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Properties of Carbon Nanotubes Under External Factors: Adsorption, Mechanical Deformations, Def ects, and External Electric Fields by Yaroslav Shtogun A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics College of Arts and Sciences University of South Florida Major Professor: Li lia M. Woods, Ph.D. Ivan Oleynik, Ph.D. Hariharan Srikanth, Ph.D. Brian Space, Ph.D. Date of Approval: February 23, 2010 Keywords: DNA molecules, radicals, nanos tructures, nanotechnology, Van der Waals interactions, electronic band structure Copyright 2010, Yaroslav Shtogun
DEDICATION To my parents, Volodymyr and Vera To my brother Lyubomyr, and grandparents To all my friends
ACKNOWLEDGMENTS I owe my deepest gratitude to my advi sor Dr. Lilia Woods for her continuous guidance and support of my Ph.D study. Her immense knowledge, motivation, and enthusiasm helped me in my Ph.D career. I could not wish having a better advisor and mentor. Besides my advisor, I would also like to thank Dr. Ivan Oleynik, Dr. Hariharan Srikanth, and Dr. Brian Space for agreeing to be in my Ph.D. committee and for your comments and advices. Finally, I thank Dr Mohamed Elhamdadi for chairing my defense. I thank the staff of the Department: Ki mberly, Mary-Ann, and Daisy for all the help during my studies and defense. I would also like to show my gratitude to the Research Computing Core at the University of South Florida, the TeraGrid Advanced Support Program at the University of Illinois, and the High Performance Co mputing facilities of DoD for their computational resources and technical supports Thanks to Brian Smith for his help and computational support. I also thank to my labmates Kevin, Jakub, and Adrian for their friendship. I would like to acknowledge my friends Balaji, Denis, and Oksana for their help and friendship over these years. Finally, thanks to my family for al l your support, love, and encouragement throughout my Ph.D. studies.
i TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES vi ABSTRACT xii CHAPTER 1 INTRODUCTION 1 CHAPTER 2 STRUCTURE OF CARBON NANOTUBES 5 2.1 Nomenclature 5 2.2 Structural Parameters 8 CHAPTER 3 APPLICATIONS CARBON NANOTUBES UNDER EXTERNAL FACTORS 12 3.1 Carbon Nanotubes Under One External Factor 12 3.1.1 Adsorption Processes 13 3.1.2 Mechanical Deformation 18 3.1.3 Mechanical Defects 25 3.1.4 External Fields 30 3.2 Carbon Nanotubes Unde r Two External Factors 32 CHAPTER 4 VAN DER WAALS INTERACTIONS BETWEEN CARBON NANOTUBES 36 4.1 Theoretical Investigations of Van der Waals Interaction 37 4.2 The Coupled Dipole Method 39
ii 4.3 Studied Systems 41 4.4 High Symmetry Stacking of Graphitic Nanostructures 44 4.5 Static Polarizability a nd Characteristic Frequency 44 4.6 Many-Body Van der Waals Energy 45 4.7 Sliding Many-Body Van der Waals Interactions 47 4.8 Van der Waals Energy Contributions 49 4.8.1 Van der Waals Energies 52 4.8.2 Effect of Structural Size on Van der Waals Energies 54 4.8.3 Sliding Van der Waals Energies 56 CHAPTER 5 DENSITY FUNCTIONAL THEORY 59 5.1 The Hohenberg-Kohn Theorems 60 5.2 The Kohn-Sham Approach 61 5.3 Exchange-Correlation Functional Approximations 62 5.4 Pseudopotentials 63 5.5 Calculation Tool 64 CHAPTER 6 DNA BASES ADSORPTION ON CARBON NANOTUBES 65 6.1 Studied Systems 66 6.2 Calculation Parameters 69 6.3 Adsorption Configurations 69 6.3.1 Adenine and Thymine Configurations 70 6.3.2 Adenine and Thymine Radical Configurations 73 6.4 Structural Changes of DNA Bases and Their Radicals 75 6.5 Adsorption Energies 75
iii 6.6 Electronic Stru cture Calculations 79 6.7 Charge Density Distributions 81 CHAPTER 7 DEFECTIVE CARBON NANOTUBES UNDER RADIAL DEFORMATION 83 7.1 Studied Systems 83 7.2 Structural Changes 84 7.3 Characteristic Energies of Deformation and Defect Formation 87 7.4 Electronic and Magnetic Properties of Carbon Nanotubes 91 7.4.1 Band Gap Modulation 91 7.4.2 Electronic Band Structures of Radially Deformed Carbon Nanotubes Without/With Stone Wales Defect 93 7.4.3 Charge Distribution in Ra dially Deformed Carbon Nanotubes Without/With Stone Wales Defect 95 7.4.4 DOS of Radially Deformed Carbon Nanotubes With Nitrogen Impurity and Mono-Vacancy 97 7.4.5 Spin Density Distribution in Radially Deformed Carbon Nanotubes With Nitrogen Impurity and Mono-Vacancy 100 CHAPTER 8 RADIALLY DE FORMED CARBON NANOTUBES UNDER EXTERNAL ELECTRIC FILEDS 102 8.1 Studied Systems 102 8.2 Deformation Energies 103 8.3 Band Gap Modulations 105 8.4 Electronic Band Structure Changes 108
iv 8.5 Charge Density Destributions 110 CHAPTER 9 CONCLUSIONS 113 9.1 Many-Body Van der Waals Interaction Forces 113 9.2 Adsorption of DNA Bases and Their Radicals 114 9.3 Combined Effect of Radial Deformation and Mechanical Defects 115 9.4 Combined Effect of Radial Deform ation and External Electric Fields 117 REFERENCES 118 ABOUT THE AUTHOR End Page
v LIST OF TABLES Table 2.1 Structural parameters for di fferent types of car bon nanotubes such as armchair, zigzag, and chiral na notubes. Detailed explanations of N and ngcd parameters are given in the text. 11 Table 6.1 Adsorption energies and equilibrium distances for adenine and adenine-radical (A-radical) a nd (6,6) CNT, and thymine and thymine-radical (T-radical) on (8,0) CNT. The equilibrium distances are measured from the surface of nanotube to the center of the hexagonal ring of each molecule. 78
vi LIST OF FIGURES Figure 2.1 The graphene sheet with lattice vectors a1 and a2, which define the graphene unit cell with tw o atoms within. The chiral Ch = 8a1+4a2 and translational T = 4a1+5a2 vectors are shown for (8, 4) nanotube. The parallelepiped made by those vectors identifies the unit cell of (8, 4) nanotube. The sh aded area with rolling direction shows the graphene layer to form (8, 4) nanotube. The chiral angle with zigzag and armchair patterns are labeled. 6 Figure 2.2 Structures of different type s of carbon nanotubes with front and top views. Single wall carbon nanotube s a) armchair (6,6), zigzag (10,0), and chiral (8,4) nanotubes; d) (14,5)/(6,6) multiwall carbon nanotube. 7 Figure 3.1 Adsorptions process in th e carbon nanotubes. Chemisorptions through A) defects, B) sidewall a nd edges. Physisorptions through C) noncovalent exohedral interac tions with surfactants, D) noncovalent exohedral interactions with polymers, E) endohedral interactions, for example, C60, adapted from . 14 Figure 3.2 Van der Waals forces in a) th e clusters of fullerene molecules , where r and r represent the separation distance between shell fullerenes and the center of fullere ne and nearest neighbor shell-toshell respectively, and b) nanoe lectromechanical switch . 16 Figure 3.3 Electromechanical measuremen ts of a partly suspended nanotube over a trench. a) Device viewed from above. The substrate has a trench of 500 nm in width and 175 nm in depth. A pair of metal electrodes (S source and D drain) is bridged by nanotube suspended over the trench. b) An atom force microscopy image of suspended nanotube. c) Side view of experimental setup when the nanotube is pushed into a trench by atom force microscopy tip. d) Experimentally measured conductan ce of nanotube as a function of strain. All figures are from . 20
vii Figure 3.4 Cross section view of (8,0) nanotubes under radial deformation. a) Perfect nanotube. b) Nanotube unde r small radial deformation with elliptical shape of the cross se ction. c) Peanut-like deformed nanotube. d) Flat-like deformed nanotube. 22 Figure 3.5 Density Functional Theory (D FT) calculations of the band gap as a function of the cross sectiona l flattering for variety of semiconductor (left panel) and metallic (right panel) carbon nanotubes. The insets on both pane ls compare tight-binding (TB) and DFT results for (10,0) and (9,0) nanotubes, respectively from . 24 Figure 3.6 Side view of (8,0) nanotube structure a) without deformation, and with different defects at yy = 20% deformation for b) a StoneWales defect, c) nitrogen subs titutional impurity, and d) monovacancy. The atoms of the defective sites are labeled. 26 Figure 3.7 Band structure and density of states of defective nanotubes. a) Band structure evolutions of (7,0) and (9,0) nanotubes upon increasing the concentration of St one-Wales defect from . b) Density of states (top) of (10,0) nanotube, (middle) where one of carbon atom is substituted by a nitrogen atom, and (bottom) where a carbon atom is substituted by a boron atom in the supercell of 120 carbon atom from . Th e nitrogen and boron defect concentration is 0.83%. c) Density of states for (8,8) and (14,0) nanotubes without/with mono-vacancy defect in their structure (solid/dashed line respec tively) from . 28 Figure 3.8 Prospective view of (8,0) nanotube under transverse external electric field (top) and ba nd structure changes under E =0 V/ and E =0.1 V/ external electric field (bottom). b) Prospective view of nanotube in the presen ce of magnetic field along nanotube axis (top) and band structure changes under magnetic field (Picture from ). The magnetic and electri c fields lift of the degeneracy of the band structure. 31 Figure 3.9 Energy band gap evolution as a function of external electric field in three directions such as +x-axis, x-axis, and +y-axis for
