . . . . . . . . . . . . . . . . 57 Figure 4.6 Cyclotron motion of electrons when a gradient magnetic is applied[55]. . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 4.7 (a) Dispersion of the graphene sheet without magnetic field. (b) Energy quantization of dispersion when a magnetic field is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 4.8 The Hall conductance as a function of magnetic field in graphene [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 4.9 (a) and (d) are the band structure of Zigzag Graphene nanorib- bon in a uniform magnetic field around the K and K’ points. (b) and (c) are the ground state as function of position for A and B type sub-lattice at K point. (e) and (f) are the ground state as function of position for A and B type sub-lattice at K’ point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Figure 4.10 (a) and (d) are the band structure of Zigzag Graphene nanorib- bon in a gradient magnetic field around the K and K’ points. (b) and (c) are the ground state as function of position for A and B type sub-lattice at K point. (e) and (f) are the ground state as function of position for A and B type sub-lattice at K’ point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 4.11 (a)Arc-bending strain induced pseudo magnetic field. (b) effective potentialV (x;k y ) fork y = 0;k y < 0, andk y > 0 with R/L=3. . . . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 4.12 ((a) Global density of states with different arc-bending strains. (b) energy levels as a function of p n. The inset shows the dependence on arc-bending strength and the corresponding energy quantization. . . . . . . . . . . . . . . . . . . . . . . 69 xi Figure 4.13 Energy bandE n (k y ) of a ZGNR (a)strain-free, (b) with arc- bending strain R=5L, and (c) with arc-bending strainR=3L 70 Figure 4.14 Local density of states along the center of a graphene dot (a) with arc-bending strain R=5L and (b) R=3L . . . . . . . . . 71 Figure 4.15 (a) and (b) are the local density of states for the inner and outer edge state. (c) shows the confinement state for various arc-bending strains. . . . . . . . . . . . . . . . . . . . . . . 72 Figure 4.16 (a) and (d) are the band structure of Zigzag Graphene nanorib- bon in a arc-bending strain around the K and K’ points. (b) and (c) are the ground state as function of position for A and B type sub-lattice at K point. (e) and (f) are the ground state as function of position for A and B type sub-lattice at K’ point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 4.17 (a) The global density of state as a function of various L/R.(b) and (c) are the local density of state at the inner and outer edges.(d) Magnetic pattern of the graphene dot. . . . . . . . 76 Figure 5.1 Decay and scattering process in graphene [8] . . . . . . . . 84 Figure 5.2 The plasmon dispersion for (a) graphene and (b) conven- tional 2DEG [34] . . . . . . . . . . . . . . . . . . . . . . . 87 Figure 5.3 Excitation frequency as a function of the chemical potential with selected energy level n=30. . . . . . . . . . . . . . . . 91 Figure 5.4 Excitation plasmon distribution for chemical potential =0 , at selected level n = 30, single impurity strength u 0 = 2, and size = 6. The corresponding excitation frequencies are (a)w = 3:48 10 4 Hz, (b)w = 4:17 10 3 Hz, (c) w = 1:43 10 2 , and (d)w = 2:13 10 2 Hz . . . . . . . 92 xii Figure 5.5 Excitation frequency as a function of the impurity strength of selected energy level n=56 . . . . . . . . . . . . . . . . . 93 Figure 5.6 Spectral density of plasmons as a function of the excitation frequency with impurity potentialu 0 =2 (black) andu 0 = 3 (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Figure 5.7 Excitation plasmon distribution for chemical potential =0eV , at selected leveln = 56, single impurity strengthu 0 = 4, and size = 6. The corresponding excitation frequencies are (a)w = 1:3 10 3 Hz, (b)w = 4:4 10 3 Hz, (c)w = 1:42 10 2 , and (d)w = 2:2 10 2 Hz . . . . . . . . . . 94 Figure 5.8 Excitation frequency as a function of the impurity size for selected energy level n=30 . . . . . . . . . . . . . . . . . . 95 Figure 5.9 Spectral density of plasmon as a function of the excitation frequency for impurity sizes = 6 (green), = 7 (red) and = 8 (black). . . . . . . . . . . . . . . . . . . . . . . . . . 95 Figure 5.10 Excitation plasmon distribution for chemical potential =0eV ,at selected leveln = 30, single impurity strengthu 0 = 2, and size = 8. The corresponding excitation frequencies are (a)w = 2:1 10 3 Hz, (b)w = 4:5 10 3 Hz, (c)w = 1:37 10 2 , and (d)w = 2:3 10 2 Hz . . . . . . . . . . 96 xiii Abstract Graphene, with its unique linear dispersion near the Fermi energy, has attracted great attention since its successful isolation from highly oriented pyrolytic graphite in 2004. Many important properties have been identified in graphene, including a remarkably high mobility at room temperature, an unusual quantum hall effect, and an ambipolar electric field effect. It has been proposed as a candidate for many applications, such as optical modulators, spintronic devices, and solar cells. Understanding the fundamental properties of graphene is therefore important. In this dissertation, I present a study of transport, magnetism and optical properties of graphene. In the first chapter, I introduce the electronic properties of mono layer and few layer graphene. In the second chapter, I present low temperature transport measurements in few layer graphene. An electric-field induced semimetal-to-metal transition is observed based on the temperature dependence of the resistance for different applied gate voltages. At small gate voltages the resistance decreases with increasing temperature due to the increase in carrier concentration resulting from thermal excitation of electron-hole pairs, as it is characteristic of a semimetal. At large gate, voltages excitations of electron-hole pairs are suppressed, and the resistance increases with increasing temperature because of the decrease in mean free path due to electron-phonon scattering, as is characteristic of a metal. The electron and hole mobilities are almost equal, so there is approximate electron-hole symmetry. The data are analyzed according to two different theoretical xiv models for few-layer graphene. A simple two band (STB) model, two overlapping bands with quadratic energy-versus-momentum dispersion relations, is used to explain the experimental observations. The best fitting parameter for the overlap energy is found to be 16 meV . However, at low temperatures, the STB suggests that the conductivity is gate independent in the small gate voltage regime, which is not observed in the data. By considering frustration of the electronic potential due to impurities from the substrate, a Gaussian-distribution puddle model can successfully describe the observed transport behavior in the low temperature, small gate voltage regime. In the third chapter, I investigate the effects of point and line defects in monolayer graphene within the framework of the Hubbard model, using a self-consistent mean field theory. These defects are found to induce characteristic patterns into the electronic den- sity of states and cause non-uniform distributions of magnetic moments in the vicinity of the impurity sites. Specifically, defect induced resonances in the local density of states are observed at energies close to the Dirac points. The magnitudes of the frequencies of these resonance states are shown to decrease with the strength of the scattering potential, whereas their amplitudes decay algebraically with increasing distance from the defect. For the case of defect clusters, we observe that with increasing defect cluster size the local magnetic moments in the vicinity of the cluster center are strongly enhanced. Fur- thermore, non-trivial impurity induced magnetic patterns are observed in the presence of line defects: zigzag line defects are found to introduce stronger-amplitude magnetic patterns than armchair line defects. When the scattering strength of these topological defects is increased, the induced patterns of magnetic moments become more strongly localized. In the fourth chapter, I theoretically study the electronic properties properties in graphene dots under mechanical deformation, using both tight binding lattice model xv and effective Dirac model. We observed an edge state, which is tunned by an effec- tive quantum well originating from a strain-induced gauge field. Applying a uniaxial strain along the zigzag or armchair directions enhances or dampens the edge state due to the development of edge quantum wells. When an arc bending deformation is applied, the inner and outer edges of graphene dot display edge states caused by the induced nonuniform gauge field. These states suggest that an effective single well potential is introduced by a strong nonuniform pseudo-magnetic field, leading to a pseudo quantum Hall effect. Furthermore, we find that introducing a Hubbard term on the mean-field level induces a strong polarization between the A and B sublattices, which provides an experimental test of the theory presented here. Finally, I study charge impurity induced plasmon resonance in graphene by using the self-consistent method within random phase approximation (RPA). I attribute the observed increase in excitation energy to the increasing carrier density due to stronger impurity potentials. On the other hand, the carrier density within low energy region is decreased when impurity size is increased, as result of lower excitation frequency. The plasmon patterns show that the dipole resonances are supported for the lower exci- tation frequency due to a simple transition process. For higher excitation frequencies, quadrapole resonance is observed because the transitions between higher energy levels become possible. With increasing impurity size, a larger spatial range of plasmons is observed. xvi Chapter 1 Introduction Carbon-based materials have been widely studied for many decades due to their unique energy band structure. Many years ago, theoretical predictions have indicated that graphene is a pristine two dimential (2D) material, which should be unstable to thermal fluctuations. However, with the improvement in processing of graphite to isolate single layer graphene, achieved by Geim and Novoselov[58], graphene research subjects has exponentially evolved, with new important results discoveres almost every week. Nearly 9000 papers were published during 2011. A number of review articles have appeared, including Castro Neto et al. [12] and Das Sarma et al. [18]. Intrigued by the potential applications such as spintronic devices, solar cells, or flexible displays, many research efforts have been devoted to understanding its fundamental properties. Figure 1.1: Illustration of Hybrid atomic orbitals sp2.[77] 1 1.1 Electric Band Structure of Monolayer and Few Layer Graphene In the isolated carbon atom, the electronic configuration is 1s 2 2s 2 p 2 . For the case of graphene, the carbon-carbon chemical bonds are hybridized orbitals generated by super- position of 2s with 2p x and 2p y orbitals, i. e. the so calledsp2 hybridization, illustrated in Figure 1.1. The remaining free 2p z orbitals present asymmetry orientations and over- lap of these orbital states between neighboring atoms plays a major role in the electronic properties. For this reason, a good approximation for describing the electric properties of graphene is to use the nearest-neighbor tight-binding (TB) approximation assum- ing that its electronic states can be simply represented by a linear combination of 2p z orbitals. The resulting Hamiltonian can be represented by H = 2 4 0 t P i f i (k) t P i f i (k) 0 ; 3 5 (1.1) wheref i (k) = e ik i , 1 = a 0 =2(1; p 3), 2 = a 0 =2(1; p 3), and 3 =a 0 (1; 0) are nearest-neighbor vectors, shown in Figure 1.2 . Figure 1.2: Graphene honeycomb lattice and its Brillouin zone.[12] 2 By diagonalizing this Hamiltonian, we obtain the energy dispersion relation of the (bonding) and (anti-bonding) bands. E (k) =t q 1 + 4 cos 2 (3k x a=2) + 4 cos( p 3k y a=2) cos(3k x a=2) (1.2) where k x and k y are components of k vector that folded into the first Brillouin zone, shown in Figure 1.2. The band structure shows that valence and conduction bands touch each other at six points located on the corners of the Brillouin zone, shown in Figure 1.3. K and K’ are named Dirac points, and are given by K = ( 4 3a ; 0);K 0 = ( 4 3a ; 0): (1.3) Figure 1.3: Electronic dispersion in the honeycomb lattice. Left: full energy spectrum. Right: zoom in of the energy bands close to one of the Dirac points[12]. Thef i (k) can be expanded around low energies and close to one of the Dirac point. The expansion gives f i (k) =f i (k F ) +r k f i (kk F ) +O(k 2 ); (1.4) 3 wheref i (k F ) = exp(iK 0 a i ) andr k f i (k) =ia i (kk F )f(k F ). Thus one gets f 1 (k F ) = exp(i( 4 3a ; 0)a(0; 1 p 3 )) = exp(0) = 1 (1.5) f 2 (k F ) = exp(i( 4 3a ; 0) a 2 (1; 1 p 3 )) = exp(i 2 3 ) = 1 2 i p 3 2 f 3 (k F ) = exp(i( 4 3a ; 0) a 2 (1; 1 p 3 )) = exp(i 2 3 ) = 1 2 +i p 3 2 r k f 1 (k) =ia(0; 1 p 3 ) (k x ;k y )f 1 (k F ) =i a p 3 k y r k f 2 (k) =i a 2 (1; 1 p 3 ) (k x ;k y )f 2 (k F ) =i a 2 (k x + 1 p 3 k y ) exp(i 2 3 ) r k f 3 (k) =i a 2 (1; 1 p 3 ) (k x ;k y )f 3 (k F ) =i a 2 (k x 1 p 3 k y ) exp(i 2 3 ) The low energy approximation of the tight binding model (Equation 1.1) can be then rewritten as H 12 =t X r k f i (k) (1.6) =i ta p 3 k y +i ta 2 (k x + 1 p 3 k y ) exp(i 2 3 ) +i ta 2 (k x 1 p 3 k y ) exp(i 2 3 ) = p 3ta 2 (k x +ik y ): The effective Dirac Hamiltonian near K and K’ with corresponding wave functions [ K A ; K B ; K 0 A ; K B ] is H =v F a 2 6 6 6 6 6 6 6 4 0 k x +ik y 0 0 k x ik y 0 0 0 0 0 0 k x +ik y 0 0 k x ik y 0 ; 3 7 7 7 7 7 7 7 5 (1.7) 4 where the Fermi velocity is v F = p 3t 2 , and the dispersion is approximated by E(k) =v F j k j. By comparing the linear energy relation of graphene with this dispersion of massless relativistic particles obtained from the Dirac equation, the electron in graphene behaves like a Dirac fermions with effective Fermi velocity (v F = 10 6 m=s) that is around 300 times smaller than the speed of light. This introduces a significant difference between graphene and other common materials with a parabolic dispersion, i. e. = ~ 2 k 2 =2m, where m is the effective mass of electron. In usual materials, the velocityv = @E=@k =~ 2 k=m, changes with momentum. In graphene, on the other hand, the velocity is independent of momentum at sufficiently low energies. Figure 1.4: Lattice structure of bilayer graphene and its band structure.[12] The band structure of few-layer graphene can be easily obtained by the third-nearest- neighbor tight binding (TB) formalism developed by Partoens and Peeters [60] and by Gr¨ uneis, et al. [27]. By considering the interlayer hopping in Bernal, i. e. ”A-B” stacking, the tight binding Hamiltonian can be easily extended to bilayer graphene, H = 2 6 6 6 6 6 6 6 4 0 v F 0 0 v F 0 1 0 0 1 0 v F 0 0 v F 0 3 7 7 7 7 7 7 7 5 ; (1.8) 5 Table 1.1: The values of the effective masses relative to the free electron mass g 0 = 2m 0 =~ 2 and of the two-dimensional densities of states g relative to the value are shown for different numbers N of graphene layers. N m =m 0 g=g 0 2 0.033 0.033 3 0.046 0.046 4 0.534 and 0.0204 0.074 5 0.0572 and 0.033 0.090 6 0.0595, 0.00412 and 0.0147 0.115 where = k x +ik y , 1 = 0:35eV is the interlayer hopping energy, shown in Figure 1.1. Diagonalization of H gives four energy bands asE =[ p ( 1 =2) 2 + (v F ~k) 2 1 =2]. Two of the bands touch at the K point, while the other two bands are split away from E = 0 by a gap of E 4 = 2 1 . A low energy approximation v F ~k 1 gives a quadratic dispersion with either no gap (E(k) = v 2 F ~ 2 k 2 = 1 ) or a gap (E(k) = (v 2 F ~ 2 k 2 = 1 + 1 )). By mapping onto Fermi liquids, the effective mass in this cases is given by 1=2m =v 2 F = 1 orm = 0:033m 0 , wherem 0 is the free electron mass. For three-layer graphene with ABA stacking order, six bands simply combine the bands for the mono-layer and bilayer case. The values of the effective masses of different number of layer are listed in Table 1.1. For three layers the effective mass is enhanced by a factor of p 2 over the bilayer case. For four layers, the two values of the effective mass are related to the two-layer case by factors of p 5 1=2. For five layers, there are again two effective masses. One is the same as for the two-layer case, and the other one is increased by a factor of p 3. These values and the resulting two-dimensional density of states for electrons (or for holes) are collected in Table 1.1. The nearest neighbor TB approximation does not give any electron-hole asymmetry nor any band overlap. A further neighbor interaction with strength called 4 does give rise to an electron-hole asymmetry[50, 87]. 