viii vacancy defect in (10,0) nanot ube with 79 carbon atoms from . The insert shows the optimized structure of vacancy defect. 34 Figure 4.1 a) Two ( 6,0) nanotubes (CNT16-CNT7); b) a (6,0) nanoribbon and (6,0) nanotube (GNR16-CNT7); c) two (6,0) nanoribbons (GNR16GNR7); d) stacking symmetry of the hexagonal rings T-top, Bbridge, and H-hollow. The subscr ipt indexes 16 and 7 denote the number of translational unit cell s along the axial di rection in each structure. The surface-to-surface distance is D0 = 3.4 . 43 Figure 4.2 Total many-body VDW energy as a function of surface-to-surface separation distance D0 for two nanotubes (CNT16-CNT7), nanoribbon and nanotube (GNR16-CNT7), and two nanoribbons (GNR16-GNR7). The studied configurati ons and the direction of displacement are also shown. 46 Figure 4.3 Total many-body VDW energy as a function of displacement for: a) two nanotubes (CNT16-CNT7), b) a nanoribbon and a nanotube (GNR16-CNT7), and c) two nanoribbons (GNR16-GNR7) in three sliding geometries. The displacement corresponds to the relative separation between the centers of mass of the considered nanostructures along the relevant axis. Parallel refers to a common axial direction for the two nanostructures, and perpendicular referrers to the case when the nanostructure axes are perpendicular. The sliding ge ometries are depicted in the inserts. The surface-to-surface distance along y -axis is kept constant 3.4 . 48 Figure 4.4 VDW energies for UTN, UT2, UT2+ UT3, and UT2+ UT3+ UT4 interactions as a function of separation distance D0 for two (6,0) nanotubes (CNT5-CNT5), (6,0) nanoribbon and (6,0) nanotube (GNR5-CNT5), and two (6,0) nanoribbons (GNR5-GNR5) with 5 translational unit cells. The relative geometries are also shown. 53 Figure 4.5 VDW energies for UTN, UT2, UT2+UT3, and UT2+ UT3+ UT4 interactions between the CNTs and GNRs nanostructures: a) two (6,0) CNTs as a function of length; b) ( n ,0) CNTs with 4 unit cells along the z -axis as a function of diameter; c) two (6,0) GNRs as a function axial length; d) ( n ,0) GNRs with 4 un it cell along z -axis as a function of width. The surface-to-surface distance is D0 = 3.4 for all cases and chiral index n = 615. 56
ix Figure 4.6 VDW energies for UTN, UT2, and UT2+ UT3 energies as a function of center of mass displacement along the z -direction for a) two nanotubes (CNT16-CNT7), b) a nanoribbon and a nanotube (GNR16-CNT7), and c) two nanoribbons (GNR16-GNR7).The inserts show the sliding geometries with the surface-to-surface separation distance D0=3.4 . 58 Figure 6.1 Chemical structure of DNA with its purines (Adenine and Guanine) and pyrimidines (Thymine and Cytosine) bases. Hydrogen bonds between DNA bases are shown by dotted lines . 68 Figure 6.2 The symmetry adsorption sites of a hexagonal ring structure: a) top symmetry site, b) bridge symmetry site, and c) hollow symmetry site. Adsorption configurations of adenine on the (6,6) CNT surface: d) Top_60b configuration w ith front and top views (the most stable configuration), e) Top_60a, f) Top_90a, g) Top_90b, h) Top_a, i) Top_b, j) Bridge_a, k) Bridge_b, l) Bridge_30, m) Bridge_60, n) Bridge_90, o) Ho llow, p) Hollow_30, r) Hollow_60, s) Hollow_90. The numbers 30, 60, and 90 represent relative angles between the longitudinal axis of adenine (dashed line) and the z -direction of the nanotube. 71 Figure 6.3 Equilibrium configurations of thymine adsorbed on (8,0) CNT: a) Top_a configuration with front and top views (most stable configuration), b) Top_b, c) Hollo w, d) Bridge_a, e) Bridge_b. 72 Figure 6.4 Equilibrium configurations of a) A-radical on (6,6) CNT and b) Tradical on (8,0) CNT with front and top views respectively after relaxation. Each complex is in Top configuration. 74 Figure 6.5 DNA base structures before and after adsorption: a) bond distance for adenine and A-radical before and after adsorption; b) bond distance for thymine and T-radical before and after adsorption; c) angles for thymine and T-radical before and after adsorption; d) angles for adenine and A-radical before and after adsorption. The dashed circles show the hydrogen atoms removed to form the deprotonated radicals. 76
x Figure 6.6 Total density of states for a) pristine (8,0) CNT, b) Thymine/(8,0) CNT, c) T-radical/(8,0) CNT, spin-up, and spin-down, d) pristine (6,6) CNT, e) Adenine/ (6,6) CNT f) A-ra dical/(6,6) CNT, spin-up and spin-down. For the radicals spin-polarization effects were included in the calculations. 80 Figure 6.7 Isodensity surfaces of the char ge density for four valence states below Fermi level with isovalue 0.023 -3 for a) A-radical/(6,6) CNT and 0.022 -3 for b) T-radical/(8,0) CNT (created with VMD ). 82 Figure 7.1 (8,0) nanotube bond length evolution as a f unction of applied strain yy for a) no defect, b) with a Stone-Wales defect, c) with a nitrogen impurity, d) and e) with a mono-vacancy. 85 Figure 7.2 The energy of a) deformation and b) defect formation as a function of yy for Peanut and Flat deforma tions in (8,0) nanotube. Crosssectional views of the deformed and defective (8,0) nanotubes are also provided. 89 Figure 7.3 Energy band gap Eg of perfect and defective (8,0) nanotubes as a function of radial deformation yy. Insert shows the semiconductormetal transition in (8,0 ) nanotube in detail. 92 Figure 7.4 The energy band structure of a) defect-free (8,0) nanotube, b) (8,0) nanotubes with Stone-Wales defect at high curvature site for several degrees of yy. The red level is associated with the StoneWales defect. 94 Figure 7.5 Total charge density plots with isosurface value is 0.0185 e /3 for a) defect-free (8,0) nanotube with yy=25%, b) Peanut structure with yy=65%, c) Flat structure with yy=65%, and d) (8,0) nanotube with Stone-Wales defect with yy=25%, e) Peanut structure with yy=65%, f) Flat structure with yy=65%. 96 Figure 7.6 a) Total Density of States for spin up and down carriers of (8,0) nanotube with a substitutiona l nitrogen impurity on its high curvature at different yy for Peanut and Flat structures. b) The magnetic moment of the (8,0) nano tube with the nitrogen impurity
xi as a function of yy. c) Total Density of States for spin up and down carriers of (8,0) nanotube with mono-vacancy defect on its high curvature at different yy for Peanut and Flat structures. 98 Figure 7.7 Spin density isosurface plot s for a) (8,0) nanotube with nitrogen impurity for yy=10% at 0.004 B/ 3, b) (8,0) nanotube with nitrogen impurity for yy=20% at 0.004 B/ 3, c) Flat (8,0) nanotube with nitrogen impurity for yy=65% at 0.0122 B/ 3, d) Peanut (8,0) nanotube with mono-vacancy for yy=65% at 0.018 B/ 3. 101 Figure 8.1 The deformation energy as a function of radial deformation yy for one unit cell of (8,0) and (9,0) nanotubes. 104 Figure 8.2 Eg as a function of yy for a) (8,0) and b) (9,0) nanotubes; Eg as a function of E for c) (8,0) at yy=0%; d) (9,0) at yy=0%; e) (8,0) at yy=10%; f) (9,0) at yy=10%; g) (8,0) at yy=20%; h) (9,0) at yy=20%; i) (8,0) at yy=25%; j) (9,0) at yy=25%. The insert indicates electric field directions as follows: Ex is along x axis, Ey along y axis, and E45 0along 450 from x axis in xy plane. 106 Figure 8.3 Energy band structure evol utions for a) (8,0) nanotube at yy=10% and b) (9,0) nanotubes at yy=25% and electric field strength E =0.1, 0.3, 0.5 V/ along x y axis, and 450 from x axis in xy plane, respectively. The Fermi level is shown as a dashed line. 109 Figure 8.4 The total charge density di fference plots for (8,0), and (9,0) CNT at E =0.1 V/. a) Applied electric field along y axis for isosurface value 0.0022 e /3; and b) along x axis for isosurface value 0.008 e /3 for (8,0) CNT. c) Applied electric field along y axis; and d) along x axis for isosurface value 0.009 e /3 for (9,0) CNT. The electron accumulation/depl etion regions are displayed in blue ()/red (+). 111
xii PROPERTIES OF CARBON NANOTUB ES UNDER EXTERNAL FACTORS: ADSORPTION, MECHANICAL DEFO RMATIONS, DEFECTS, AND EXTERNAL ELECTRIC FIELDS YAROSLAV SHTOGUN ABSTRACT Carbon nanotubes have unique electronic, optical, m echanical, and transport properties which make them an important element of nanoscience and nanotechnology. However, successful application and in tegration of carbon nanotubes into new nanodevices requires fundamental understandi ng of their property changes under the influence of many external factors. This di ssertation presents qualitative and quantitative theoretical understand ing of property changes, while car bon nanotubes are exposed to the deformations, defects, external electric fi elds, and adsorption. Adsorption mechanisms due to Van der Waals dispersion forces are anal yzed first for the interactions of graphitic materials and biological molecules with carbon nanotubes. In particular, the calculations are performed for the carbon nanotubes and gr aphene nanoribbons, DNA bases, and their radicals on the surface of carbon nanotubes in terms of binding energies, structural changes, and electronic prope rties alterations. The results have shown the importance of many-body effects and discrete na ture of system, which are commonly neglected in many
xiii calculations for Van der Waals forces in the na notube interactions with other materials at nanoscale. Then, the effect of the simultaneous appli cation of two external factors, such as radial deformation and different defects (a Stone Wales, nitrogen impurity, and monovacancy) on properties of carbon nanotubes is studied. The results reveal significant changes in mechanical, electrical, and ma gnetic characteristic s of nanotubes. The complicated interplay between radial deforma tion and different kinds of defects leads to the appearance of magnetism in carbon nanotub es which does not exist in perfect ones. Moreover, the combined effect of radial deform ation and external electric fields on their electronic properties is shown for the first time. As a result, metal-semiconductor or semiconductor-metal transitions occur and are strongly correlated with the strength and direction of external elec tric field and the degree of radial deformations.
1 CHAPTER 1 INTRODUCTION Carbon1 nanotubes are quasi-one dimensi onal nanostructures with many unique properties [1,2]. Their diverse and exemplary properties have motivated a wide range of their potential applications in rectifiers, fiel d effect transistors, memory elements, field emission sources, reinforcement systems, conductive films, spintronics, and spectrometers [3,4]. Therefore, these materi als are considered as promising building blocks in nanoelectronics and nanodevice engi neering. However, potenti al applications of carbon nanotubes have strongly re lied on their extraordinary ability of property changes under external factors such as adsorption, inte ractions with nanostructures, mechanical deformations, mechanical defects, and applied external fields. Adsorption properties of nanotubes and possibility of nanotube interaction with other objects have drawn a significant interest in many other fields of science such as biology, chemistry, and medicine. Many studie s have shown successful application of nanotubes for the detection of biological and chemical substances, as a main element of bio-sensors and gas-sensors [5,6]. Carbon na notubes have also been considered as potential materials for hydrogen storage due to their large surface area and hollow structure . This can lead to the developmen t of more efficient batteries for cell phones, notebooks, and electric cars and to resolve th e problem of energy storage. In medicine, carbon nanotubes are wildly inve stigated in the area of drug delivery system and cancer
2 therapy, since they demonstrate high drug lo ading capacities, excelle nt cell penetrations qualities, and selective cell targeting abilities [8,9]. All those promising applications of nanotubes are governed by complex adsorption mechanisms in the carbon nanotube s, which are generally classified as chemisorptions and physisorptions. The chem isorption is characterized by forming the covalent bonding between nanotube and the adsorption substance, while the physisorption is described by weak van der Waals forces between a nanotube and other substance such as atoms, molecules, clusters or even other nanotube s. It has been shown that these van der Waals forces are signifi cant not only for physisor ptions mechanisms, but also for interaction and structural stab ility of many systems such as nanoclusters carbon nanotube itself, nanoribbons fullerene crystals, boron nitride nanotubes, proteins, and DNA molecules [10-15]. They also becam e a crucial component in nanotechnology for the proper functionality and design of new nanodevices . Therefore, it is required to extend our knowledge of Van der Waals fo rces in order to explain the nanoscale phenomena. Furthermore, the adsorption process can be facilitated by applied external factors such as mechanical deformations, mechanical defects, and external fields [17-20]. For example, mechanical deformations of carbon nanotubes change the binding energy of adsorbed molecules . Therefore, it can l ead to binding some molecules to nanotubes that are usually not bounded to graphite su rface . When defect s, such as a Stone Wales, substitutional impurities, and vacancies are introduced in the nanotube structure, they create highly reactive sites for the adso rption of other molecules. Applied external electric field can be considered as a driv ing force for encapsulation of DNA molecules
3 into carbon nanotubes  for drug delive ry systems. Moreover, researchers have realized that these external factors can be used in new me thods to control and/or tune carbon nanotube properties for new nanodevices. However, the development and design of current and new carbon nanotube based nanodevices has led to the search of additional ways to modify and tailor their properties. To achieve this flexibility, re searchers have begun to study the influence of more than one external factor simultane ously. This is a relatively new area which opens up more possibilities for new devices. Thus, the knowle dge of physical propert y changes in carbon nanotubes under several extern al factors is crucial fo r fundamental science and technological prospects. The goal of this work is to provide a framework for better understanding of adsorption mechanisms in carbon nanotubes, and how their elec tronic and magnetic properties change under several external f actors applied simultaneously. We consider theoretical studies of physisor ption processes, which are de scribed by dispersion Van der Waals forces, in the adsorption of the fin ite length carbon nanot ubes and nanoribbons on the surface of other carbon nanotubes. Our results show, for the first time to our knowledge, the importance of the many-body Van der Waals interactions in carbon nanostructures with a discrete nature of the nanostructures taken into account, which are usually neglected in many studi es. In addition, we evalua te how these Van der Waals interactions change as a function of the ge ometry, dielectric response, and size of the system, and how these results agree with the data from the other methods, which are now commonly used for van der Waals force calculat ions. Moreover, we have investigated the adsorption processes of DNA bases, such as adenine and thymine, and their radicals on