6 Figure 1.5: Geometry of zigzag graphene nanoribbons[12] 1.2 Zigzag Graphene Nanoribbons So far we have discussed the bulk properties of graphene. Nevertheless, it is also inter- esting to study electrons confined to edges. There are two basic configurations for graphene nanoribbons (GNR), namely, armchair and zigzag edges. Here we only focus on the zigzag GNR which has tight-binding (TB) Hamiltonian[15] derived from Equa- tion. 1.1 with wave function A;n (k y ) and B;n (k y ) in line n H = 2 4 0 T (k y ) T (k y ) 0 3 5 ; (1.9) where T (k y ) = 2 6 6 6 6 6 6 6 6 6 6 4 0 2t cos( p 3k y a 0 =2) 0 ::: 2t cos( p 3k y a 0 =2) 0 t ::: 0 t 0 ::: 0 0 2t cos( p 3k y a 0 =2) ::: : : : ::: 3 7 7 7 7 7 7 7 7 7 7 5 (1.10) 7 Different from the bulk graphene, a remarkable new feature arises in the energy dispersion, shown in Figure 1.6. The highest valence band and lowest conduction band now not only touch at Dirac point, but also form a flat band area which depends on the width of the zigzag GNR. The corresponding wave functions are localized at the edges, i. e. so called edge state. Figure 1.6: Energy spectrum of zigzag graphene nanoribbon We can also quantitatively understand this unite flat band by using the Dirac equation (Equation 1.7) by introducing the proper boundary condition[9, 12] for zigzag edges, K A (0) = K B (L) = K 0 A (0) = K 0 B (L) = 0 (1.11) Then apply Hamiltonian twice to the wave function. (@ 2 x +k 2 y ) A (x) = ~ " 2 A (x) (1.12) (@ 2 x +k 2 y ) B (x) = ~ " 2 B (x) (1.13) 8 where ~ " ="=ta. The solution is A (x) =A(e zx e zx ) withz = p k 2 y ~ " 2 . (@ x k y ) A (x) = ~ " B (x) (1.14) We get B (x) = iA ~ " f(zk y )e zx + (z +k y )e zx ). Since B (L) = 0, (zk y )e zL + (z +k y )e zL = 0: (1.15) Then we can get the transcendental relation as k y z k y +z =e 2zL (1.16) When k y > k c y = 1=L, the solution of z is real. Therefore the corresponding wave function is decay exponentially as reflected in the flat band area, shown in the Figure 1.3 (a) and (d) for K and K’, respectively. At the K point, the wave function for both A and B sublattices in the flat band region exponentially decay into the bulk within scales of few lattice constant and do not contribute to the conductance due to their zero group velocity. When slightly away from the flat band area (k y = 0:078 2 a 0 ), the wave function with Gaussian distribution, called bulk state, propagates through the bulk area toward the positive direction. Similarly at K’, a bulk state contributes to the conductance and travels in the opposite direction. 9 1.3 Outline of This Work The outline of this work is as follow. In chapter 2, we discuss low temperature transport behavior of few layer graphene. A metal-insulator transition has been observed. Two different theory models, a simple two band model and a paddle model, are applied to fit the experimental data. In chapter 3, we study magnetism in a graphene sheet with defect by using a mean field Hubbard model. We carefully examine the effect of size, strength, and configurations of line defects on the density of states and the magnetic pat- terns. In chapter 4, we discuss the electronic properties of graphene in a strain induced pseudo magnetic field through the density of states. We compare the quantum Hall effect observed in graphene and in conventional two-dimensional materials in a gradient magnetic field. In chapter 5, charge impurity induced plasmons in graphene are studied by using a self consistanct method within the random phase approximation(RPA). 10 -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 E(eV) K' -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 E(eV) K 0 40 80 120 160 0.0 0.2 0.4 0.6 ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=-0.078 ky=-0.157 ky=-0.2356 ' x x 0 40 80 120 160 200 0.0 0.1 0.2 0.3 0.4 0.5 x x ky=-0.078 ky=-0.157 ky=-0.2356 ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 0 40 80 120 160 200 0.0 0.1 0.2 0.3 0.4 0.5 ky=-0.078 ky=-0.157 ky=-0.2356 ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 (x) x (a) (b) (c) (d) (e) (f) 0 40 80 120 160 0.0 0.2 0.4 0.6 0.8 ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=-0.078 ky=-0.157 ky=-0.2356 ' (x) x Figure 1.7: (a) and (d) are band structures around the K and K’ points. (b) and (c) are the ground state as a function of position for A and B type sublattices at the K point. (e) and (f) are the ground states as function of position for A and B type sublattices at the K’ point. 11 Chapter 2 Electric-field Induced Semimetal-to-metal Transition in Few-layer Graphene In contrast to bulk graphite, electrons in few layer graphene (FLG) system are con- fined along one crystallographic direction, as a consequence of the 2D character of its electronic response. Due to overlapping conduction and valence bands, FLG has been shown to behave like mixed carrier semi-metallic systems. Moreover, it features strong gate dependent transport behavior, a long mean-free path (l 0:4um), and high mobil- ity, making FLG a promising material for nanoelectronic devices. A recent experimental study of trilayer graphene was performed by Craciun et al. [16]. Based on an earlier developed simple two band (STB) model, they interpreted the residual conductivity at low temperatures assuming that the carrier density is constant across the sample. How- ever, impurities fromSiO 2 substrate can introduce local puddles, leading to fluctuations in the carrier density. This phenomenon has been observed by scanning single-electron tunneling (SET) [51] and scanning-tunneling microscopy (STM) [19, 20, 88] measure- ments. In this chapter, we first present experimental results on the temperature depen- dent resistance of FLG. We then compare the our results by using both STB model and local puddle model. We show that the effects of the impurities cannot be ignored at low temperatures. We also determine the number of layers through the effective mass calculated from the local puddle model. 12 2.1 Experiment set up Few-layer grapheme samples were extrapolated from highly oriented pyrolygic graphite (HOPG) by the peeling off process as described in reference [58]. After several peeling processes, micron-sized few-layer graphene samples with thicknesses ranging from 1 to 4 nm were fabricated and then transferred onto a SiO2 (300 nm)/Si substrate with align- ment marks that were patterned using photolithography. Sheets of few-layer graphene were located by an optical microscope. By using an atomic force microscope (AFM), the thickness of this sample was measured to be 1.5 nm, which corresponds to between four and five layers of graphene. The interlayer spacing in graphene is 0.335 nm. However, the thickness measurement, which was not very accurate, could include a dead layer or a layer of water. We will later show that our sample has three layers. Raman scattering shows that the sample has more than two layers but cannot conclude the exact number. In fact our sample was slightly p-doped due to absorption of humidity [10, 58]. Standard e-beam lithography was adopted to fabricate electrodes. Pd (10 nm)/Au (60nm) contacts were deposited by an e-beam sputter after development. The resistance was measured as a function of temperature and of the voltage applied to a gate electrode. The resistivity measurements were carried out in a cryostat over a temperature range from 4.5 to 300 K. 13 -40 -20 0 20 40 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 240K 124K 70K R( ) Vg(V) 4.5K Figure 2.1: Measured resistance of trilayer graphene as a function of the gate voltage at different temperatures (4.5, 10, 30, 50, 70, 90, 100, 124, 155, 170, 220, and 240 K). The upper left inset shows an optical image of the device. The current is injected between probes 1 and 4, and the voltage is measured between probes 2 and 3. The distance between probes 2 and 3 isL = 4:3m. The upper right insert shows the band structure for the simple two-band model with a band overlapE 0 of width shaded. 14 2.2 Results and Discussions For our measurements of the electrostatic-field effect in few-layer graphene we used a sample with width and length as shown in the inset of Figure 2.1. Because of the irregu- lar shape of our sample the value of the width is approximate, but this does not affect the temperature and gate voltage dependence of our measurements. The results of our resis- tance measurements are plotted in Figure 2.1 as a function of the gate voltage at different temperatures. Figure 2.2 shows the same data as a function of temperature for different gate voltages. While temperature was varied from 4.5K to 240K, we observed two dif- ferent types of temperature dependent behavior. For small gate voltage, the resistance decreases with increasing temperature ( @ @T < 0); while it increases with temperature for large gate voltage ( @ @T > 0). This transition could be explained by different temperature dependent behavior of the mobility and of the carrier concentration in the literature [35]. Following we discussed the carrier density calculated from STB model and local puddle model. 15 0 1 0 0 2 0 0 3 0 0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0 R ( ) T ( K ) Figure 2.2: Temperature dependence of the resistance at several gate voltages ranging from -40V to 0 V . The points are data. The solid lines are fits to the overlapping-band theory, while the dotted lines are fits to the continuous-puddle model. 2.2.1 Simple Two Band(STB) Model We first assume the band structure with a small overlap, denoted byE 0 as shown in the inset of Figure 2.1. The Dirac bands are not important for the conductivity for samples with more than one layer owing to their small density of states. This gives a simple two-band model that is similar to the one proposed by Klein [41] to explain the temper- ature dependence of the resistance in the parent compound bulk graphite. In the case of graphite this approximation represents an average over the band structure along a cor- ner edge of the three-dimensional Brillouin zone as found in theoretical Slonzcewski- WeissMcClure [53] model. The main correction that we make to Kleins approximation is to reduce the density of states by a factor of two, because half of the quadratic bands 16 are shifted to higher energies and are not relevant to our low energy measurements. The dispersion of conduction and valence band are approximated as a parabolic form as E h = E 0 2 ~ 2 k 2 =2m h ; (2.1) E e =~ 2 k 2 =2m e E 0 2 whereh ande represent the valence band and conduction band, E 0 is the overlapping energy, andm is the effective mass of carrier. The carrier density from the single band is then given by [42] n e (T ) =g e k B T ln[1 + exp(( E 0 2 + F ))] (2.2) n h (T ) =g h k B T ln[1 + exp(( E 0 2 F ))] whereT is the temperature,k B is the Boltzmann constant, = 1=k B T , and F is the chemical potential. The density of statesg for electrons and for holes is given by g e=h = g s g v 2~ 2 X m e=h (2.3) whereg s is spin degeneracy andg v is valley degeneracy. By using the tight binding(TB) model described in the introduction, the effective mass for three layer graphene are shown in the third column of Table 1.1. When a potential difference V g is applied between the sample and the gate electrode, it induces a charge ofCV g =A in the sample, whereC is the capacitance between the gate electrode and the sample. The capacitance per area isC=A = r 0 d , with r = 3:9 for theSiO 2 dielectric and thed = 300nm for separation between the sample and the gate electrode. An extra electron can either go into the conduction band and increase n e or go into the valence band and reduce n h . 17 Consequently, with charge equilibriumn e =n h for zero gate voltage, the total induced density of charges is CV g A = (n e n h )e (2.4) =gek B T (ln[1 + exp(( E 0 2 + F ))] ln[1 + exp(( E 0 2 F ))]) Equation 2.4 gives us the quadratic equation to solve for F (V g ;T ) as F (V g ;T ) =k B T ln( A 1 (V g ;T ) 1 + p (1A 1 (V g ;T )) 2 + 4A 1 (V g ;T )B 1 (T ) 2 ) (2.5) whereA 1 (V g ;T ) = exp( CVg 5gk B Te ) andB 1 (T ) = exp( E 0 k B T ). For numerical accuracy it is convenient to use Equation 2.5 only for positive values of V g and then to use the fact that F (V g ;T ) = F (V g ;T ) for negative values ofV g rather than taking the differ- ence between two large approximately equal numbers. Note that we have ignored the nonuniform distribution of charge in the direction perpendicular to the film caused by electrostatic screening[4, 43, 52, 87]. Figure 2.3 shows the carrier concentration as a function of the gate voltage with different temperatures. When a small gate potential is applied, the carrier concentration is smaller at low temperatures, and thermal excita- tion of electron-hole pairs gives rise to a rapid increase in the carrier concentration and therefore of the conductivity. When the gate potential is large the density of carriers is almost independent of temperature, and the resistance increases with temperature due to the decrease in the mobility caused by increasing electron-phonon scattering. 18 -40 -20 0 20 40 0.0 2.0x10 15 4.0x10 15 6.0x10 15 n Vg(V) n total n e n h -40 -20 0 20 40 0.0 2.0x10 15 4.0x10 15 6.0x10 15 n Vg(V) n total n e n h -40 -20 0 20 40 0.0 2.0x10 15 4.0x10 15 6.0x10 15 n Vg(V) n total n e n h -40 -20 0 20 40 0.0 2.0x10 15 4.0x10 15 6.0x10 15 n Vg(V) n total n e n h ( a ) ( b ) ( c ) ( d ) Figure 2.3: The carrier concentration of electron(blue), holes(red) and total carrier with temperature (a) 4.5K, (b)70K, (c)170K, and (d) 240K as functions of gate voltage. At large gate potential, the conductivity is dominated by only one type of carrier because the chemical potential has been shifted far out of the band overlap region shown in the upper right inset of Figure 2.1 and its value is given by = e=h CV g =A. We can then estimate e=h from the sloped=dV g . The results for the mobilities are plotted in Figure 2.4. Both mobilities are well fit by an expression of the functional form(T ) = 0 =(1+ aT 2 ). The two mobilities differ from each other by less than 10%. This small difference might be interpreted to imply that the effective masses of the electrons and holes are approximately equal. However, it is known experimentally that the holes are about 30% to 40% heavier than the electrons in bilayer graphene [11, 87]. Nevertheless, we will use a model in which the two effective masses are taken to be the same, as was done in reference [16], since this allows us to analyze the dependence of our experimental results on gate voltage analytically. 