4 the metallic and semiconducting carbon nanot ubes in terms of st ructural changes, equilibrium distances and energies, and electr onic property changes. Such Van der Waals forces and DNA bases interactions are of pa rticular interest in many fields of the nanotechnological, biologi cal, and medical applica tion of carbon nanotubes. Furthermore, we present extensive studi es in the new area of modulating carbon nanotube properties under combined effect of two external factors. In particular, we consider the simultaneous application of a ra dial deformation and mechanical defects, such as a Stone Wales defect, nitrogen impur ity, and mono-vacancy, along with the radial deformation and external electric field. We perform the analysis of structural changes in defective carbon nanotubes under two types of radial deformations. One is the deformation achieved by squeezing of a na notube between two ha rd narrow surfaces, which have the width smaller than the nanotub e cross section. The other one is achieved by squeezing of the nanotube between two hard surfaces with the width larger than the nanotube cross section. As a result, we show the characteristic energies of deformation and defect-formation. Further, we show first principle calcu lations of band gap modulations, band structure evolutions, and ch anges in magnetic properties of carbon nanotubes in terms of the types of defects an d degrees of radial deformation. A special consideration is given to the mechanically induced magnetic properties. Finally, we present the combined effect of radial defo rmation and external electric field on the nanotube electronic structure and band gap mo dulations. Such combined effect of two external factors on mechanical, electronic, and magnetic properties of nanotubes provides a test ground for greater capab ilities in the applications of nanotubes in new devices.
5 CHAPTER 2 STRUCTURE OF CARBON NANOTUBES 2.1 Nomenclature Carbon nanotubes (CNTs) can be viewed as infinite cylindrically rolled graphene sheets as shown in Figure 2.1. Rolling this graphene sheet into a hollow cylinder forms single walled carbon nanotubes (SWNT) (Fi gure 2.2a-c). However, if a nanotube is comprised of several concentrically rolled grap hene sheets, such nanotube is refered to as multiwall carbon nanotube (MWNT) (Figur e 2.2d). Single wall carbon nanotubes are studied in theory and experiment more th an multiwall carbon nanotubes because of their well defined structure and electronic prop erties since the structure and electronic properties of multiwall carbon nanotubes are more complex and less defined due to numerous possibilities of layering and arrangeme nts of different concentric shells. Single wall carbon nanotubes can be synthesized by several methods such as laser ablation, high-pressure CO conversion, and arc-di scharge method [21-23]. These techniques produce SWNT with average diameter of 10 -15 . During the growth process carbon nanotubes usually aggregate into hexagonal-pa cked bundles because of the dispersion Van der Waals (VDW) interaction. The separa tion distance between two nanotubes in the bundles or multiwall carbon nanotubes is close to interlayer distance of graphite (3.42 ). Such assembling of nanotubes into bundles cr eates a problem of their separation since nanotubes are unsolvable in many solvents.
6 Figure 2.1 The graphene sheet with lattice vectors a1 and a2, which define the graphene unit cell with two atoms within. The chiral Ch = 8a1+4a2 and translational T = 4a1+5a2 vectors are shown for (8, 4) nanotube. Th e parallelepiped ma de by those vectors identifies the unit cell of (8, 4) nanotube. The shaded area with rolling direction shows the graphene layer to form ( 8, 4) nanotube. The chiral angle with zigzag and armchair patterns are labeled.
7 Figure 2.2 Structures of differe nt types of carbon nanotubes with front and top views. Single wall carbon nanotubes a) armchair (6,6), zigzag (10,0), and chiral (8,4) nanotubes; d) (14,5)/(6,6) multiwall carbon nanotube.
8 2.2 Structural Parameters The structure of carbon nanotubes is defined in terms of graphene lattice vectors because nanotubes are viewed as a derivativ e from graphene sheet. Figure 2.1 shows the infinite graphene shee t with lattice vectors a1 and a2 which define the unit cell of graphene and their lengths are | a1| = | a2| = a0 = 2.46 respectively. The unit cell consists of two carbon atoms at positions ( a1 + a2)/3 and 2( a1 + a2)/3. Specific direction of graphene rolling into nanotube is characterized by a chiral vector Ch = na1+ma2, which becomes the circumference of nanotube, and integer numbers ( n m ) is the chiral index of nanotube (Figure 2.1). Ch vector uniquely defines a particular nanotube and their electronic, optical, and mechan ical properties. Mo reover, even if nanotubes have the same length or/and diameter but different Ch vectors, their proper ties are very different. For example, (9,0) nanotube is metallic w ith 36 carbon atoms in the unit cell while the next closest (9,1) is a semiconductor and consists of 1820 carbon at oms per unit cell. The chiral vector Ch = 8a1+4a2 of (8,4) nanotube is shown at Figure 2.1. The direction of this vector is determined by the chiral angle which is the angle between a1 and Ch vectors and it is equal to 2 2 1 12 / | | | | ) cos( m nm n m n C a C ah h (2.1) In the nomenclature of carbon nanotubes, it is common to consider the chiral indexes ( n m ) of nanotubes such as n m 0, which is a counterpart of 00 300, because of the six-fold rotational symmetry of graphene which makes the ot her indexes equivalent to chiral indexes ( n m 0). Among all chiral nanotubes, there are two achiral types zigzag and armchair nanotubes. The names of zigzag and armchair come from the
9 circumference pattern when a nanotube is form ed (Figure 2.1). Each type has a specific chiral indexes and an angle. For instance, zigzag nanotubes have ( n 0) chiral indexes and = 00, while armchair nanotubes have ( n n ) chiral indexes and = 300. Figure 2.2a-c shows an example of (6,6) armchair, ( 10,0) zigzag, and (8,4) chiral nanotubes. In addition, structural parameters of nanotubes such as diameter, translation vector, number of atoms in a unit cell, and the shape and si ze of Brillouin zone are determined in terms of lattice vectors of graphene and their chiral vector [1,2]. The diameter of nanotubes is derive d from the formula of the circle circumference, which is the length of chiral vector Ch for nanotube: N a m nm n a C dh 0 2 2 0| | (2.2) where N= n2 + nm + m2. Any vector perpendicular to Ch defines the dire ction of carbon nanotubes axis. However, the smallest T vector determines the translation period T of nanotube. For instance, the translation vector of (8,4) nanotube is T = 4a1+5a2 as shown in Figure 2.1. Moreover, the translation vect or and period of carbon nanotubes are also defined by chiral indexes ( n m ) as has been shown for the diameter of nanotubes: 2 gcd 1 gcd2 2 a R n m n a R n n m T (2.3) T = 0 gcd 0 gcd 2 23 ) ( 3 | | a R n N a R n m nm n T (2.4) where ngcd is the greatest common divisor of ( n m ), and R = 1 if ( n m )/3 n is odd and R = 3 otherwise. Therefore, the unit cell of a nanotube is a cylinder with diameter d (| Ch|) and height T (Figure 2.1). The formulas for diameter and a translation vector (Eqs. 2.2 and 2.3) can be simplified for zigzag and armchair nanotubes as follows:
10 3 | | 3 | |0 0 0 0n a d a T n a d a TA A Z Z (2.5) respectively. The unit cell of a nanotube also determines the number of atoms within it. To calculate the number of atoms per unit cell NC, one can take the ratio between the surface area of a nanotube Th TC S and the area of gr aphene unit cell 2 / 32 0 a Sg with consideration of the fact that there ar e two atoms in graphene unit cell, which leads to the following formula: R n m nm n NC gcd 2 2) ( 4 (2.6) The structural parameters of different type s of carbon nanotubes are summarized in Table 2.1. The structure of carbon nanotubes is comple tely characterized by chiral indexes ( n m ). Besides the structural parameters, these chiral indexes also define other properties such as mechanical, electronic, and opti cal. For instance, 2/3 of all nanotubes are semiconductors while only 1/3 of nanotubes are metallic. Moreover, all armchair nanotubes are metallic. However, zigzag nanotubes can also be metallic or semiconductor. The general rule for nanotubes to be metallic is that their chiral indexes satisfy the following condition: ( n + m )/3 is integer number, other nanotubes are semiconductors. Our interest and the purpose of this work lies in the study of carbon nanotube electronic properties. We will further discuss how this properties change under external factors in more detail. These re sults are important for modulating nanotube properties in various ways to achieve different nanotubes functional devices.