19 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 1 .2 1 .3 1 .4 1 .5 1 .6 1 .7 1 .8 1 .9 2 .0 M o b ility (m 2 /V s ) T (K ) H o le E le c tro n Figure 2.4: The mobilities e for electrons and h for holes as functions of temperature. For data fitting we take E 0 and g as tuning parameters that are temperature inde- pendent. At the charge neutral point where n e = n h , the value of conductivity ( =e(n e e +n h h )) is near its minimum and F = 0 at all temperatures. At very low temperatures we haven e =n h =gE 0 =2 from Equation 2.2 and obtain min (0) = egE 0 2 [ e (0) + h (0)] (2.6) while at a finite temperature we have min (T ) =egk B T [ e (T ) + h (T )] ln(1 + exp(E 0 =2)) (2.7) By taking the ratio of equation 2.7 to equation 2.6, the factor g drops out and E 0 was solved. g was determined by equation 2.6. We find that E 0 = 16meV , and the density of states isg = 0:051g 0 , which is in good agreement with the theoretical value 20 listed in Table 1.1 for three layers, which leads us to the conclusion that our sample has in fact three layers. Our value of the band overlap is only about half as large as the value 28 meV given in reference[16], which shows thatE 0 is not an intrinsic property of trilayer graphene. At the maximum gate voltageV g the shift of the chemical potential is F = 60meV . Figures 2.5 shows the results at a low and a high temperature. The minimum conductivity is offset by about 3 volts from zero gate voltage, possibly due to instrumental offset or surface contamination of our sample[27, 58]. At large gate voltage, the slight disagreement between theory and experiment data on slope, shown in Figure 2.5, is due to the difference between the electron and the hole mobilities, while in theory we assume they are the same. Compared with the work of Morozov et al. [54] on bilayer graphene, the shapes of the conductivity versus gate voltage curves are similar. At small gate voltage, there is a noticeable discrepancy at low temperature. Theory curve is flat while experimental data is curved. The STB theory predicts that the total density of carriers is constant in the band overlap region where E 0 =2 < F < E 0 =2 at zero temperature. Next we discuss a puddle model which consider the inhomogenous carrier density introduced by the impurity. 21 -4 0 -2 0 0 2 0 4 0 1 2 3 4 5 6 7 -4 0 -2 0 0 2 0 4 0 2 3 4 (b ) G (1 0 6 ) (a ) G (1 0 6 ) V g (V ) Figure 2.5: (Conductivity(black dot) and STB fitting(red dash) versus gate voltage at (a)T = 4:5K and (b)T = 240K. 2.2.2 Puddle Models SET and STM works [19, 20, 51, 88] have shown that impurities in the substrate cause significantly uneven distributions of charge, called puddles, in the graphene plane, as consequence of a fluctuation of electrostatic potential. First we consider the simplest two-puddle model which was introduced by Zhu et al. [89, 90]. The energies of the particles are shifted up by a potential in half of the sample and down by in the 22 other half, illustrated in Figure 2.6. The band overlap E 0 is set equal to zero. Then equation 2.2 is changed to n e (T ) = gk B T 2 (ln(1 + exp(( + F )) + ln(1 + exp(( + F ))) (2.8) n h (T ) = gk B T 2 (ln(1 + exp(( F )) + ln(1 + exp(( F ))) Figure 2.6: Illustration of electrostatic potential fluctuation due to impurities[90]. Because the total density of states including both electrons and holes is constant and independent of the band shift when the band overlapE 0 is set equal to zero. Together with equation 2.4 CV g A = gek B T 2 ln 1 + exp(( F ) 1 + exp( F ) (2.9) gek B T F F =CV g =(geA) 23 Chemical potential is simply linearly depend on gate voltage in this model. Fitting the conductivity data as before using the same values for the mobilities we find the best fit parameters as = 12meV andg=g 0 = 0:62. At low temperatures the fit to the data, as shown in the lowest curve in Figure 2.7, is exactly the same as for the STB model. It is still flat at small gate voltage. The only a small difference between the puddle models and STB model is that two-puddle models gives the upper curve at higher temperatures shown in Figure 2.8. We then consider generalizing the two-puddle model to a three-puddle model where we divide the film into three regions, one shifted by , one shifted by, and one unshifted. Then Equation 2.8 is replaced by n e (T ) = gk B T 3 ln[(1 + exp(( + F ))(1 + exp( F ))(1 + exp(( + F )] n h (T ) = gk B T 3 ln[(1 + exp(( F ))(1 + exp( F ))(1 + exp(( F )] (2.10) Again using Equation 2.9, the new values found for the parameters are = 19meV and g=g 0 = 0:60. At low temperatures the fit to the data is improved as shown by the middle curve in Figure 2.6. It now consists of two straight line segments at small gate voltage. However, the model is still not curved like the data. The best fit that we get to our data is using the continuous puddle model developed by Hwang and Das Sarma [35]. A continuous distribution of puddle potentials is taken with a Gaussian distribution. n e (T ) = gk B T 0 p Z 1 1 d exp[(= 0 ) 2 ] ln[(1 + exp(( + F )] (2.11) n h (T ) = gk B T 0 p Z 1 1 d exp[(= 0 ) 2 ] ln[(1 + exp(( F )] 24 Table 2.1: The values of the effective massesm e =m 0 for electrons andm h =m 0 for holes relative to the free electron mass are shown for different numbers N of graphene layers. The normalized densities of states shown in the last column are the averages of the values shown in columns two and three. N m e =m 0 m e =m 0 g=g 0 2,theory[60] 0.026 0.039 0.033 2, theory[27] 0.041 0.055 0.048 2, theory[11] 0.041 0.040 0.035 3, theory[60] 0.030 0.049 0.055 4, theory[27] 0.044 0.066 0.067 4, theory[27] 0.024 and 0.060 0.035 and 0.088 0.104 Using Equation 2.9 as before we now obtain the parameter values = 22meV and g=g 0 = 0:60. The fitting result, shown in upper curve in Figure 2.7, now matches well with the data at small gate voltage. The value of 0 is expected to depend on the particular sample being measured, while the value of g provides a measurement of the average of the electron and hole effective masses. g from STB model and the puddle model are somewhat higher than the valueg=g 0 = 0:46 shown in Table 1.1 for a trilayer film. There are two possible reasons for this discrepancy. First, the width of our film, which is used to convert resistance to resistance per square R A = Rw=L , is approximated by average value from the AFM image shown in inset of Figure 2.1. Second, Table 1 is constructed by only considering nearest-neighbor interactions. The values of the effective masses are somewhat higher at the bottom of the band obtained from calculations with non-nearest-neighbor interactions. Now we compare effective masses from different calculation methods. By fitting band structure in Figure 8b of reference [60] and Figure 12b,d, and f of reference [27], the effective masses for few layer graphene are listed in Table 2.1, together with the minimum effective masses from the experimental data of reference [35] for bilayer graphene. Our experimental result is midway between the two theoretical values shown for trilayer graphene. 25 Our result forg is obtained based on the temperature dependence of the resistance at the charge-neutral point F = 0. The screening effect does not play a role at this point. When the electron and hole effective masses are not equal, the effective value ofg is an average value of these two masses weighted with the corresponding mobilities. g g 0 = e m e + h m h ( e + h )m 0 (2.12) Since our electron and hole mobilities are almost equal, it is proper to compare our result for g=g 0 with the theoretical results listed in Table 2.1. Note that for trilayer graphene, the value ofg also higher than one calculated from TB model, listed in Table 1.1. Figure 2.7: (Conductivity versus gate voltage at T = 4:5K. The lowest curve shows the fit to the two-puddle theories. The middle curve shows the fit to the three-puddle theory displaced up by 2 mS. The top curve shows the fit to the continuous-puddle theory displaced up by 4 mS. The same data points are shown three times with the corresponding upward displacements. 26 Figure 2.8: Conductivity versus gate voltage at . The bottom curve shows the fit to the overlapping-band theory. The top curve shows the fit to the continuous-puddle theory displaced up by 2 mS. The same data points are shown twice, the higher values being displaced up by 2 mS for comparison with theory. 27 2.3 Conclusions We have performed measurements on trilayer graphene films, illustrating the depen- dence of the resistance on temperature and gate voltage. An insulator to metal transition is observed as a function of gate voltage. We carefully examine data using several the- ories. At low temperatures, both the STB model and the two puddle model predict conductance behavior gate independently at low electrical field region. The three pud- dle model has a two segment feature, which does not fit the curved data. We find that the best fit is to a continuous-puddle model. We then conclude that the residual conduc- tivity at low temperature is due to impurities in the substrate, and the impurity potential is distributed according to a Gaussian function. In addition, we get the density of stateg and the average effective mass for electron and hole for trilayer graphene by fitting and have a great agreement with other references. 28 Chapter 3 Defect Induced Resonances and Magnetic Patterns in Graphene In situ formation of atomic size defects has recently been observed in graphene layers, using transmission electron microscopy.[31, 39, 59] Specifically, it was demonstrated that certain topological defects can be induced by irradiation with electrons beams, thus raising the possibility that more complex impurity structures, such as specifically tai- lored line defects, can in principle be achieved using similar experimental techniques. The effects of such designer impurity structures on the nanoscale are interesting, as they can have profound effects on the electronic properties of the material. Similar to anisotropic superconductors, graphene is known to have a reduced electronic phase space close to its Fermi surface, resulting in a linear low-energy density of states.[6] For the case of unconventional superconductors, the introduction of such defects has been shown to have profound consequences, such as formation of low-energy bound states (or quasi-bound states) and localization of quasi-particles.[30] In this chapter, we show that these features have an analogue in metallic graphene, in particular when magnetic fluctuations are taken into account. The effects of impurities in graphene are of particular interest because their presence has been shown to strongly reduce the otherwise remarkably high electronic mobility in this compound[1, 70] and to change its electronic band structure[15, 17, 37]. Fur- thermore, impurity induced local puddles of charge carriers have been proposed to be responsible for the observed minimum conductivity.[1] Very recently, impurity induced 29 bound states have been experimentally observed using scanning tunneling microscopy. [79] Specifically, it was shown that the tunneling current amplitude of these single impurity bound states decays inversely with the square of the distance from the defect. Because of this algebraic dependence, they are in fact quasi-bound states. It was also suggested that such defects induce local magnetic moments[17, 32, 37], which in turn can cause global ferromagnetic instabilities with a transition temperature that scales as the square root of the impurity concentration. [79] A number of properties of graphene sheets with point impurities have already been established [63, 65, 69, 79, 85, 86]. In particular, in a recent experiment extended one-dimensional defects were realized [45], demonstrating that the creation of designer defects in graphene sheets is becoming realistic. In this chapter, we investigate the effects of such topological defects on the local densities of state and on the magneti- zation patterns. Specifically, we study the impurity-doped Hubbard model[2, 66] on a graphene sheet geometry, where we consider the cases of single impurities and one- dimensional impurity clusters. This chapter is organized as follows. In the following section, we discuss the model, the approximations used, and the quantities we investigate. In the subsequent section, we show results for the induced density of states and for the magnetization in the impu- rity doped Hubbard model. We conclude with a section summarizing our results and discussing possible experimental implications. 30 3.1 Mean-field Hubbard Model In the standard Hubbard model[76], one simplifies the situation considerably by assum- ing that each atom in the lattice has only one electron orbit which is non-degenerate, shown in Figure 3.1. When the atoms are brought together, the interaction between the neighboring atoms can be simplified to a single orbital model. We denote the creation operator byc y i which indicates that an electron of spin =";# is created at atom sitei . In turn, the annihilation operatorc i represents an electron of spin which is destroyed at atom sitei.n i =c y i c i is the number operator. Figure 3.1: Schematic of the single orbit tight binding model.(a) a single atom with electrons in different orbits. (b) when atoms are brought together to form a solid. The outermost orbit becomes global, while the second outermost orbit is still localized. (c) The second outermost orbit is the most important feature when we are only interesting in the low energy physics. (d) The simplified model which describes the electrons live on the lattices site and hop between sites.[76] 31 The Hamiltonian of Hubbard model is represented as a sum of the hopping and on-site interaction Hamiltonians. H = N X <i;j>; t i;j (c y i c j +h:c:) +U N X i; n i n i ; (3.1) Heret i;j represents the quantum mechanical amplitude that an electron hops from site i toj. The interaction Hamiltonian represents a nonlinear interaction which increases the system energy byU when two electron occupy a single orbital state at site i. The Hubbard model is a simplification as it ignores the Coulomb interaction from electrons in different orbitals. In Hartree-Fock (HF) mean field theory, an infinite system exhibits a Mott-Hubbard transition from a paramagnetic semimetal to an antiferromagnetic (AF) insulator with a critical value U c = 2:23t [1, 70, 73]. Figure 3.2 shows the magneti- zation as a function of U/t on the two sublattices. A phase transition is observed. In this proposal, we choose U/t=1.2, leading to a paramagnetic semimetallic phase with a conical dispersion, as observed in pristine graphene. Figure 3.2: The total magnetization of the A and B sublattices as a function of the on-site Coulomb repulsion. 32 In order to solve the Hubbard model without any approximations, one uses exact diag- onalization of a small systems or run Quantum Monte Carlo simulations for system of larger scale[23]. Alternatively, mean field theory is an economic approximation which is applicable for any system size. The interaction Hamiltonian is rewritten as H mf =U N X i hn i" in i# +hn i# in i" hn i" ihn i# i + (n i" hn i" i)(n i# hn i# i): (3.2) Assuming the variations in the number of spin up and spin down electrons are small, the last term can be neglected. This approximation states that a spin up electron at site i interacts with an average number of spin down electrons at the same site. The Hamiltonian matrix is composed of spin-up and spin-down blocks, in which the diagonal elements depend on the unknown average number of spin-down (spin-up) electrons, respectively. A random initial guess of thehn i i is first given.hn i i are updated after diagonal- ization until allhn i i are converged. The mean fieldhn i i is computed from hn i i = Z dEg i (E)f(EE f ); (3.3) whereg i (E) = P j i (E j ) i (E j )(EE j ) is the local electronic density of states and f(EE f ) is the Fermi function. This self-consistent solution provides the local density of states and the spin densitiesM i = (hn i ihn i i)=2 on each atom. When a potential impurity is created in the graphene sheet at site i, an additional term is introduced H d =U d X c y i c i ; (3.4) where U d is the scattering strength of the impurity. When a strong scattering impu- rity, such as a vacancy, is created, the scattering strength U d goes towards infinity. 33 Figure 3.3: (a) Zero-energy local density of states in a graphene sheet with a single vacancy. (b) Local density of states at a lattice site next to the vacancy (red dotted line) and at a site far away from it (black solid line). (c) Spatial dependence of the intensity of the low-energy peak in the local density of states, corresponding to an impurity induced bound state. The solid red curve is a fit to ar 2 decay. (d) Resonance energy at impurity site as a function of the scattering strengthU d . The red solid curve is a fit to the where W is 5.6eV 34 3.2 Results and Discussion 3.2.1 Density of States Single Defect Let us start by considering the effects of a single impurity in graphene. Figure 3.3(a) shows the calculated zero-energy local density of state for a graphene sheet in the vicin- ity of a vacancy. A scattering induced bound state is observed to form in the vicinity of the defect site, similar to what has been reported by Pereira et. al.[64], with a charac- teristic triangular spatial pattern that is commensurate with the lattice symmetry. Here U d =t = 1000 (corresponding to a vacancy), the energy of the induced bound state is at -0.1eV . In Figure 3.3(b), we compare the local densities of states at a site next to the vacancy and at another site far away from it. The bound state is clearly absent in the latter case, which instead shows the well known Dirac cone shape. Note that the linear dispersion is slightly smeared out by the finite broadening ( /t=0.083) obtained as the delta functions ing i (E) are replaced by Lorentzians. Next we examine the spatial decay of the amplitude of the defect induced bound state. As observed in Figure 3.3(c), the magnitude of the impurity peak can be fitted well by a power law proportional to the squared inverse of the distance from the vacancy. This is the same algebraic decay which has recently been reported by scanning tunneling experiments[79]. Similar power-law decay has been observed for bound states around non-magnetic impurities in anisotropic superconductors along certain directions[5, 63]. 35 Figure 3.4: Lattice structure with respect to the defect site. [63] In Pereira’s paper[63], the wave functions at the sites in close proximately the vacancy and on the left plane are denoted as L 0;1 = A P k e ik(n+1) ; L 0;0 = A P k e ikn ; L 0;1 = A P k e ik(n1) ; shown in Figure 3.4, where k is in unit a=2. For E=0, the total components of the wave function over nearest-neighbor sites should vanish, i. e. L 0;1 + L 0;0 + L 1;0 = 0; L 0;1 + L 0;0 + L 1;1 = 0; and L 1;0 + L 1;1 + L 2;0 = 0: Therefore the L 1;0 ; L 1;1 ; L 2;0 are found to be A P k 2cos(k=2)e ik(n+1=2) ;A P k 2cos(k=2)e ik(n1=2) ; andA P k [2cos(k=2)] 2 e ikn , respectively: A general form of the wave function on site (l;j) is given by L l;j = A P k [2cos(k=2)] l exp(ikj +l=2):Replacingx =l3a=2;y =a p 3(j +l=2); the wave function can be approximated by L (x;y) =A Z 4=3 2=3 dk 2 cos( k 2 ) 2x=3 e iky= p 3 A exp(4iy=3 p 3) x +iy +A exp(2i(x +y= p 3)=3 xiy ; (3.5) 36 when the lattice site (x;y) is in the different sublattice of the vacancy. L (x;y) is zero on sites in the same sublattice of the vacancy. The local density of states is proportional to the ( L (x;y)) 2 ; from which follows 1/r 2 decay. In Figure 3.4(d), we study the dependence of the resonance energy on the magni- tude of the impurity scattering strength. The observed dependence is in agreement with the resonant scattering behavior reported by Skrypnyk et. al.[72] and Wehling et. al [83, 84]. In Skrypnyk’s paper[72], a Green’s function method was adopted to study impurity induced resonance states in two-dimensional system with a Dirac spectrum. The diagonal element of the Green’s function(GF) b G=( b H) 1 on the impurity site is given by G 0 = g 0 1U d g 0 (3.6) whereg 0 =( c H 0 ) 1 is the diagonal element of the GF in the host. g 0 can be obtained by g 0 = 1 N X k 2 2 (k) a 2 2 Z 2 p =a 0 kdk 2 (tak) 2 = 4t 2 ln( 2 4t 2 2 )i jj 4t 2 : (3.7) It is convenient to set W=2 p t. The local density of states at the impurity site is 0 = 1 ImG 0 = jj f1U d F ()g 2 + ( U d W ) 2 ; (3.8) where F() = W 2 ln( 2 W 2 2 ):This implies a bound state present at resonance (E r ), satis- fying U d = W 2 E r ln E 2 r W 2 E 2 r ; (3.9) 37 When a vacancy is created,U d =t is infinite, and the resonance peak is close to the Dirac point. The fitting to the asymptotic behaviorE r = 1exp( W 2 2U d ) in the strong coupling regime indicates that for the present case the bandwidth is 5.6eV . Our fitting value W is smaller than the value reported by Wehling (6.06eV), because the impurity configuration is slightly different. In Ref. [83, 84], the impurity only acts on one C-atom. In our case, the impurity is embedded in the graphene sheet which forms chemical bonds with all three nearest-neighbor carbon atoms. When the on-site Coulomb repulsion U/t is increased, we find that the resonance peak continues to follow the same dependence as in Figure 3.4(d). The fit shown here is only for the regime of scattering strengths larger thanU d =t = 4. Line Defect Next, let us examine the effects of more extended defects in graphene on the electronic density of states. In Figure 3.5 we study the local densities of states in the vicinity of a zigzag line defect, consisting of 13 contiguous vacancies in a graphene sheet. The Hamiltonian parameters are chosen to be the same as for the point defect discussed above. We observe, similar to the case of a single vacancy, that the local density of states shows a pronounced low-energy peak close to the impurity, indicating localization of the charge carriers. As expected, the amplitude of this induced resonance peak in the local density of states is enhanced with respect to the case of a single impurity. For example, at the central site next to the line defect for the given parameters it is amplified approximately by a factor of 2.2, a direct consequence of constructive interference of the joint point defects. Furthermore, we find that the spatial decay in the local density states at the Dirac point is the same as for the point defect when moving away from the line defect in a perpendicular direction. This is similar to the localized state of zigzag edge[25, 81, 82]. As shown in the upper inset of Figure 3.5(a), it falls off with an inverse 38 square power law along the path indicated in the lower inset. In contrast, the zero energy local density of states parallel to the line defect does not behave monotonically. In Figure 3.5(b), we plot the local density of states along a path parallel to the line defect. It is evident that the magnitude of the defect induced bound state varies by two orders of magnitude along this cut. At resonance energy, on the other hand, g i# (E r =0:4eV ) shows a Gaussian behavior with the maximum point located at the center of the impurity cluster, indicating a constructive interference. Figure 3.5: (a) Local density of states at various distances from a zigzag line defect in a graphene sheet. The upper inset shows the intensity of the zero energy local density of states as a function of the position along the direction perpendicular to the defect, indicated by the red line in the lower inset. (b) Same as part (a), but along the direction parallel to the defect, as shown in the lower inset. Impurity Clusters and Defect Concentration The size of impurity cluster is another important issue affecting the conductivity in the graphene sheet. It is believed that the carrier concentration is dramatically affected by extended defects in the system[3]. When the number of vacancy sites (n d ) in the impurity cluster is gradually increased from 1 to 13, the first significant feature is that the intensity of the spin-up resonance peak increases due to increased spectral weight in the vicinity of the Dirac point, as shown in Figure 3.6. This is in agreement with a prediction of Pereira et. al.[63]. Second, the resonance peak is slightly blue-shift with increasing 39 size of the impurity cluster. This is due to the presence of on-site coulomb interactions, leading to stronger spin polarization in the vicinity of vacancies cluster. Noticed that the resonance peak is not present in the spin-up LDOS of the impurity cluster with n d =2 and 4, reflecting the absent of induced local magnetic moment around these vacancies. This is in agreement with the observation of Kumazaki et. al.[44] Figure 3.6: The local density of states on the site next to the vacancy. The impurity cluster contains a single vacancy (bottom) up to 13 vacancies (up). Next, the vacancies are randomly distributed in the graphene sheet. Figure 3.7(a) shows the spin up(down) global density of states as a function of defect concentration. At low energy region, intensity of the resonance peak (E r 0.36 eV) is exponentially increased with the increasing the defect concentration, shown in Figure 3.7(b). This indicates that the global density of states is not simply superimposed by the LDOS surrounding the vacancies. The space between the vacancies is reduced when the defect concentration increased. The contributions from the localized states are therefore reduced. This implies that a maximum conductivity will be observed when a critical 40 defect concentration is formed in the graphene sheet. Notice that, another relatively smaller resonance peak is located at E r -0.36 eV , which originates from the uneven number of spin up and spin down electrons in the system, resulting from the random distribution of vacancies sites. In the high energy region, the softening of the van Hove singularity located at2:7eV and the development of Lifshitz tails at the band edge are induced by increasing the defect concentration[64]. Figure 3.7: (a)The gloabal density of state with differnt defect concentration.(b) The intensity of resonance peak located at 0.25eV as a function of defect concentration. 3.2.2 Magnetic Pattern Vacancy Defects In 1986, Lieb proved a fundamental theorem for the half-filled Hubbard model. On a bipartite lattice, i.e. a system composed of two sublattices of atoms A and B. The Lieb’s theorem states that if a half-filled system contains N A and N B , numbers of A and B atoms, the ground state has spin angular momentumS = 1 2 (N B N A ) in the repulsive case (U > 0). In a system withN B 6=N A , the ferromagnetic state should be expected. [24, 47]. 41 Next, we examine the magnetic patterns induced by these defects. Within the self- consistent mean field calculation the numbers of spin-up and spin-down electrons are fixed during the iteration process, whereas the total number of electrons is kept equal to the number of carbon atoms. As the honeycomb lattice of graphene is composed of two sublattices, containing atoms A and B, the presence of a single vacancy defect implies that the numbers of A atoms (N A ) and B atoms (N B ) are not equal. Therefore magnetic moments are induced, consistent with Lieb’s theorem[47], i.e. the total spin of the ground state is S = (N A -N B )/2. Let us first examine the magnetic pattern in a graphene sheet induced by a single defect, shown in Figure 3.8(a). The total magnetic moment in this case is 0.5/960, since only one A atom in the 960-site sheet is missing. The magnetic moment is localized around the vacancy resembling the LDOS, as shown in Figure 3.3(a). If the vacancy introduced into the A sublattice, the magnitude of the induced magnetic moment in the at B sublattice is larger than in the A sublattice with maximum valueM B =0:058 B and M A = 0:0088 B . This indicates that the interaction between nearest-neighbor spins is antiferromagnetic, and the interaction between spins on the same sublattice is ferromagnetic. This is in agreement with the previous observation [7, 44, 49, 71]. When a zigzag type line defect is introduced, a pronounced localized magnetic pat- tern is formed close to the defect, as observed in Figure 3.8(b). By taking out 7 A atoms and 6 B atoms, the induced magnetic moment magnitudes in the B sublattice are larger than the A sublattice with maximum magnetic moment M B =0:151 B and M A = 0:146 B . Although the total magnetic moment remains 0.5/960, the local mag- netic moment is 2.6 times larger than for a single vacancy in the spin-down case and 16.6 times larger for the spin-up case. This can be understood by comparing the intensities of the LDOS resonance peaks in these cases. The magnetic moment shows a similar Gaus- sian spread analogous to the resonance local density of states, shown in Figure 3.5(b). 42 Figure 3.8(c) shows the magnetic pattern of a graphene sheet in the presence of an arm- chair line defect. Compared to the zigzag line defect and the single defect, the induced local magnetic moment is much weaker. This is because the bound state doesn’t exist in the vicinity of the armchair line defect which is similar to the armchair edge indicated by previous work[82]. Therefore, instead of localizing around the vacancies, the magnetic moments distribute uniformly in the graphene sheet . Figure 3.8: Magnetic patterns in a graphene sheet induced by (a) single defect placed at the center of the sheet, (b) zigzag line defect, and (c) armchair line defect. The positions of the impurity atoms are denoted by white symbols. The color scale ranges from dark blue (negative magnetization) to yellow (positive magnetization). Weak Impurity Scattering So far, we have focused on the limit of very strong impurity scattering, corresponding to vacancy defects. When these vacancy sites are replaced by impurity atoms, the resulting scattering strengths are typically smaller, i.e. of the orderU d t. In Figure 3.8, we study the evolution of the magnetic patterns induced by a zigzag line defect as a function of increasing impurity scattering strength, while leaving U/t=1.2. Examining the spatial structure of these magnetic patterns, we observe that they become more localized with increasing impurity scattering strength, which is expected. More interesting, however, is the evolution of these patterns. For weak impurity scatter- ing (U d =t)=1, the local magnetic moment is not strongly localized around the impurity sites. When the scattering strength is increased toU d =t = 3, the local magnetic moment localizes more strongly around the edge of the impurity cluster with maximum induced 43 magnetic moment are M B = 0:0187 B and M A = 0:0024 B . For even stronger scattering, i.e. U d =t = 5; the local magnetic moment localizes close to the center of impurity cluster, with maximum induced magnetic moment areM B =0:0586 B and M A = 0:0088 B . A Gaussian shape along the direction parallel to the zigzag defect is observed, which was already seen in Figures 3.9(b) for the case whenU d =t tends to infinity. When the scattering strength of impurity increased, a stronger localized state is formed around the impurity sites, and hence a stronger local magnetic moment. 44 Figure 3.9: Magnetic patterns in a graphene sheet induced by a zigzag line defect placed at the center of the sheet. The positions of the impurity atoms are denoted by white sym- bols. Here, we consider various impurity scattering strengths (a)U d =t = 1, (b)U d =t = 3, and (c)U d =t = 5, while keeping U/t=1.2. The color scale ranges from green (negative magnetization) to yellow (positive magnetization). 45 3.3 Conclusions In summary, we have examined the Hubbard model on a two-dimensional honeycomb lattice to study the effects of point, zigzag and armchair defects on the electronic and magnetic structure in graphene sheets. In the vicinity of these defects, we observe pro- nounced impurity induced scattering resonances in the electronic density of states. In the case of a point vacancy defect, the amplitude of the impurity induced local density of states is found to decay inversely proportional to r 2 , and its frequency is found to converge asjE r j 1=U d with increasing impurity scattering strength. The local elec- tronic density of states around line defects is found to be strongly enhanced as well. The amplitudes of the impurity induced scattering resonances decay with a power law similar to the case of a point defect, and otherwise their spatial dependence is rather fea- tureless, with the exception of a local minimum in the local electronic density of states appearing near the center of zigzag line defects, indicating destructive interference. The magnetic patterns of the armchair line defect are found to be rather uniformly distributed, inducing a weak modulation of the local magnetic moment. In contrast, stronger magnetic patterns were observed in the vicinity of the single defect and zigzag line defects. For the case of single defects, a threefold symmetric magnetic pattern is observed. In the case of zigzag line defects, the amplitudes of the defect induced mag- netic moments are strongest at the center of the line defect and weaker at its end. Gener- ally, the impurity induced magnetic patterns of the zigzag line defect display a Gaussian shape along the direction of the line defects. This strong orientational magnetic pattern is found to persist down to fairly small impurity scattering strengths ofU d =t = 5, below which the induced patterns become more uniform. 