11 Table 2.1 Structural parameters for different types of carbon nanotube s such as armchair, zigzag, and chiral nanotubes. Detailed explanations of N and ngcd parameters are given in the text. Chiral index NC Diameter d Translation period T Chiral angle Armchair ( n n ) 4 n / 30n a 0a 300 Zigzag ( n 0 ) 4 n /0n a 30a 00 Chiral ( n m ) 4 N/ ( ngcdR ) /0N a ) /( 3gcd 0R n N a ] ) 2 / arccos[( N m n
12 CHAPTER 3 APPLICATIONS CARBON NANOTUBE UNDER EXTERNAL FACTORS This chapter provides background info rmation regarding changes in carbon nanotube properties under severa l external factors. For the beginning, we discuss studies of one external factor such as an adsorp tion, mechanical deformations, mechanical defects, applied external electric fields or applied external magnetic fields altering nanotube properties. Then, the results of two external fact ors simultaneously applied for the modification of the nanotube properties are presented. 3.1 Carbon Nanotubes Und er One External Factor In many earlier theoretical calculations, carbon nanotubes were considered to have a perfect structure, wher e ideal graphite plane was rolle d up into a cylindrical tube. Depending on a particular way of graphene rolling, nanot ubes can be metallic or semiconducting. It has been revealed that these nanostructures have many interesting mechanical, optical, transport, and electroni c properties strongly depended on chirality, which immediately drew intere st to their application in areas of nanotechnology, biology, chemistry, and medicine. However, experimental measurements have shown that the nanotube structure is not perf ect [24-26]. External factors, such as an adsorption, mechanical deformations, mechanical defect s, or applied external fields can induce
13 various changes in nanotube properties. On th e other hand, it has been realized that such external factors can be applie d to achieve the desirable func tionality of carbon nanotubes in the device design. 3.1.1 Adsorption Processes The great challenge for scientists that appeared after discovering carbon nanotubes  has been to find the effici ent method of carbon nanotube separation. As the result of synthesis, carbon nanotubes aggr egate into the bundles by disperse Van der Waals (VDW) forces which makes it difficu lt of studying carbon nanotube properties. The bundle formation of nanotubes is governed mainly by adsorption mechanisms which are a consequence of surface energy minimizatio n. In general, the adsorption nature has two distinctive forms of physisorptions and chemisorptions. The physisorption is commonly described by weak Van der Waals force, while the chemisorption is characterized by the formation of covalent bonding. These both type s of the adsorptions have been considered in the search of new separation met hods of nanotubes . It has been shown that the adsorptio n in/on nanotubes can occur th rough the defect-group, inner or outer sidewall, and with or without the formation of covalent biding. For example, Figure 3.1 shows chemisorption with a c ovalent adsorption and physisorption with noncovalent adsorption in the nanotubes . Many studies have shown the importa nce of physisorption processes for application in the carbon nanotube based devices In particular, a signi ficant role of Van der Waals forces has been emphasized, which describe these processes. The VDW forces are well known in many physical, chemical and biological phenomena. They are
14 Figure 3.1 Adsorptions proce ss in the carbon na notubes. Chemisorptions through A) defects, B) sidewall and e dges. Physisorptions through C) noncovalent exohedral interactions with surf actants, D) noncovalent exohedral interactions with polymers, E) endohedral interactions, for example, C60, adapted from .
15 originated from the quantum mechanical fl uctuation of electroma gnetic field of each constituent in the entire system. Such cumula tive contributions from each entity of the structure are also known as many-body interact ions. Usually these many-body effects are neglected, and whole structure is consider ed as continuous body. However, as the dimensional size of the new systems decrea ses to the nanoscale, such many-body effects become a dominant factor in the interactions and in the stability of the nanostructures, nanoclusters, biological mol ecules, and nanocolloids . The significance of Van der Waals forces has also been shown in ma ny theoretical and expe rimental studies of carbon structures such as graphite, fullere ne, carbon nanotubes, and carbon nanoribbons [12,13,33]. For instance, the cr ystalline structure of fullere ne molecules is governed by Van der Waals interactions, as shown in Fi gure 3.2a . Carbon nanotube aggregation, structure, and solubili ty strongly depend on those interac tions. In addition, Van der Waals interactions also affect the dispersion of nanoclusters and nanocolloids in the suspension . Moreover, efficient func tionality of microelectromech anical systems (MEMS) and nanoelectromechanical systems (NEMS) also depends on such interactions [35-37]. Figure 3.2b shows operational nanoelectromec hnical switch based on carbon nanotubes where its functionality is a ffected by Van der Waals for ces . The study of the interaction forces in nanoclusters is consider ed to be a new field of research with many opportunities. Thus, the study of Van der Waals fo rces is important for the application of nanoclusters, nanotubes, and other nanom aterials in nanoelectronics and new nanodevices.
16 Figure 3.2 Van der Waals forces in a) the cl usters of fullerene molecules , where r and r represent the separation distance betw een shell fullerenes and the center of fullerene and nearest neighbor shell-to-shell respectively, and b) nanoelectromechanical switch .