46 Chapter 4 The Magnetism of Graphene Under Strain The linear dispersion of graphene is very sensitive to lattice deformation. Since strain in graphene, originating from coupling to a substrate and from the fabrication process cannot be completely avoided, such effects on its band structure, electric properties and magnetism are therefore important from an application point of view. Recently, experiments have demonstrated that the graphene sheet is capable of toler- ating strains as high as 18% without breaking the C-C bonds.[40] It is therefore promis- ing to use strain engineering to control graphene’s electrical and magnetic properties. Theoretical studies have indicated that the Dirac points K and K’ shift in opposite direc- tions due to the lattice deformation[62]. When the strain is sufficiently large, K and K’ can collapse onto each other, leading to the opening of a gap. This energy band gap transforms the graphene sheet from a semimetal to a semiconductor, resulting from a suppression of the conductivity [13, 14, 62]. This gap opening mimics the response to a strong physical magnetic field applied to the graphene sheet. Guinea et. al have proposed that a non-uniform strain bending a graphene ribbon into a circular arc could generate the pseudo-magnetic fields exceeding 10T, leading to the observation of a quan- tum Hall effect.[29] This strain induced energy quantization was experimentally realized in a graphene nano-bubble[46]. The pseudo-magneto-oscillation observed in the local density of states indicates the presence of Landau quantization due to strain coming from the topography. Many theoretical and experimental results have demonstrated that 47 strain can strongly modulate the electronic spectrum of graphene. Intuitively, the mag- netism of graphene should also be affected by the change of band structure. Few studies, however, have so far been conducted on the strain modulated magnetization, especially under non-uniform strain. In the following, we explore the magnetic ground state of graphene in the presence of uniaxial and nonuniform strain by using the mean field Hubbard model described in chapter 3. We consider a graphene dot under two types of strain, uniaxial and arc- bending. This chapter is organized as follows. First, we discuss the development of edge states under the uniaxial strain along two different directions, zigzag and arm- chair. To compare with the gradient strain-induced pseudo magnetic field, we discuss the Quantum Hall effect on both a conventional finite 2DEG strip and a zigzag graphene nanoribbon with a physical uniform and gradient magnetic field. We further study the edge states and the confinement state in the presence of a strain induced gradient pseudo magnetic field. Last, the confinement state is also discussed by comparing with analyti- cal results from the non-uniform pseudo magnetic field model. 48 4.1 Uniaxial Strain Figure 4.1(a) shows the magnetic pattern in a relaxed graphene dot. The presence of the edge local magnetic moment at the zigzag edge has been studied in much detail[86]; it can be explained in terms of a localized edge state resulting from the nonbonding bonds. The electrons in localized edge state are strongly polarized in the correlated systems, resulting in a shift of the low energy density of state peak for both spin up and spin down electrons. This behavior has been experimentally observed by scanning tunneling microscopy(STM)[75]. First, we consider a graphene dot in the presence of an applied uniaxial strain. The general strain tensor is expressed as[62] 0 = 0 @ cos 2 () sin 2 () (1 +) cos() sin() (1 +) cos() sin() sin 2 () cos 2 () 1 A ; (4.1) where is the angle between the zigzag direction and the tension direction, is the tensile strain, and = 0.165 is the isotropic elastic parameter for graphite. The displacement of the atom position is given by ! r = (1 + 0 ) ! r 0 ; (4.2) where ! r 0 is the relaxed equilibrium position of a C atom. This deformation causes a change in the bond length i : 1 = 3 =a( 3 4 (cos 2 sin 2 ) + 1 4 (sin 2 cos 2 )) (4.3) 2 =a(sin 2 cos 2 ); 49 wherea = 1:42 A is the equilibrium C-C bond length. This change leads to a different hopping integralV i pp ; which is exponentially decaying with increasing bond length. V i pp =te 3:37( i ) ; (4.4) wheret = 2:7eV is the hopping integral without bond stretching. Figure 4.1 (b) and (c) shows the magnetic patterns induced in a graphene dot when a 20% uniaxial strain is applied along the zigzag(ZZ) ( = 0) and the armchair(AC) ( = 2 ) directions, respec- tively. Comparing with the magnetic pattern in a relaxed graphene dot, our calculation indicates that the development of edge local magnetic moments (ELMM) is strongly dependent on the direction of the applied strain. As can been seen, applying a strain in the ZZ direction amplifies the ELMM. Conversely, strain applied in the AC direc- tion has the opposite effect of reducing the ELMM. This behavior can be understood by looking at the local density of states(LDOS) at the edges, shown in Figure 4.2 (a) and (b). The intensity of the low energy peak indicates that electrons tend to localize at the edge, and are therefore strongly spin polarized, when a ZZ strain is applied. In contrast, edge localized states under an AC strain are weaker, resulting in a smaller polarization. These effects are a direct consequences of the different behaviors of the ELMM, shown in the inset of Figure 4.2. The reason for this strain-direction dependent edge localiza- tion behavior is rooted in a gauge field theory which was first developed by Sasaki et. al.[36]. In this theory, the strain induced gauge field depends on the modification of the hopping energy which is related to the strain direction. We analyze the uniaxial strain induced gauge field as follows; when the deformation is small, the leading order of the change in hopping energy is given by t i t a ( 1 ; 2 ; 3 ): (4.5) 50 The effect of the change in hopping integral introduces an effective gauge field A = (A x ;A y ) in the Hamiltonian. K+k A jHj K B =v F e X j X i (t +t i )(f i (k) +rf i (k))e ikr ; (4.6) where v F is the Fermi velocity and K A=B is the Bloch wave function for A/B at the K point. From Equation 1.5, one can get the deformed Hamiltonian when the deformation is small. K+k A jH deform j K B =v F e X j X i t i f i (k)e ikr (4.7) =v F e X j e ikr (t 1 +t 2 ( 1 2 +i p 3 2 ) +t 3 ( 1 2 i p 3 2 ) =v F e X j e ikr (t 1 1 2 (t 2 +t 3 ) +i p 3 2 (t 2 t 3 ) =v F ee ikr (A x +iA y ): The gauge field from the deformation is v F eA y (r) =t 1 (r) 1 2 (t 2 (r) +t 3 (r)); (4.8) v F eA x (r) = p 3 2 (t 2 (r)t 3 (r)): When two different directions of strain are applied, the gauge field is 51 v F eA y (x) = (1 ( 1 4 3 4 ))t x=0 3 4 (1 +)t x>0 armchair; (4.9) v F eA y (x) = (1 ( 3 4 1 4 ))t x=0 3 4 (1 +)t x>0 zigzag . When a gauge field is present, the Dirac equation for electrons around the K and K’ points can be written as H =~v F 2 6 6 6 6 6 6 6 4 0 @ x k y +e A(x) ~ 0 0 @ x k y +e A(x) ~ 0 0 0 0 0 0 @ x +k y e A(x) ~ 0 0 @ x +k y e A(x) ~ 0 : 3 7 7 7 7 7 7 7 5 (4.10) The corresponding wave functions are ( K A (x); K B (x); K 0 A (x); K 0 B (x)) which satisfy (~ 2 @ 2 x +f( E v F ) 2 [U(x) 2 +~@ x U(x)]g) K A (x) = 0; (4.11) whereU(x) =~k y +eA(x). In the lowest energy state, the electron experiences an effective potential which is V (x) =U(x) 2 +~@ x U(x): (4.12) Around the K point, the first term is nearly gauge field independent, so we only consider the second term which is strong dependent on to change of the gauge field(A(x)). Since the gauge field is a constant in the bulk, electrons travel freely. However, the gauge field sharply changes near the edges (x = 0, andx = L), resulting in the formation of effective quantum wells. Equation 4.12 shows that the potential well becomes deeper as 52 strain is applied along the ZZ direction, whereas it becomes shallower under the effect of a strain applied in the AC direction. (a ) (b ) (c ) Figure 4.1: Magnetic pattern of (a) a relaxed graphene dot, (b) in the presence of a Zigzag strain, and (c) in the presence of an armchair strain. 53 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -0.1 0.0 0.1 0 0.05 0.1 0.15 0.2 LDOS@edge E(eV) 0.00 0.05 0.10 0.15 0.20 0.10 0.12 0.14 LMM Strain -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 -0.2 -0.1 0.0 0.1 0.2 LDOS@edge E(eV) 0 0.05 0.1 0.15 0.2 0.00 0.05 0.10 0.15 0.20 0.15 0.20 0.25 0.30 LMM Strain (a) (b) Figure 4.2: (a) and (b) are the local density of states at the edge with uniaxial strain along the zigzag and armchair directions, respectively. The insets shows the local magnetic moments as a function of strain. For the above mentioned calculation, we used a weak Coulomb interactionU = 1:2t, i. e. assuming no phase transition. However, a second order phase transition could be induced by strain[80], especially when the strain is applied along the zigzag direction as 54 a consequence of the strong enhancement of the local magnetic moments as discussed above. In order to clarify that our calculation remains in the ferromagnetic regime, we identify the criticalU c by studying the total magnetization as a function of the U/t, shown in Figure 4.3. The phase transition happens at smallerU c as the applied uniaxial strain is increased. This is in agreement with the Viana-Gomesetal. observation which indicates that theU c <t()>. At a ZZ strain of = 0:2,U c is still higher than the U we used in the above calculation. Figure 4.3: Variation of the total magnetic moment of the A and B sublattices as a function of the Hubbard interactionU=t. 55 4.2 Non Uniform Strain Induced Pseudo Magnetic Field When a uniaxial strain is applied, only the electrons localized to the edges encounter a pseudo magnetic field. Now we turn our attention to nonuniform strains. This also introduces a gauge field in the bulk area. To better understand the effects of strain induced pseudo magnetic fields, we first compare the quantum Hall effect (QHE) in a conventional 2DEG strip and in a zigzag graphene ribbon with a physical magnetic field for both the uniform and the gradient case. We then study the pseudo magnetic field induced quantum Hall effect at the end of this section. 4.2.1 Conventional Quantum Hall Effect in Graphene Let us first consider the QHE in a conventional 2DEG. The cyclotron motion of electrons with magnetic lengthl B = q ~ B is illustrated in Figure 4.4. The dispersion of energy as function of the one-dimensional momentumk is shown in Figure 4.4. Note that the horizontal axis is the position along x, due to the relation x =l 2 B k in the Landau gauge. The energy levels are bend due to the edge confining potential. The charges flow along the edges with opposite directions and same group velocities. In the bulk region, electrons do not propagate and do not contribute to the conductance. The LL has energy levelsE n =~w c (n + 1=2), where n=0,1,2,... andw c = eB=mc,m being the effective mass. 56 U Magnetic field E(k) K(y) Figure 4.4: Cyclotron motion of electrons in the presence of a uniform magnetic was applied Figure 4.5: The Hall conductance as a function of magnetic filed in conventional 2DEG [18]. The Hall conductance is given by xy =gn e 2 h (4.13) 57 shown in Figure 4.5, where g is the spin degeneracy. Now, when the gradient magnetic field (B = B 1 x) is applied in a conventional 2DEG, the classical picture of two-dimensional electron trajectories becomes asymmet- ric. The propagation can be understood by a classical argument [55]. The quasi-classical electrons in the bulk area travel in spiral trajectories in the plane according to the gra- dient of the magnetic field, and drift along the direction perpendicular gradient field, as shown in Figure 4.6. This state can be found in the band structure around finite positive k. The corresponding effective potential is a single well with the position of its minimal potential to p k. Similar to systems in a uniform magnetic field, electrons near sample edge are scattered by the rigid walls and are forced into skipping orbits, and travel in the opposite direction if the sign of magnetic field doesn’t change. The magnitude of the group velocities is now different at the two edges. It is larger at the edge with the smaller magnetic field (k < 0) and becomes smaller at edge with the larger one (larger k > 0) . The LL is therefore not flat in the gradient magnetic field. The average energy band can be roughly expressed as E n ' (n + 1) ~ m r eB 1 ~k 2c (4.14) One can expect that the Hall conductance will not have a sharp step-like behavior shown in Figure 4.5 because of the absence of electron cyclotron motion. 58 E(k) k B x Figure 4.6: Cyclotron motion of electrons when a gradient magnetic is applied[55]. For graphene, we revisit Equation 1.7 with gauge field A = (0;Bx; 0), is applying a constant magnetic field with magnitudeB along the positive z direction. The effective low-energy Hamitonian close to the nodal K and K’ points in terms of K A , K B , K 0 A , and K 0 B is of the form H =~v F 2 6 6 6 6 6 6 6 4 0 @ x k y +e A(x) ~ 0 0 @ x k y +e A(x) ~ 0 0 0 0 0 0 @ x +k y +e A(x) ~ 0 0 @ x +k y +e A(x) ~ 0 3 7 7 7 7 7 7 7 5 (4.15) Different from the pseudo-magnetic field in Equation 4.10, in the case of a physical non-uniform magnetic field the sign of gauge field does not change from K to K’. Thus time reversal symmetry is broken in this case. By acting with the Hamiltonian of the K point on the wave function and replacingk = p } , one can gets p 2 y } 2 K A + ( By } P x } ) 2 K A = [( E v F ) 2 + B } ] K A : (4.16) 59 To map this into the Quantum Harmonic Oscillation problem, one can replacew c = B ~ , andy 0 = px B , p 2 y } 2 K A +w 2 c (yy 0 ) 2 K A = [( E v F ) 2 + B } ] K A : (4.17) One thus can obtain the Landau level as } 2 2 [( E v F ) 2 + B } ] =w c (n + 1 2 ) (4.18) = B ~ (n + 1 2 ) E n =sgn(n)v F p }Bn (4.19) and the corresponding wave function K A _e ikxx h n (y), with h n (y) = i n 2 n n!l B exp[ 1 2 ( yy 0 l B ) 2 ]H n ( yy 0 l B ); (4.20) wherel B = q ~ B is the cyclotron length, andH n () is Hermite polynomial H n () = (1) n e 2 d n d n e 2 (4.21) Note that the solution of K B _e ikxx h n1 (y). This originates from the spin degeneracy. The corresponding wave function is of the form 2 6 6 6 6 6 6 6 4 K A K B K 0 A K 0 B 3 7 7 7 7 7 7 7 5 _ e ikxx 2 6 6 6 6 6 6 6 4 h n (y) h n1 (y) h n1 (y) h n (y) 3 7 7 7 7 7 7 7 5 : (4.22) The corresponding electronic band structure is shown in Figure 4.7. For the lowest Landau level (LLL), the wave function change from the cosine to a Gaussian shape 60 when increasing magnetic field atk x = 0. This is agree with the classical argument, i. e. electrons orbit with smaller cyclotron radii when larger magnetic fields are applied. The Hall conductance is xy =g s g v (n + 1 2 ) e 2 h ; (4.23) whereg s andg v are spin and Vally degeneracies, shown in Figure 4.8. Different from the conventional 2DEG, the Hall plateau at xy = 0 is absent. This can be explained by the lowest Landau level being now atn = 0 which is an extended state at this energy [67, 68]. n=0 n=1 n=2 K E n=4 n=3 Figure 4.7: (a) Dispersion of the graphene sheet without magnetic field. (b) Energy quantization of dispersion when a magnetic field is applied. 