17 Furthermore, the development and improve ment of adsorption processes for the nanotubes immediately trigger a significant intere st from other fields of science such as chemistry, biology, and medicine. The exceptio nal mechanical and el ectronic properties and large surface area of carbon nanotubes ma ke them an excellent candidate for chemical and biological sensors [5, 38]. In a ddition, the adsorption properties of carbon nanotubes with other organic molecules such as polymers, proteins, amino acids, and DNA raise great hope for medical applications [8,39,40]. Some realization of carbon nanotubes in biomedical applications has been achieved in orthopedics and neuroscience [41-43]. However, further prog ress in that direction requir es better understanding of the adsorption process between carbon nanot ubes and biological complexes. In spite of promising results, carbon nanot ubes have shown a toxicity effect on living cells which should be eliminated for th eir further application in biosystems . The toxicity effect is the result of the pres ence of remaining metal catalysis and unsolved materials. One way it can be avoided is by functionalization methods On the other hand, applications of carbon nanotubes with other mo lecule complexes are considered for drug, antigens, and gene delivery systems . Theref ore, much more research has to be carried out in this direction. In addition, a significant attention is given to DNA/nanotube complexes because the interaction of DNA with nanotubes is viewed as a separation method for nanotubes . The successful deployment of this me thod has been shown on sorting nanotubes by their diameter and chirality [46,47]. More over, DNA/nanotube complexes have been shown to be useful for DNA conforma tion transformation, DNA hybridization, electrochemical detection of DNA, and DNA sensors [48-51]. Several studies have
18 shown that DNA usually wraps around carbon na notubes, but it can also be encapsulated inside carbon nanotubes due to Van der Waals forces [52-54]. These successes demonstrate the usefulness and promis e of DNA-carbon nanotube complexes for biological and medical usage. However, the deeper studies are n eeded for fundamental understanding of the nature and importa nt factors of the DNA-carbon nanotube interactions. 3.1.2 Mechanical Deformation Carbon nanotubes have shown remarkable elastic-mechanical properties. Their high elastic modulus, strength, flexibility, and low mass dens ity make them an excellent candidate for polymer composite materials an d nanodevices [55-57]. Th ey are rigid in the axial direction, while they are flexible in th e radial direction. However, their mechanical properties are strongly affected by the environment. In act ual experiments, nanotubes undergo structural modificati ons such as bending, twisti ng, stretching, and radial deformation [58-60]. There have been many theoretical and experimental studies to evaluate these mechanical prope rties for practical use, especially when it had been shown that carbon nanotubes can have the large Youngs modulus ranging between 0.1 and 2.0 TPa . As the result of those observations, it has been suggested that the most common mechanical deformation is radial squeezing. Such deformation takes place in bundles of nanotubes, under applied hydrostatic pressure, in interaction of nanotubes with surfaces, nano-indentation experiments, and atom force microscopy (AFM) squeezing of nanotubes [62-64]. Moreover, carbon nanotubes are considered to be outstanding materials for AFM tip . Because of sma ll diameter of carbon nanotubes and high
19 elasticity, it significantly increases the image resolution and it is successfully applied to biological studies [66, 67]. In addition, carbon nanotubes are manipulated by an AFM tip in many experiments to achieve their desirabl e localization. This ha s a major effect not only on the mechanical, but also on the electr onic and transport prope rties of nanotubes. For instance, pressing suspended nanotube over trench by an AFM tip causes a local deformation in the nanotube structure (Figure 3.3). Subseq uently, the deformed region affects the electron flow between the source and drain electrodes, and as a result, the conductance of carbon nanotubes decreases several times as compared to the theoretically predicted one [68, 69]. Thus, carbon nanotube properties can be controlled, but such process is complex and requi res better understanding.
20 Figure 3.3 Electromechanical measurements of a partly suspended nanotube over a trench. a) Device viewed from above. The subs trate has a trench of 500 nm in width and 175 nm in depth. A pair of metal electrodes (S source and D drain) is bridged by nanotube suspended over the trench. b) An atom force microscopy image of suspended nanotube. c) Side view of experimental set up when the nanotube is pushed into a trench by atom force microscopy tip. d) Experimenta lly measured conductance of nanotube as a function of strain. All figures are from .
21 During AFM manipulations or device engineering, carbo n nanotubes undergo structural changes in their cross section. These change s of carbon nanotube geometry under radial deformation are char acterized by two applied strains xx and yy along x and y axes, respectively, and they are defined as follows: 0 0D a D xx (1.1) and 0 0D b D yy (1.2) where D0 is the diameter of the undeformed nanotube, a and b are the major and minor axes of the ellipse (Figure 3.4a,b). In some literature, it can be found that strains xx and yy areas are defined as (dimensionless parameter) with respect to direction of deformation. However, for discussion purpos es and convenience of understanding here, we use percentage (%) valu es of the applied strains xx and yy. Depending on the degree of radial deformation, the radial cross sec tion of the nanotube can take a different form. When the applied strain is small, the nanotube has an elliptical-like shape. However, if the applied strain continues to increase, th en nanotube cross section undergoes Peanutlike or Flat-like forms. The Peanut-like fo rm corresponds to squeezing of the nanotube between two hard surfac es with a cross section smaller than its radial one, while the Flatlike form corresponds to radial squeezing betw een two hard surfaces with a cross section larger than its radial one (Fi gure 3.4c,d). The Peanut and Flat deformations can also occur during experimental measurements when the size of AFM tip is smaller or larger than the cross section of the nanotube.
22 Figure 3.4 Cross section view of (8,0) nanot ubes under radial deformation. a) Perfect nanotube. b) Nanotube under small radial deformation with elliptical shape of the cross section. c) Peanut-like deformed nanotube. d) Flat-like deformed nanotube.
23 Structural changes of carbon nanotubes unde r radial deformati on affect not only mechanical, but also their electronic propert ies of carbon nanotubes. It has been shown that the electronic properties are sensitive to the degree of radial deformation. Moreover, the understanding of those propert ies under radial deformation is crucial element in the functional operation of NEMS and MEMS de vices. Nevertheless, the predictions from first principle calculations have shown th at the band gap of zigzag semiconductor nanotubes decreases with the increase of radi al deformation, while for zigzag metallic nanotube it increases and then decreases (Fi gure 3.5). At the same time, the armchair metallic nanotubes can show two responds unde r radial deformation. If the mirror symmetry of armchair carbon nanotubes is not broken, they have similar dependence as the one for zigzag metallic nanotube. Otherwis e, they remain metallic regardless of the degree of the radial deformati on [71, 72]. Therefore, the appl ication of radial deformation causes chirality dependent semiconductor-metal and metal-semiconductor transitions in carbon nanotubes. There are several factors re sponsible for these transitions, such as hybridization of the atomic or bitals located on the high cu rvature regions, interaction between low curvature regions, breaking mi rror symmetry of carbon nanotubes, and geometry deformations [73-77]. Thus, one can reach a great diversity in the properties of carbon nanotubes just by changing the degree of the radial deformation. In addition, high elastic properties of carbon nanotubes provide the opport unity to restore original properties after radial deformation vanishes.
24 Figure 3.5 Density Functional Theory (DFT) cal culations of the band gap as a function of the cross sectional flattering fo r variety of semiconductor (lef t panel) and metallic (right panel) carbon nanotubes. The insets on both panels compare tight-binding (TB) and DFT results for (10,0) and (9,0) na notubes, respectively from .
25 3.1.3 Mechanical Defects The presence of mechanical defects in carbon nanotube structure drastically affects all their properties such as trans port, mechanical, magnetic, electronic, and optical. These defects occur during growth, purif ication, alignment, or device application processes. They can also be introduced intentionally to modify carbon nanotube properties. Common defects ar e Stone Wales defects (Fi gure 3.6b), substitutional impurities (Figure 3.6c), and vacancies (Fi gure 3.7d). A Stone Wales defect takes place in a carbon network when one C-C bond is rotated by 900 degree . Such rotation creates two pairs of a pentagon and a hepta gon around the rotated C-C bond. It is also known as a 5-7-7-5 defect . These pairs of pentagon and heptagon create a local disturbance which serves as a reactive center for adsorption of various atoms, nanoparticles, and molecules [80-82]. Incr easing the concentra tion of Stone Wales defects leads to semiconductor-metal transitions in zigzag nanotubes. It also increases the density of states ar ound the Fermi level ( EF) for armchair nanotubes due to new defect states, thus, making the tubes more metallic  (Figure 3.7a). In addition, it has been shown that the Stone Wales defects can cau se phonon scattering in the wide frequency region which leads to significant reducti on in the thermal conductivity of carbon nanotubes and graphite .
26 Figure 3.6 Side view of (8,0) nanotube stru cture a) without deformation, and with different defects at yy = 20% deformation for b) a St one Wales defect, c) nitrogen substitutional impurity, and d) mono-vacancy The atoms of the defective sites are labeled.
27 Another well-known defect in carbon nanotubes is a substitutional impurity, which happens when carbon atoms are substitu ted by the foreign ones (Figure 3.6c). It has been considered that the common dopi ng atoms in carbon nanotubes are boron and nitrogen, which can appear during laserablation and arc-disc harge synthesizing processes or during substitutional reaction met hods [85-87]. Since their atomic radii have similar size to the carbon atom, they create a small perturbation in the nanotube structure in comparison to the perfect one. However, ot her atoms such as Li, K, Br, and Ni, which can create a larger disturbance, are also considered for the new nanodevice application of nanotubes [88-90]. The boron and nitrogen serve as acceptors and donors of electrons in the nanotubes, respectively, since boron ha s one electron less, and nitrogen has one electron more than the carbon at om. Theoretical calculations have revealed that such impurities lead to the shift of the Fermi level, additional impurity states in the band structure, and a break of the nanotube mirror symmetry [91-93]. Figure 3.7b shows evidence of new impurity levels in the density of states (DOS) of (10,0) nanotube around the Fermi level under nitrog en and boron doping. The impurity effect on electronic properties becomes more evid ent if their concentration increases. Nitrogen doping has drawn substantial interest due to the one extra electr on provided by a nitrogen atom, which causes magnetic effects in nanotubes, transforms semiconductor nanotube into metallic one, and serves as a reactive center in the interaction of nanotubes with other molecules . Thus, the doping of nanotubes by boron and nitrogen leads to the changes in electronic structure of nanotubes because of the presence of the additional acceptor and donor states.
28 Figure 3.7 Band structure and density of states of defective nanotubes. a) Band structure evolutions of (7,0) and (9,0) nanotubes upon increasing the concentr ation of Stone Wales defect from . b) Density of states (top) of (10,0) na notube, (middle) where one of carbon atom is substituted by a nitrogen atom, and (bottom) where a carbon atom is substituted by a boron atom in the superce ll of 120 carbon atom from . The nitrogen and boron defect concentration is 0.83%. c) Density of st ates for (8,8) and (14,0) nanotubes without/with mono-vacancy defect in their structure (solid/dashed line respectively) from .
29 Furthermore, removing a carbon atom leaves a vacancy defect with three dangling carbon bonds in the nanotube ne twork . These three da ngling bonds are unstable and undergo recombination to make a chemi cal bond between two of them forming a pentagon ring and one remaining dangling bond in a nonagon ring (Figure 3.6d) . The existence of vacancies has been demonstr ated experimentally . It occurs during synthesis of nanotubes or under intentional i on or electron irradiat ion techniques [97-98]. The formation of vacancy and its orientati on depends on the radius and chirality of nanotubes. Such a disturbance causes significan t changes in their transport, magnetic, mechanical, and optical properties [99-101]. Fi gure 3.7c shows the density of state of (8,8) and (14,0) carbon nanotubes without a vaca ncy (solid line) and with vacancy defect (dashed line) . The vacancy defect gives the sharp peak in the density of states above the Fermi level, which consists of localized states from the C dangling bond. Such imperfections can reduce the mean-free path a nd mobility of the free carrier in nanotubes. This leads to the decrease in the nanotube conductivity. On the other hand, the dangling bond can participate in the intera ction with other molecules as well as it can be the center of functionalization of nanotubes. Theoretica l studies have shown that carbon nanotubes with vacancies demonstrate ferromagneti c ordering due to the dangling bond of the nonagon ring . Moreover, the magnetic mome nt depends on the defect concentration. Thus, it opens up opportunities for the applic ation of carbon nanotube s in spintronics, logical, and memory storage devices.
30 3.1.4 External Fields The successful application of carbon na notubes in nanodevices, such as fieldeffect-transistors , rectif iers , p-n junctions , or sensors , requires the knowledge of changes in electronic proper ties of nanotubes under external electric or magnetic fields. However, it is also important to realize that car bon nanotube properties can be tuned by such fields to reach a desira ble functionality. The cy lindrical structure of nanotubes suggests two high symmetry directions for application of those external fields: parallel and perpendicular to the nanotube ax is (Figure 3.8). Theore tical and experimental studies have shown that external electric a nd magnetic fields have a drastic effect on electronic and magnetic propert ies of carbon nanotubes [107-109] The general pattern of external electric and magnetic fields is an electronic band stru cture modulation with semiconductor-metal or metal-semiconductor tran sition, as well as lifting the degeneracy of sub-bands (Figure 3.8). As Figure 3.8 shows, the energy band ga p decreases, and the degeneracy of the energy bands around EF disappears with the increase of the strength of applied fields. Application of the external elec tric field parallel to the nanotube axis has attracted a lot of attention because of the excellent field emission properties of carbon nanotubes . Such properties have effectively been used in prototype devices such as flat panel display , x-ray tube , and scanning x-ray source . Neverthele ss, the search for new devices is ongoing. Current ly, researchers are exploring the perpendicular direction of field applications as ne w opportunities of controllin g electronic properties of nanotubes.
31 Figure 3.8 Prospective view of (8,0) nanotube under transverse external electric field (top) and band structure changes under E=0 V/ and E=0.1 V/ external electric field (bottom). b) Prospective view of nanotube in the presence of magnetic field along nanotube axis (top) and band structure changes under magnetic field (Picture from ). The magnetic and electric fields lift of the degeneracy of the band structure.
32 The application of external magnetic field leads to metal-insulator transition in nanotubes. Moreover, the carbon nanotubes ca n have diamagnetic or paramagnetic characteristics which depend on their chiral ity and the orientation of magnetic field . When the magnetic field is applied along the na notube axis (Figure 3. 8b), oscillations in the magnetoresistance or in the band ga p can be observed due to the Aharonov-Bohm effect, which is a periodic behavior of the wave-function phase factor in the cylindrical geometry [115-116]. The Zeeman effect has been found to cause metallization of carbon nanotubes at certain values of the magnetic flux . Howeve r, perpendicularly applied magnetic field has a different e ffect on the electronic structur e of nanotubes. It opens the band gap at a weak strength and closes that ba nd gap at a high strength of magnetic field for zigzag nanotube. When the magnetic length eH c l / is smaller than the nanotube circumference, the formation of Landau level at high magnetic fields closes the band gap, while those levels do not appear at small magnetic fields . The application of external electric or magnetic fields has also been used in experimental alignment of carbon nanotube samples [119-120]. 3.2 Carbon Nanotubes Under Two External Factors Many devices associated with differen t applications are based on the unique properties of carbon nanotubes and the opportunity of controlling them. The search for new methods to modulate nanotube properties ha s led to the introduction of deformation, mechanical defects, and external fields. Reaching the desirable properties of nanotubes under one external factor, however, can be di fficult in device engi neering. To overcome such difficulties, researchers now consider how more than one extern al factor can alter
33 the carbon nanotube properties. Thus, it is important to build f undamental understanding of the combination of more th an one external stimulus by starting with applying two stimuli simultaneously. Studies have shown that the combination of radial deformation and impurity or vacancy, for example, can facilitate tuni ng electronic properties of nanotubes [121-122]. As we have shown above, the radial defo rmation causes the closing of band gap in semiconductor nanotubes, but how fast this band gap is closed can be controlled by various defects, such as impurities and vacan cies . Radial deformation produces two regions with high and low curvatures (Figur e 3.4b). As a result, the atoms on the high curvature tend to have sp3 configuration, and they are more strained than the ones on the low curvature with sp2 configuration. It ma kes it easier to create the defect on the high curvature in comparison to the low one. The fl exibility in localizati on of the defect in radially deformed nanotubes gives more control over nanotube properties. Band gap modulation can also be achieved by combining external electric fields and defects in nanotubes. This is of particular interest in a pplication of nanotubes in fieldeffect transistors since the current can be tu rned off or on by applying external electric field. Such combination provides an opportunity to identify the presence of defects in carbon nanotubes. Rotating nanotube in exte rnal electric fiel d provides different responses in different directions . Figure 3.9 shows the presence of a vacancy defect in (10,0) nanotubes with an applied external electric field for various directions. We can see that the value of the band gap can be controlled by the presence of a vacancy, strength, and direction of the electric field. It is inte resting that semiconducting tubes are usually used in nanoelectronics. Metallic na notubes have been discarded because their
34 Figure 3.9 Energy band gap evol ution as a function of external electric field in three directions such as +x-axis, x-axis, and +y-axis for vacancy defect in (10,0) nanotube with 79 carbon atoms from . The insert shows the optimized st ructure of vacancy defect.