61 Figure 4.8: The Hall conductance as a function of magnetic field in graphene [18]. Now we turn our attention back to the zigzag graphene nanoribbon (ZGNR) in a uni- form magnetic field. Zigzag boundary condition (Equation 1.11) are introduced in Equation4.15. Without magnetic field, this boundary supports low-energy edge state which exponentially decay from the edge on the length scale of the lattice spacing. This is much shorter than the magnetic lengthl B , which is usually 100nm. The edge states therefore are not sensitive to the finite magnetic field. Figures 4.9 (a) and (d) show the dispersion of the ZGNR as a finite magnetic field was applied at K and K’ points. Both lowest energy levels at the K and K’ points shift to negative k due to the broken time reversal symmetry. At the K point, the ground state for spin up electrons ( K A ) becomes a Gaussian in the bulk area and propagates toward the negative y direction in the bulk area. Meanwhile, the spin down electrons ( K B ) travel in the same direction along the left edge, shown in Figures 4.9(b) and (c). At the K’ point, on the other hand, the spin up electrons ( K 0 A ) move in the positive y direction along the edge, and the spin down electrons ( K 0 B ) travel in the bulk area, shown in Figure 4.9. The electron trajectory is similar to Figure 4.4. The only difference is the spin degeneracy due to the nature of the 62 AB sublattices. Despite this, the Hall conductance is similar to Figure 4.8. Figure 4.10 shows the dispersion of the ZGNR when a gradient magnetic field (B = B 1 xB 1 L) is applied. The edge state is still independent with the gradient magnetic field and embedded in the flat band area. At the K point, the K A moves in the negative y direction with a Gaussian form. Conversely K B still moves along the edge with exponentially decay form. At the K’ point, both K 0 B and K 0 A move through the bulk area with Gaussian form. Note that the higher Landau level are not flat due to the gradient magnetic field, which is same as in the conventional 2DEG case, shown in Figure 4.6. This means that the electrons with higher energy do not have a cyclotron motion and always move and contribute to the conductance. One can expect that the Hall conductance will not have the step-like behavior, shown in Figure 4.8. 63 -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 E(eV) K' 0 40 80 120 160 200 0.0 0.2 0.4 ' x ky=-0.2356 ky=-0.157 ky=-0.078 ky=0 ky=0.078 ky=0.157 x -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 E(eV) K 0 40 80 120 160 200 0.0 0.2 0.4 0.6 0.8 ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=0 ky=-0.078 ky=-0.157 ky=-0.2356 x x 0 40 80 120 160 200 0.0 0.2 0.4 0.6 0.8 x x ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=0 ky=-0.078 ky=-0.157 ky=-0.2356 (d) (e) (f) (a) (b) (c) 0 40 80 120 160 200 0.0 0.2 0.4 ' x x 0.384 0.314 0.235 0.157 0.078 0 -0.078 -0.157 -0.235 -0384 Figure 4.9: (a) and (d) are the band structure of Zigzag Graphene nanoribbon in a uni- form magnetic field around the K and K’ points. (b) and (c) are the ground state as function of position for A and B type sub-lattice at K point. (e) and (f) are the ground state as function of position for A and B type sub-lattice at K’ point 64 -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 2.5 E(eV) K 0 40 80 120 160 200 0.0 0.2 0.4 0.6 0.8 ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=0 ky=-0.078 ky=-0.157 ky=-0.2356 x x 0 40 80 120 160 200 0.0 0.2 0.4 0.6 0.8 x x ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=0 ky=-0.078 ky=-0.157 ky=-0.2356 -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 E(eV) K' (d) (e) (f) (a) (b) (c) 0 40 80 120 160 200 0.0 0.2 0.4 ' x x 0.384 0.314 0.235 0.157 0.078 0 -0.078 -0.157 -0.235 -0384 0 40 80 120 160 200 0.0 0.2 0.4 ' x x 0.384 0.314 0.235 0.157 0.078 0 -0.078 -0.157 -0.235 -0384 Figure 4.10: (a) and (d) are the band structure of Zigzag Graphene nanoribbon in a gradient magnetic field around the K and K’ points. (b) and (c) are the ground state as function of position for A and B type sub-lattice at K point. (e) and (f) are the ground state as function of position for A and B type sub-lattice at K’ point. 65 4.2.2 Strain Induced Pseudo Magnetic Field Now we consider an arc-bending of the graphene lattice which induces an effective gradient magnetic field. The resulting deformation is written as[28] u x = (R +y) sin()x; (4.24) u y = (R +y) cos()Ry; where(x) = 2x L arcsin( L 2R ); L is the width of the graphene dot, and R is the radius of the circle. The modification of the bond length is[12] 1 = 3 = ! ab a [ ! u a ! u b ]; (4.25) =a( p 3 4 sin((x)) 1 4 cos((x)) + 1); where ! u a ( ! u b ) is the displacement of the A(B) atom, and ! ab is the vector between the A and B atoms in the relaxed situation. Figure 4.11(a) shows the arc-bending strain induced pseudo magnetic field in a the graphene dot of length (L = 120a 0 ). For a small strain (larger R), the magnetic field is relatively uniform. When reducing R, the induced magnetic field varies linearly with position B s (x) =C arcsin( L R )(1 R +x L arcsin( L 2R )) (4.26) =B 1 x +B 0 : A s (x) = 1 2 B 1 x 2 +B 0 x 66 The corresponding effective potential (Equation 4.12) is shown in Figure 4.11 for spin up electrons with different arc-bending strains. As one can see, the effective potential is asymmetric. Whenk y > 0,V (x;k y ) grows with the position, while fork y < 0, a single potential well opens up and its minimum shifts toward outer edge with decreasingk y . This implies that the electrons are confined in the bulk. Figure 4.15 shows the electronic spectrum for different arc-bending strains. For a strain free system, an edge state appears in the region0:75 a 0 <k y <0:4 a 0 . As the strain increases, the corresponding flat band becomes wider,0:98 a 0 <k y <0:15 a 0 . The pseudo magnetic field causes cyclotron motion of electrons and leads to flat bands at higher energy level fork y <0:75 a 0 . This flat bands reveals that the energy level is quantized and yields the peaks observed in the density of states. For weak strain, i. e. L R 1, Equation 4.26 gives thatB 1 s ( L R ) 2 andB 0 s ( L R ). The magnetic field is relatively uniform and Landau level energy are given asE n _ p nB 0 t q Ln R , shown in Figure 4.12. 67 0 40 80 120 160 -140 -120 -100 -80 -60 Max strain~10% B(T) Position x (angstrom) Max strain~20% 0 40 80 120 160 0 1 2 3 4 5 6 k y <0 Effective potential V(10 18 /m 2 ) Position x (angstrom) k y =0 k y >0 (b) (a) Figure 4.11: (a)Arc-bending strain induced pseudo magnetic field. (b) effective potential V (x;k y ) fork y = 0;k y < 0, andk y > 0 with R/L=3. 68 -0.8 -0.4 0.0 0.4 0.8 0 20 40 60 R=3L R=3.6L R=4.5L R=5.1L DOS E(eV) -2 -1 0 1 2 -0.8 -0.4 0.0 0.4 0.8 En(eV) n 1/2 R=3L R=3.3L R=3.6L R=4.2L R=4.5L R=4.8L R=5.1L R=5.4L -1.6 -1.4 -1.2 -1.2 -1.1 -1.0 -0.9 -0.8 Ln(En/n 1/2 ) Ln(L/R) y=0.568x-0.2569 (a) (b) Figure 4.12: ((a) Global density of states with different arc-bending strains. (b) energy levels as a function of p n. The inset shows the dependence on arc-bending strength and the corresponding energy quantization. 69 -1 .0 -0 .5 0 .0 -1 .0 -0 .5 0 .0 0 .5 1 .0 (a ) E (e V ) k y (2 π/a 0 ) -1 .0 -0 .5 0 .0 -1 .0 -0 .5 0 .0 0 .5 1 .0 (b ) E (e V ) k y ( 2π/a 0 ) D e n s ity o f s ta te -1 .0 -0 .5 0 .0 -1 .0 -0 .5 0 .0 0 .5 1 .0 (C ) k y ( 2π/a 0 ) E (e V ) D e n s ity o f s ta te Figure 4.13: Energy bandE n (k y ) of a ZGNR (a)strain-free, (b) with arc-bending strain R=5L, and (c) with arc-bending strainR=3L 70 Next, we examine the local density of states along the centerof the graphene dot, as shown in Figure 4.14. To simplify the problem, we turn off the on-site Coulomb interaction. As one can see, the inner edge state is independent of the arc-bending strain, while the outer edge, on the other hand, increases. This can be understood by the gauge field which is independent of R at the inner edge and strengthens at the outer edge, as we discussed before. Figure 4.14: Local density of states along the center of a graphene dot (a) with arc- bending strain R=5L and (b) R=3L 71 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .0 0 .2 0 .4 0 .6 (a ) L D O S E (e V ) R = 3 L R = 3 .6 L R = 4 .2 L R = 4 .8 L R = 5 .4 L -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .0 0 .2 0 .4 0 .6 L D O S E (e V ) R = 3 L R = 3 .6 L R = 4 .2 L R = 4 .8 L R = 5 .4 L (b ) -2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 0 .0 0 0 0 .0 0 5 0 .0 1 0 0 .0 1 5 0 .0 2 0 R = 3 L R = 3 .6 L R = 4 .2 L R = 4 .5 L R = 5 .1 L L D O S x (C ) Figure 4.15: (a) and (b) are the local density of states for the inner and outer edge state. (c) shows the confinement state for various arc-bending strains. The confinement state of the A sublattice shows up as a peak in the LDOS. For the Continuum description, we revisit the Dirac equation (Equation 4.10) with zigzag 72 boundary conditions (Equation 1.11) and gauge field from Equation 4.26. The calculated energy band is shown in Figure 4.16. For a strain free system, the wavefunction decay exponentially from the edges as e kyx for both spin up and spin down electrons. At k y = 0, the differential of the spin up(down) wavefunction (@ x A=B ) becomes zero at the inner(outer) edge in order to meet the boundary conditions. For values k y < 0, the Dirac equation supports confinement state. However, in the tight binding model, one only observes the edge state which is close to zero energy but does not see the confinement state due to the its high energy. When the arc-bending strain is applied, the pseudo magnetic field makes the flat band wider, which agrees with the result from the tight binding calculation. The corresponding energy of confinement states is reduced and therefore is captured by the tight binding model. For spin down electrons, on the other hand, the wavefunction moves towards the outer edge and contribute to the growth of the edge state, as shown in Figure 4.15(b). This behavior is identical for both Dirac cones. 73 -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 2.5 E(eV) K 0 40 80 120 160 200 0.0 0.2 0.4 0.6 0.8 ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=0 ky=-0.078 ky=-0.157 ky=-0.2356 x x 0 40 80 120 160 200 0.0 0.2 0.4 0.6 0.8 x x ky=0.38 ky=0.314 ky=0.235 ky=0.157 ky=0.078 ky=0 ky=-0.078 ky=-0.157 ky=-0.2356 -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 E(eV) K' 0 40 80 120 160 200 0.0 0.1 0.2 0.3 0.4 0.5 ky=-0.2356 ky=-0.157 ky=-0.078 ky=0 ky=0.078 ky=0.157 ky=0.235 ky=0.314 ky=0.38 ' x x 0 40 80 120 160 200 0.0 0.2 0.4 0.6 0.8 ky=-0.2356 ky=-0.157 ky=-0.078 ky=0 ky=0.078 ky=0.157 ky=0.235 ky=0.314 ky=0.38 x ' x (d) (e) (f) (a) (b) (c) Figure 4.16: (a) and (d) are the band structure of Zigzag Graphene nanoribbon in a arc- bending strain around the K and K’ points. (b) and (c) are the ground state as function of position for A and B type sub-lattice at K point. (e) and (f) are the ground state as function of position for A and B type sub-lattice at K’ point. 74 Now we turn on the Coulomb repulsionU = 1:2t 0 . Figure 4.17(a) shows the density of states (DOS) under various applied strains for spin up (positive) and spin down (neg- ative) electrons. The presence of the peak feature indicates energy quantization. The shift of the peak at the lowest Landau level indicates that the system is polarized. The local density of states of spin polarized edge states are shown in Figure 4.17(c)/(d) when various levels of arc-bending strains are applied. The spin polarization is enhanced at the outer edge, while it is reduced at the inner edge due to the expansion and constric- tion of the bond lengths. We now examine this development of edge localized states using the gauge field theory. Assuming thatL=R << 1 , the change in bond length is 1 p 3a 4 x R :The gauge field is then written as v F eA y (x) = p 3t 4 x R : (4.27) From equation (12), the effective depth of the edge potential well is given byV / (1 + p 3 2 x R ): For the inner edge (x < 0), the quantum well becomes shallower when the arc-bending strain(L/R) increases; the opposite effect is observed at the outer edge. The modification of the potential well is in agreement with the evolution of the localized state given by our calculation. We also observe that local magnetic moments develop in the bulk. This can be explained by the confinement state with reducing energy when the arc-bending strain is applied. The electrons are now localized at the bulk and the on-site coulomb repulsion contributes to the local magnetic moments at the bulk area. The local magnetic moments are therefore modified into a non-symmetric distribution, shown in Figure 4.17 (d). 75 -2 -1 0 1 2 -2 0 0 2 0 D O S E (e V ) R = 1 0 L R = L (a ) -2 -1 0 1 2 -0 .1 2 -0 .0 8 -0 .0 4 0 .0 0 0 .0 4 0 .0 8 0 .1 2 R = 1 0 L L D O S @ in n e r e d g e E (e V ) R = L (b ) -2 -1 0 1 2 -0 .3 -0 .2 -0 .1 0 .0 0 .1 0 .2 0 .3 L D O S @ o u te r e d g e E (e V ) R = 1 0 L R = L (c ) (d ) Figure 4.17: (a) The global density of state as a function of various L/R.(b) and (c) are the local density of state at the inner and outer edges.(d) Magnetic pattern of the graphene dot. 76 4.3 Conclusions The main result of this work is that in the presence of uniaxial strain the edge localized states are strongly dependent on the direction of the strain via the mechanism of a strain induced gauge field. Since the gauge field is sharply modified at the edge, an effective quantum well is induced. The depth of this quantum well becomes deeper when a uniax- ial strain is applied along the zigzag direction and decreased when it is applied along the armchair direction. When an on-site Coulomb repulsion is considered, the edge local magnetic moments are formed. The magnitude of the ELMM depends on the intensity of the localized states. When an arc-bending strain is applied, a pseudo magnetic field is introduced. This leads to energy quantization. When the arc-bending strain is small, the pseudo mag- netic field is uniform. The energy level is quantized as a function of q Ln R ; while the arc-bending strain becomes larger (R=L > 3), the energy quantization start to deviate from the relation due to the presence of a non-uniform pseudo magnetic field. Different from the uniaxial strain case, the edge localized states develop opposite features at the inner and outer edges due to the non-uniform strain induced gauge field. It gives rise to an increase and decrease of the effective quantum well at the right edge and the left edge, respectively. As a result, electrons prefer not to be localized at the inner edge, and instead develop into a confinement state. This phenomenon can be understood as an effective potential well created by the non-uniform strain-induced magnetic field. The Dirac Hamiltonian, by introducing the the proper zigzag boundary is conditions and a gradient magnetic field, can reproduce these results of the tight binding calculation. It shows that the energy of the confinement state is reduced. Therefore it is consistent with the tight binding model in the low energy region. When an on-site Coulomb interaction is considered, the local magnetic moments behave the same way as the localized states within the low interaction region. This 77 phenomenon can be directly observed by using magnetic force microscopy or scanning tunneling microscopy. For transport measurements, the pseudo magnetic field induces an anomalous quantum Hall effect which can be distinguished from a physical magnetic field as it the preserves time reversal symmetry. 78 Chapter 5 Polarization of Graphene Graphene has been reported as a promising candidate for photo detectors capable of detecting rates up to 10 Gbit/s[21] and for tunable terahertz metamaterials[38]. This has stimulated intense experimental and theoretical efforts to study the optical response of graphene. Screening properties in graphene exhibit significantly different behavior from conventional 2D systems due to the linear dispersion of electrons and holes near its nodal K and K’ points in the Brillouin zone. Plasmon resonance in graphene can been experimentally observed by using inelastic electron scattering spectroscopy and inelastic scanning tunneling microscopy. Infrared absorption in doped charge systems is directly related toT CNP T=T CNP , whereT CNP andT are the transmission coeffi- cients at the charge neutral point (CNP) and the gated sample. While tuning the gate, the absorption peak shifts toward higher energies due to the change of the interband transi- tion originating from the increase of the carrier concentration. More experiments have been carried out. It has been reported that the width and edges structure of graphene ribbons can also affect the plasmon resonance due to the modification of the band struc- ture. Theoretical predictions on plasmons in graphene describe collective excitations of the Dirac fermions, taking into account dynamic screening of Coulomb interactions and many-body effects. These phenomena have been widely studied theoretically using a self-consisted field approach[34, 78]. In the following, I start with introducing this method. 79 5.1 Self-consisted Field Approach The self-consistent field method (SCF) was developed by Ehrenreich and Cohen[22]. They compare this method to a many body approach and indicate that the two approaches are equivalent under certain approximation. This provides a simple way to study the many body problem. The single particle problem Liouville equation is given by i~ @ @t = [H;]; (5.1) whereH =H 0 +V (x;t).H 0 =p 2 =2m is the kinetic energy of free electron.V (x;t) is the self-consistent potential in the Fourier expansion as V (x;t) = X V (q 0 ;t)e q 0 x : (5.2) The charge density operator can be expanded as = (0) + (1) . The unperturbed density has the property (0) jki =f 0 (E k )jki, wheref 0 (E k ) is the Fermi distribution. Equation 5.1 becomes i~ @ @t = [H 0; (0) ] + [H 0; (1) ] + [V ; (0) ] + [V ; (1) ]: (5.3) By neglecting the product ofV (1) which is equivalent to first-type self consistent per- turbation theory, Equation. 5.3 can be approximated as i~ @ (1) @t = [H 0; (1) ] + [V ; (0) ]: (5.4) 80 The matrix element between statesk andk +q is i~ @ @t k (1) k +q = k [H 0; (1) ] k +q + k [V ; (0) ] k +q (5.5) = k H 0 (1) (1) H 0 ] k +q k (0) VV (0) k +q = (E k E k+q ) k (1) k +q + (f 0 (E k+q )f 0 (E k ))hkjVjk +qi: The last term can be expressed byV (q;t), including the externalV 0 (q;t) and screening potentialV s (q;t). Using Poisson’s equation,r 2 V s =4e 2 n, with the induced charge density n =Trf(rx) (1) g (5.6) = 1 X q e qx X k 0 k 0 (1) k 0 +q By substitutingn into Poisson’s equation, we thus find r 2 V s = 4e 2 X q e qx X k 0 k 0 (1) k 0 +q (5.7) V s (q;t) = 4e 2 X k 0 k 0 (1) k 0 +q ZZ d 2 xe qx = 4e 2 q 2 X k 0 k 0 (1) k 0 +q =v q X k 0 k 0 (1) k 0 +q wherev q = 4e 2 q 2 . By substitutingV s obtained in Equation 5.7, Equation 5.5 becomes i~ @ @t k (1) k +q = (E k E k+q ) k (1) k +q (5.8) + (f 0 (E k+q )f 0 (E k ))V q X k 0 k 0 (1) k 0 +q : 81 The definition of the dielectric constant is P (q;t) = ((q;w) 1) 4 "(q;t): (5.9) The polarization P (q;t) is related to the induced charge density byr P = en or iqP (q;t) = en(q;t), and the electric field is given by e"(q;t) = qV (q;t). By assuming k (1) k +q has the same time dependence asV 0 (q;t) which ise t e wt , it can be obtained as k (1) k +q =V (q;t) X k f 0 (E k+q )f 0 (E k ) E k+q E k ~w +i~ : (5.10) The dielectric constant can be found as (q;w) = 1 + 4 P (q;t) "(q;t) (5.11) = 1 4 e 2 q 2 n(q;t) V (q;t) = 1v q P k k (1) k +q V (q;t) = 1v q V (q;t) P k f 0 (E k+q )f 0 (E k ) E k+q E k ~w++i~ V (q;t) = 1 lim !0 v q X k f 0 (E k+q )f 0 (E k ) E k+q E k ~w +i~ : This is known as the Lindhard formula [48],(q;w) = 1v q (q;w), where the polar- izability (q;w) is (q;w) = lim !0 X k f 0 (E k+q )f 0 (E k ) E k+q E k ~w +i~ : (5.12) 82 By using the Fourier transformation, the real space representation of the polarizability is written as (r;r 0 ;w) = lim !0 X i;j f 0 (E i )f 0 (E j ) E i E j ~w +i~ i (r) i (r 0 ) j (r 0 ) i (r); (5.13) where i (r) andE i are obtained by solving the tight binding model. The induced charge density due to an external field of frequency is given by ind (r;w) = Z (r;r 0 ;w) tot (r 0 ;w)dr 0 ; (5.14) where tot (r 0 ;w) = ind (r 0 ;w) + ext (r 0 ;w). The induced electric field is ind (r;w) = Z ind (r 0 ;w) jrr 0 j dr 0 : (5.15) The process begins with initial setting ind (r;w) = 0. Equation 5.14 and Equation 5.15 are solved self-consistently by iterating ind (r;w) and ind (r;w). This method has been widely applied on many material, such as atomic cluster[26, 57]. 83 5.2 Optical Properties of Graphene Plasmon is known as one of the quasiparticle due to the collective excitation of elec- tron gas. By finding the polesw from Equation 5.11, the plasmon dispersion of three dimensional(3D) electron gas in the long wavelength limit is characterized by w 3D (q! 0) =w p + 3v 2 F 10w p q 2 ; (5.16) wherev F is the Fermi velocity.w p = p n 3D e 2 =m" 0 where" 0 ,n 3D ,e, andm are dielec- tric constant in vacuum, charge density, electron charge, and electron mass, respectively. For two dimensional (2D) material, the plasmon dispersion becomes[74] w 2D (q! 0) = r n 2D e 2 2m" q: (5.17) In the finite wave length region, conventional 2D plasmons are found to follow w 2 2D (q) n 2D e 2 2m" q + 3q 2 v 2 F 4 q 2 (5.18) Figure 5.1: Decay and scattering process in graphene [8] 84 Since the unique linear dispersion of graphene is different from the conventional 2DEG (parabolic dispersion), the excitation process of graphene is shown in Figure 5.1. The polarizability in Equation 5.12 for graphene can be rewritten as (q;w) = g s g v L 2 X kss 0 f sk f s 0 k 0 E sk E s 0 k w +i F ss 0(k;k 0 ); (5.19) where L is the k 0 = k +q, s;s0 =1, and F ss 0(k;k 0 ) = (1 +ss 0 cos( kk 0))=2. The Fermi distribution function isf sk (T! 0) = (k F k). Whens =s0 is the inter-band transition, inter (q;w) = g s g v 2L 2 X f f k+ f k+q E k+ E k+q w +i + f k f k+q+ E k E k+q+ w +i g (5.20) (1 cos( kk 0)): Whens =s0 is the intra-band transition intra (q;w) = g s g v 2L 2 X f f k+ f k+q+ E k+ E k+q+ w +i + f k f k+q E k E k+q w +i g (5.21) (1 + cos( kk 0)): WithE ks = sv F k, and fork F > 0, leading to thef k+ = (k F k),f k+q+ = (k F (k +q)),f k = 1;f k+q = 1. inter (q;w) = g s g v 2L 2 X f (k F k) 1 2v F k +v F qw +i + 1 (k F (k +q)) 2v F kv F qw +i g (5.22) (1 cos( kk 0)); intra (q;w) = g s g v 2L 2 X (k F k) (k F (k +q) v F qw +i (1 + cos( kk 0)): (5.23) 85 The plasmon dispersion is obtained by finding the poles of the dielectric constant in Equation 5.11. Within the long wave length limit (q! 0), Equation 5.22 and Equation 5.23 can be reduced to the following forms in the high and low frequency regimes[34]: (q;w) = 8 < : D 0 2 q 2 2 q 2 ( 1 w 2 1 4E 2 F ) ( q< w< 2E F ) D 0 (1 +i w q ) (w< q) (5.24) whereD 0 = p g s g v n== is the density of states at the Fermi energy and is the band parameter. By setting dielectric constant to zero, 1 + 2e 2 k D 0 2 q 2 ( 1 w 2 p 1 4E 2 F ) = 0; (5.25) The excitation dispersion can be obtained as w 2 p = 4E 2 F Aq Aq 4E 2 F ; (5.26) whereA = 2e 2 k 2 2 p gsgvn . Therefore w 2 p s 2e 2 k 2 2 r g s g v n q; (5.27) Different from conventional 2D plasmons, graphene showsw 0 /n 1=4 due to the quan- tum relativistic nature of Dirac fermions, shown in Figure 5.2. The plasmon dispersion in graphene is sensitive to many external factors, including the doping level, strain, and impurities. In the finite wavelength regime, since the electronic band is no longer lin- ear, the excitation dispersion behavior is more like conventional 2DEG when the doping level is higher[33]. Pellegrino et. al [61] reported a modification of the plasmon disper- sion when an uniaxial strain is applied. The square root dependence behavior in the low 86 frequency regime is still robust. The finite wavelength region, on the other hand, a non- monotonic behavior is observed. Next, we study plasmons by introducing the impurities by using the Random Phase Approximation. Figure 5.2: The plasmon dispersion for (a) graphene and (b) conventional 2DEG [34] 87 5.3 Plasmon in Graphene with Impurity We model graphene by a tight binding Hamiltonian for which the polarization operator is computed through the Random Phase Approximation (RPA). RPA is the most popular attempt to go beyond HartreeFock(HF) approximation including the quantum fluctua- tions of the interaction by including virtual particle-hole pairs. A limitation of the RPA is the high density system, i. e. a small Wigner-Seitz radius r s = n 1=2 measures the ratio of the potential to the kinetic energy in an interacting quantum Coulomb system. For graphene,r s 0:5[34]. The value of ther s makes RPA an excellent approximation for graphene. The excitation frequency is computed by diagonalizing the polarization operator (w). This method provides full information of the plasmon excitations including polarizability, dielectric function, and local spectral density of states. In the RPA, the Coulomb interaction is treated as a perturbation to the dielectric function. The Hamilto- nian is written as H =H 0 +V =t X <i;j> (c y i c j +c i c y j ) + X i u i (x)(c y i c i ) + X abcd V abcd c y a c y b c c c d ; (5.28) where t = 2:7eV is the hopping integral, u i (x) is the impurity potential which has a Gaussian form, u i (x) =u 0 exp( jxx 0 j 2 2 ); (5.29) wherex 0 is the location of the impurity, is the size of the impurity andu 0 is the strength of the impurity potential.V abcd is the Coulomb interaction between atoms within energy levelsa ,b,c andd. V abcd = e 2 2 Z dx Z dx 0 ' a (x)' b (x 0 )' c (x 0 )' d (x) jxx 0 j (5.30) 88 After the non-interacting Hamiltonian H 0 is diagonalized, the Coulomb interaction V is considered as a perturbation. Revisiting Equation 5.13, the polarization operator (r;r 0 ;w) is (r;r 0 ;w) = X abmn a (x) b (x) ab;mn (w) m (x) n (x): (5.31) For a non-interacting polarization, 0 (r;r 0 ;w) = X abmn a (x) b (x) 0 ab;mn (w) m (x) n (x); (5.32) 0 ab;mn (w) = 4n ab;mn 4E ab;mn Iw ; where4E and4n are diagonal matrices with elements. n ab;mn = am bn (n 0 a n 0 b ); (5.33) E ab;mn = am bn (E 0 a E 0 b ): Considering the Coulomb interaction, the polarization operator can be written as (w) = 0 (w) (1 0 (w)V ) = n (EwI nV ) : (5.34) By finding the poles of the polarization operator through solving eigen equation (E nV ) =w: (5.35) Spectral density functionA(w) is then computed through the definition A(w) = 1 Im R (w); (5.36) 89 where the R (w) = (w +i). By applying the relation lim !0 1 yi =} 1 y i(y); (5.37) A(w) = 1 Im n (E (w +i)IV n) (5.38) =n(EwIV n): By solving the eigenproblem to find the poles of trA(w) =n(ww ) (5.39) = n (ww ) 2 + 2 : The induced charge density is proportional to A(w). In the real space representation, one gets A(r;w) = Z dr 0 (r)A(w) (r 0 ): (5.40) By introducing a single charge impurity located at the center of the graphene sheet with 1600 atoms, according to Muniz et al [56], there are several features present in the global density of states. An extra state around zero energy is a direct consequence from the localized wavefunction around the impurity. We first examine the effect of doping level or equivalently gate effect by tunning the chemical potential from0:04t to +0:04t. To reduce the computational effort, we only consider the lower energy levels ranging fromn i =15 ton f = 15. Figure 5.3 shows the excitation frequency as a function of the chemical potential. As one can see, the excitation frequency is independent of the chemical potential due to the small range of the doping level. 90 Figure 5.3: Excitation frequency as a function of the chemical potential with selected energy level n=30. Figure 5.4 shows the excitation plasmon pattern for selected frequencies. The red/blue represent positive /negative induced plasmons. For the lowest excitation fre- quency (Figure 5.4(a)), the induced plasmons behave like dipole resonances; while for slightly higher frequencies (Figure 5.4(b)), they become more quadrapole like. This is in agreement with observations in the atomic clusters[57]. This can be explained by the fact that the dipole resonances are associated with the transition between the highest occupied energy level and lowest vacant energy level. In contrast with dipole resonance, the collective excitation with higher excitation frequencies are more sophisticated, since charge oscillations are supported. The changing chemical potential should introduce more electrons or holes into the excitation process. Therefore one should expect that the excitation frequencies supporting dipole resonances should be higher. In our case, the carrier density does not change significantly. Therefore the change of the excitation energy is not observed. However, one can expect that the excitation frequency should be higher due to the increase of the carrier density. Since the excitation frequency does 91 not change significantly, the plasmon patterns remain the same within this small range of doping level. -2 .8 8 0 E -5 -2 .5 9 2 E -5 -2 .3 0 4 E -5 -2 .0 1 6 E -5 -1 .7 2 8 E -5 -1 .4 4 0 E -5 -1 .1 5 2 E -5 -8 .6 4 0 E -6 -5 .7 6 0 E -6 -2 .8 8 0 E -6 0 2 .8 8 0 E -6 5 .7 6 0 E -6 8 .6 4 0 E -6 1 .1 5 2 E -5 1 .4 4 0 E -5 1 .7 2 8 E -5 2 .0 1 6 E -5 2 .3 0 4 E -5 2 .5 9 2 E -5 2 .8 8 0 E -5 -2 .8 0 0 E -4 -2 .5 2 0 E -4 -2 .2 4 0 E -4 -1 .9 6 0 E -4 -1 .6 8 0 E -4 -1 .4 0 0 E -4 -1 .1 2 0 E -4 -8 .4 0 0 E -5 -5 .6 0 0 E -5 -2 .8 0 0 E -5 0 2 .8 0 0 E -5 5 .6 0 0 E -5 8 .4 0 0 E -5 1 .1 2 0 E -4 1 .4 0 0 E -4 1 .6 8 0 E -4 1 .9 6 0 E -4 2 .2 4 0 E -4 2 .5 2 0 E -4 2 .8 0 0 E -4 -0 .0 0 5 2 0 0 -0 .0 0 4 6 8 0 -0 .0 0 4 1 6 0 -0 .0 0 3 6 4 0 -0 .0 0 3 1 2 0 -0 .0 0 2 6 0 0 -0 .0 0 2 0 8 0 -0 .0 0 1 5 6 0 -0 .0 0 1 0 4 0 -5 .2 0 0 E -4 0 5 .2 0 0 E -4 0 .0 0 1 0 4 0 0 .0 0 1 5 6 0 0 .0 0 2 0 8 0 0 .0 0 2 6 0 0 0 .0 0 3 1 2 0 0 .0 0 3 6 4 0 0 .0 0 4 1 6 0 0 .0 0 4 6 8 0 0 .0 0 5 2 0 0 (d ) (c ) (b ) -0 .0 0 4 9 0 0 -0 .0 0 4 4 1 0 -0 .0 0 3 9 2 0 -0 .0 0 3 4 3 0 -0 .0 0 2 9 4 0 -0 .0 0 2 4 5 0 -0 .0 0 1 9 6 0 -0 .0 0 1 4 7 0 -9 .8 0 0 E -4 -4 .9 0 0 E -4 0 4 .9 0 0 E -4 9 .8 0 0 E -4 0 .0 0 1 4 7 0 0 .0 0 1 9 6 0 0 .0 0 2 4 5 0 0 .0 0 2 9 4 0 0 .0 0 3 4 3 0 0 .0 0 3 9 2 0 0 .0 0 4 4 1 0 0 .0 0 4 9 0 0 (a ) Figure 5.4: Excitation plasmon distribution for chemical potential =0 , at selected level n = 30, single impurity strengthu 0 = 2, and size = 6. The corresponding excitation frequencies are (a)w = 3:48 10 4 Hz, (b)w = 4:17 10 3 Hz, (c)w = 1:43 10 2 , and (d)w = 2:13 10 2 Hz Next we examine the effects of impurity potentials by changing u 0 from -4t to 4t. Figure 5.5 shows symmetry for positive and negative potentials. Within the strong potential region (u 0 > 1teV ), the distribution of excitation frequencies has a peak position located at 0:35eV with higher impurity potentials. Figure 5.6 shows that the max excitation energy increases up to 0.4eV when the impurity potential is increased. This can be explained by the enhancement of localized states around the impurity. This may increase the carrier density involved in the transition, therefore leading to the increase of the excitation frequency described in Equation 5.27. However, by choosing limited lowest energy, the transition from the higher energy level is missing, which 92 makes the impurity of higher potentialu 0 = 4 not showing the increasing tendency. -4 -3 -2 -1 0 1 2 3 4 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 w (e V ) u 0 0 1 .1 5 0 2 .3 0 0 3 .4 5 0 4 .6 0 0 5 .7 5 0 6 .9 0 0 8 .0 5 0 9 .2 0 0 Figure 5.5: Excitation frequency as a function of the impurity strength of selected energy level n=56 0 .0 0 .2 0 .4 0 .6 0 2 4 6 8 1 0 A (w ) w (e V ) -2 -3 Figure 5.6: Spectral density of plasmons as a function of the excitation frequency with impurity potentialu 0 =2 (black) andu 0 =3 (red). Figure 5.7 shows the excitation plasmon distribution with an impurity potential of u 0 = 4 with selected excitation frequency. The plasmon distribution in this case is more sophisticated than for the case of weak impurity potential. For lower excitations, 93 shown in Figure 5.7(a), the excitation pattern is similar to a dipole resonance. For higher excitation frequencies, the pattern is again quadrapole like. -4 .1 2 0 E -6 -3 .7 0 8 E -6 -3 .2 9 6 E -6 -2 .8 8 4 E -6 -2 .4 7 2 E -6 -2 .0 6 0 E -6 -1 .6 4 8 E -6 -1 .2 3 6 E -6 -8 .2 4 0 E -7 -4 .1 2 0 E -7 0 4 .1 2 0 E -7 8 .2 4 0 E -7 1 .2 3 6 E -6 1 .6 4 8 E -6 2 .0 6 0 E -6 2 .4 7 2 E -6 2 .8 8 4 E -6 3 .2 9 6 E -6 3 .7 0 8 E -6 4 .1 2 0 E -6 -1 .0 0 0 E -5 -9 .0 0 0 E -6 -8 .0 0 0 E -6 -7 .0 0 0 E -6 -6 .0 0 0 E -6 -5 .0 0 0 E -6 -4 .0 0 0 E -6 -3 .0 0 0 E -6 -2 .0 0 0 E -6 -1 .0 0 0 E -6 0 1 .0 0 0 E -6 2 .0 0 0 E -6 3 .0 0 0 E -6 4 .0 0 0 E -6 5 .0 0 0 E -6 6 .0 0 0 E -6 7 .0 0 0 E -6 8 .0 0 0 E -6 9 .0 0 0 E -6 1 .0 0 0 E -5 -1 .0 0 0 E -5 -9 .1 4 5 E -6 -8 .2 9 0 E -6 -7 .4 3 5 E -6 -6 .5 8 0 E -6 -5 .7 2 5 E -6 -4 .8 7 0 E -6 -4 .0 1 5 E -6 -3 .1 6 0 E -6 -2 .3 0 5 E -6 -1 .4 5 0 E -6 -5 .9 5 0 E -7 2 .6 0 0 E -7 1 .1 1 5 E -6 1 .9 7 0 E -6 2 .8 2 5 E -6 3 .6 8 0 E -6 4 .5 3 5 E -6 5 .3 9 0 E -6 6 .2 4 5 E -6 7 .1 0 0 E -6 -6 .0 0 0 E -5 -5 .4 0 0 E -5 -4 .8 0 0 E -5 -4 .2 0 0 E -5 -3 .6 0 0 E -5 -3 .0 0 0 E -5 -2 .4 0 0 E -5 -1 .8 0 0 E -5 -1 .2 0 0 E -5 -6 .0 0 0 E -6 0 6 .0 0 0 E -6 1 .2 0 0 E -5 1 .8 0 0 E -5 2 .4 0 0 E -5 3 .0 0 0 E -5 3 .6 0 0 E -5 4 .2 0 0 E -5 4 .8 0 0 E -5 5 .4 0 0 E -5 6 .0 0 0 E -5 Figure 5.7: Excitation plasmon distribution for chemical potential =0eV , at selected level n = 56, single impurity strength u 0 = 4, and size = 6. The corresponding excitation frequencies are (a)w = 1:3 10 3 Hz, (b)w = 4:4 10 3 Hz, (c) w = 1:42 10 2 , and (d)w = 2:2 10 2 Hz Now we turn our attention to the effect resulting from adjusting the size of the impurity from 0 to 8a 0 . Figure 5.8 shows the spectral intensityA(w) as a function of the impurity size. When the size of the impurity is increased, the excitation frequency with maximum plasmon intensity shifts towards lower energies, shown in Figure 5.9. This can be explained by decreased of the localized state around zero energy, leading to a decrease of the carrier density at the low energy level. When the size of the impurity is smaller than 2a 0 , the excitation frequency is no different from the impurity-free system. 94 Figure 5.8: Excitation frequency as a function of the impurity size for selected energy level n=30 0 .0 0 .1 0 .2 0 .3 0 1 2 3 4 5 σ=6 A (w ) w (e V ) σ=8 Figure 5.9: Spectral density of plasmon as a function of the excitation frequency for impurity sizes = 6 (green), = 7 (red) and = 8 (black). Figure 5.10 shows the excitation plasmon pattern of a larger impurity size = 8 for various excitation frequencies. The increase of the impurity size spreads the plasmon distribution over the graphene sheet. The intensity of the excitation is higher than the case with smaller impurity side. 95 -5 .1 0 0 E -4 -4 .5 9 0 E -4 -4 .0 8 0 E -4 -3 .5 7 0 E -4 -3 .0 6 0 E -4 -2 .5 5 0 E -4 -2 .0 4 0 E -4 -1 .5 3 0 E -4 -1 .0 2 0 E -4 -5 .1 0 0 E -5 0 5 .1 0 0 E -5 1 .0 2 0 E -4 1 .5 3 0 E -4 2 .0 4 0 E -4 2 .5 5 0 E -4 3 .0 6 0 E -4 3 .5 7 0 E -4 4 .0 8 0 E -4 4 .5 9 0 E -4 5 .1 0 0 E -4 -7 .7 0 0 E -6 -6 .9 3 0 E -6 -6 .1 6 0 E -6 -5 .3 9 0 E -6 -4 .6 2 0 E -6 -3 .8 5 0 E -6 -3 .0 8 0 E -6 -2 .3 1 0 E -6 -1 .5 4 0 E -6 -7 .7 0 0 E -7 0 7 .7 0 0 E -7 1 .5 4 0 E -6 2 .3 1 0 E -6 3 .0 8 0 E -6 3 .8 5 0 E -6 4 .6 2 0 E -6 5 .3 9 0 E -6 6 .1 6 0 E -6 6 .9 3 0 E -6 7 .7 0 0 E -6 -5 .7 0 0 E -4 -5 .1 3 0 E -4 -4 .5 6 0 E -4 -3 .9 9 0 E -4 -3 .4 2 0 E -4 -2 .8 5 0 E -4 -2 .2 8 0 E -4 -1 .7 1 0 E -4 -1 .1 4 0 E -4 -5 .7 0 0 E -5 0 5 .7 0 0 E -5 1 .1 4 0 E -4 1 .7 1 0 E -4 2 .2 8 0 E -4 2 .8 5 0 E -4 3 .4 2 0 E -4 3 .9 9 0 E -4 4 .5 6 0 E -4 5 .1 3 0 E -4 5 .7 0 0 E -4 -3 .0 0 0 E -5 -2 .7 0 0 E -5 -2 .4 0 0 E -5 -2 .1 0 0 E -5 -1 .8 0 0 E -5 -1 .5 0 0 E -5 -1 .2 0 0 E -5 -9 .0 0 0 E -6 -6 .0 0 0 E -6 -3 .0 0 0 E -6 0 3 .0 0 0 E -6 6 .0 0 0 E -6 9 .0 0 0 E -6 1 .2 0 0 E -5 1 .5 0 0 E -5 1 .8 0 0 E -5 2 .1 0 0 E -5 2 .4 0 0 E -5 2 .7 0 0 E -5 3 .0 0 0 E -5 Figure 5.10: Excitation plasmon distribution for chemical potential =0eV ,at selected level n = 30, single impurity strength u 0 = 2, and size = 8. The corresponding excitation frequencies are (a)w = 2:1 10 3 Hz, (b)w = 4:5 10 3 Hz, (c) w = 1:37 10 2 , and (d)w = 2:3 10 2 Hz 96 5.4 Conclusions We have theoretically studied the plasmon of graphene with charge impurities within level of RPA. We find that excitation frequencies of maximum intensity increase with an impurity of stronger potential, resulting from the increasing carrier density from the localized states. On the other hand, the carrier density within low energy regime is decreased due to the increasing impurity size. The excitation frequency is there- fore decreased accordingly. Further, the plasmon pattern shows that the dipole reso- nance is supported for lower excitation frequencies due to the simple transition process. For higher excitation frequencies, quadrapole resonances are observed as the transition between higher energies becomes available. When increasing the impurity size ,plas- mons are distributed over larger areas in the graphene sheet. The induced plasmon intensity increases with increasing impurity size and potential. 97 Chapter 6 Conclusions In this dissertation, I discussed an electric-field induced semimetal-to-metal transition in few layer graphene, based on the temperature dependence of the resistance for different applied gate voltages. A semimetal-like temperature dependent behavior, i. e. decreas- ing resistance with increasing temperature, was observed at low gate voltages, due to an increase in the carrier concentration resulting from thermal excitation of electron- hole pairs. A metal-like temperature dependent behavior, i. e. resistance increases with increasing temperature, was observed at large gate voltages due to suppression of the excitations of electron-hole pairs, and due to decrease in mean free path resulting from electron-phonon scattering. Two different theoretical models were adopted to analyze this transport behavior in few layer graphene. (i) a simple two band (STB) model which describes the system by two overlapping parabolic bands. This model can only explain the gate-dependent conductivity at high temperatures and gives the best fitting parame- ter for the overlap energy which is found to be 16 meV . However, at low temperatures, the STB model predicts that the conductivity is gate independent in the small gate volt- age regime, which is not observed in the data. (ii) By considering frustration of the electronic potential due to impurities from the substrate, a Gaussian-distribution puddle model can successfully describe the gate-dependent conductivity in the low temperature, small gate voltage regime. The magnetism of graphene with defects was studied within a framework of the self-consistent mean-field Hubbard model. The electronic density of states shows char- acteristic patterns due to the presence of defects, leading to non-uniform distributions of 98 magnetic moments in the vicinity of the impurity sites. Specifically, defect induced res- onances in the local density of states were observed at energies close to the Dirac points. The magnitudes of the frequencies of these resonance states were shown to converge as j E r j= 1=U d with the strength of the scattering potential, whereas their amplitudes were found to decay inversely proportional tor 2 from the defect. For the case of defect clusters, the local magnetic moments in the vicinity of the cluster center are strongly enhanced with increasing defect cluster size. The impurity induced magnetic pattern of zigzag line defect displays a Gaussian shape along the direction of the line defects. This strong orientation magnetic pattern was found to persist down to fairly small impurity scattering strengthU d =t = 5, below which the induced patterns become more uniform. Furthermore,the zigzag line defects were found to introduce stronger-amplitude mag- netic patterns than armchair line defects. Electronic properties properties in graphene dots under mechanical deformation were studied by using both a tight binding lattice model and an effective Dirac model. The localized states around edge were found to be enhanced when an uniaxial strain is applied along zigzag direction due to the development of effective edge quantum wall. We also found that a pseudo magnetic field can be generated by applying an arc- bending strain. The effective edge quantum wall at inner(outer) edge states becomes swallower(deeper), as a result of reduction(increase) of the edge localized states. When the arc-bending strain is sufficiently small, this pseudo magnetic field is relatively uni- form, leading to the quantization of energy level asE n / q Ln R . Furthermore, a con- finement state were found to develop into the bulk area with increasing strength of strain due to a reduction of its corresponding energy. Furthermore, we found that introducing a Hubbard term on the mean-field level induces a strong polarization between the A and B sublattices, which provides an experimental test of the theory presented here. Plasmon resonances in graphene were studied by using a self-consistent method 99 within the random phase approximation (RPA). The observed increase in excitation energy can be attributed to the increasing carrier density due to stronger impurity poten- tials. With increasing impurity size, the carrier density within low energy region is decreased, as a result of lower excitation frequency. The dipole resonances can be found at lower excitation frequency due to its simple transition process. For higher excita- tion frequencies, quadrapole resonances were observed because the transitions between higher energy levels become possible. When the size of the impurity increases, we observed a larger spatial range of plasmons. 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Abstract (if available)
Abstract
Graphene, with its unique linear dispersion near the Fermi energy, has attracted great attention since its successful isolation from highly oriented pyrolytic graphite in 2004. Many important properties have been identified in graphene, including a remarkably high mobility at room temperature, an unusual quantum hall effect, and an ambipolar electric field effect. It has been proposed as a candidate for many applications, such as optical modulators, spintronic devices, and solar cells. Understanding the fundamental properties of graphene is therefore important. In this dissertation, I present a study of transport, magnetism and optical properties of graphene. In the first chapter, I introduce the electronic properties of mono layer and few layer graphene. ❧ In the second chapter, I present low temperature transport measurements in few layer graphene. An electric-field induced semimetal-to-metal transition is observed based on the temperature dependence of the resistance for different applied gate voltages. At small gate voltages the resistance decreases with increasing temperature due to the increase in carrier concentration resulting from thermal excitation of electron-hole pairs, as it is characteristic of a semimetal. At large gate, voltages excitations of electron-hole pairs are suppressed, and the resistance increases with increasing temperature because of the decrease in mean free path due to electron-phonon scattering, as is characteristic of a metal. The electron and hole mobilities are almost equal, so there is approximate electron-hole symmetry. The data are analyzed according to two different theoretical models for few-layer graphene. A simple two band (STB) model, two overlapping bands with quadratic energy-versus-momentum dispersion relations, is used to explain the experimental observations. The best fitting parameter for the overlap energy is found to be 16 meV. However, at low temperatures, the STB suggests that the conductivity is gate independent in the small gate voltage regime, which is not observed in the data. By considering frustration of the electronic potential due to impurities from the substrate, a Gaussian-distribution puddle model can successfully describe the observed transport behavior in the low temperature, small gate voltage regime. ❧ In the third chapter, I investigate the effects of point and line defects in monolayer graphene within the framework of the Hubbard model, using a self-consistent mean field theory. These defects are found to induce characteristic patterns into the electronic density of states and cause non-uniform distributions of magnetic moments in the vicinity of the impurity sites. Specifically, defect induced resonances in the local density of states are observed at energies close to the Dirac points. The magnitudes of the frequencies of these resonance states are shown to decrease with the strength of the scattering potential, whereas their amplitudes decay algebraically with increasing distance from the defect.For the case of defect clusters, we observe that with increasing defect cluster size the local magnetic moments in the vicinity of the cluster center are strongly enhanced.Furthermore, non-trivial impurity induced magnetic patterns are observed in the presence of line defects: zigzag line defects are found to introduce stronger-amplitude magnetic patterns than armchair line defects. When the scattering strength of these topological defects is increased, the induced patterns of magnetic moments become more strongly localized. ❧ In the fourth chapter, I theoretically study the electronic properties properties in graphene dots under mechanical deformation, using both tight binding lattice model and effective Dirac model. We observed an edge state, which is tunned by an effective quantum well originating from a strain-induced gauge field. Applying a uniaxial strain along the zigzag or armchair directions enhances or dampens the edge state due to the development of edge quantum wells. When an arc bending deformation is applied, the inner and outer edges of graphene dot display edge states caused by the induced nonuniform gauge field. These states suggest that an effective single well potential is introduced by a strong nonuniform pseudo-magnetic field, leading to a pseudo quantum Hall effect. Furthermore, we find that introducing a Hubbard term on the mean-field level induces a strong polarization between the A and B sublattices, which provides an experimental test of the theory presented here. ❧ Finally, I study charge impurity induced plasmon resonance in graphene by using the self-consistent method within random phase approximation (RPA). I attribute the observed increase in excitation energy to the increasing carrier density due to stronger impurity potentials. On the other hand, the carrier density within low energy region is decreased when impurity size is increased, as result of lower excitation frequency. The plasmon patterns show that the dipole resonances are supported for the lower excitation frequency due to a simple transition process. For higher excitation frequencies, quadrapole resonance is observed because the transitions between higher energy levels become possible. With increasing impurity size, a larger spatial range of plasmons is observed.
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Chang, Yi Chen
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Core Title
Modeling graphene: magnetic, transport and optical properties
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College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
06/04/2012
Defense Date
04/30/2012
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graphene,Hubbard model,OAI-PMH Harvest,tight binding model
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English
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Haas, Stephan W. (
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), Bickers, Nelson Eugene (
committee member
), Daeppen, Werner (
committee member
), Däppen, Werner (
committee member
), Mak, Chiho (
committee member
), Thompson, Richard S. (
committee member
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graphene
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