35 resistance is not sensitive to gate voltage or transverse electric field . New views on metallic nanotube application are considered due to the combined effect of external electric field and impurities, which allows tuning the resistance by two orders of magnitude. The origin of this change come s from the backscattering of the conduction electron by impurities, and it strongly depends on the strength and orientation of the applied electric field . Nowadays, most magnetic materials are based on d and f elements. However, as we mentioned above, sp2 and sp3 hybridization due to mechan ical defects (vacancy and impurity) can also induce magnetism in carbon na notubes. This is of great importance for nanotube applications in magnetic and spin tronic devices. Magnetism can be further influenced by the combined effect of mech anical deformation and defects. The strong magneto-mechanical coupling has been shown in defective nanotubes under mechanical deformation . It can increase or decrea se the magnetic moment and cause changes in the spin transport. To further investigate this, we will explore energetic, structural, magneto-mechanical, and electronic changes in carbon nanotubes under combined effect of radial deformation and different defects and under co mbined effect of radial deformation and external electric field.
36 CHAPTER 4 VAN DER WAALS INTERACTIONS BE TWEEN CARBON NANOTUBES The Van der Waals (VDW) forces originate from electromagnetic fluctuations of the electrons in the interacting objects, and are of quantum mechan ical and dispersive nature. They are found to depend strongly on th e size, geometry, dielectric response, and material discreteness of an involved syst em. The importance of the VDW forces in adsorption processes and in nanostructure d devices has led to many theoretical investigations using a wide array of methods. These theo retical studies have been performed on a diversity of materials and ge ometries that closely resemble current nanodevices and their components. For exam ple, the efficient functionality of microelectromechanical and nanoelectromechan ical systems is directly related to VDW interactions [35-37]. Moreover, the stability of many nanostr uctures, more specifically multiwall carbon nanotubes, nanotube rope s and bundles, and graphene nanoribbons (GNR) is effectively due to th ese dispersive forces. VDW for ces are also responsible for the successful operation of many devices which involve carbon nanotubes (CNT). Some examples are different carbon oscillators, ro tators, and nanotube ba sed AFM applications [65, 127-129]. However, most such theoretical investigations of CNT interactions have been done for infinitely long structures. These us ually involve calculating the VDW interaction using the macroscopic approximations which ignore the discrete atomic nature or
37 collective effects (many-body effects). On the other hand, the progress of the nanotechnology leads to shrinkage of new device dimensional characteristics where many-body effects and discrete na ture of materials have a significant impact of device functionalities. Besides, the re al experimental and actual na nodevices have to deal with finite systems. As a result, one can expect different properties of finite structures in comparison with infinite ones. A few results have been reported for the VDW interactions between finite length CNTs in regards to the CNT oscillator concept, but they also neglect many-body effects and discreteness of CNTs [130,131]. Thus, understanding the nature of the VDW dispersion forces in a finite structure of carbon nanotubes is of great importance for the desc ription and prediction of various phenomena and devices involving graphitic ma terials. In this regards, we have carried out the study of VDW interaction between the finite lengt h nanotubes. For comparative purpose, we also investigate the VDW in teractions between nanotubes and graphene nanoribbon and two nanoribbons. We are not aware of any i nvestigations of GNR VDW interactions. 4.1 Theoretical Investigation of Van der Waals Interaction The VDW interaction is usually calculated within the pairwise summation approximation. This method utilizes London s formula for the in teracting potential 6 6/ij ijr C U between atoms i and j derived from perturbation theory  where rij is the distance between these atoms and / ] ) ( ) ( [ 30 6 d i i Cj i where i( i ) and j( i ) are polarizability of those atoms evaluated at imaginary frequency i For structures with a relatively small number of atoms, the VDW interaction is calculated by
38 summing over all possible atomic pairs ij and taking C6 from quantum calculations. Thus, it assumes that interaction between those two atoms is not affected by the rest of the atoms. However, it is difficult to perform such summation for large systems, and, for this reason, it is assumed that each object is not a discrete collection of atoms anymore but rather it is a continuous medium. Therefore, the summation is substituted with integration over the entire structure the Hamaker method . Although these two methods are relatively easy to apply, both of them ignore many-body and screening effects. To improve the pairwise London approxi mation for the interaction between nanostructures, researchers have adapted th e Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) method , which was originally developed for macroscopic objects. This approach is also known as Lifshitz theory and takes in to account many-body and screening effects, but it has been applied only to specific geomet ries such as planar, cylindrical, or spherical because of the difficulties arising from solvi ng the Maxwells equations for systems with more complicated boundary conditions. Alt hough the DLP method shows the importance of many-body effects and gives the interactions for large distances and all types of materials, it ignores the discrete atomic natu re of the structures and assumes that the dielectric response properties are the same as the ones for macroscopic objects. Moreover, when the objects are very close to each other, and they are not semi-infinite structures anymore such as nanoclusters or colloids, the DL P-Derjagun has been developed . This method assumes that at such close distances the objects can be represented as a collection of parallel plates where for each pa rallel plate the macroscopic DLP approach is applied. Then, many-body VD W interactions are just the contribution from each of those parallel plates. Hence, this is an inherent additive pairwise approach,
39 which ignores the discrete atomic struct ures and assumes the dielectric response properties are the same as for the bulk. At the same time, many reports have shown that such microscopic nature of the nanostruc tures is particularly important in VDW interactions [136-137]. 4.2 The Coupled Dipole Method To overcome the difficulties of Hamaker, DLP, and DLP-Derjagun approximations for finite nanostructures in WVD calculations, the coupled dipole method (CDM) has been deployed. Within the CDM, the dispersive forces are formulated as a result of the collective interaction between all atoms of the system. The advantages of this approach are that all many-body effects, the types of different atoms and their specific locations are taken into account simu ltaneously. In addition, the method allows the total energy to be represented as a su m of various many-body terms. Thus, one can determine the importance of all contributions to the total VDW interactions separately. At this point of time, however, CDM is currently only available for nonretarded case, where the interaction among atoms is instantaneous, and the speed of light is neglected . The retardation effect appears for the sepa ration distance larger than the size of the system and where the finite speed of light ca uses the attenuation of VDW interaction (> 20 nm) [138-139]. The successful application of CDM to the VDW interaction has been already demonstrated for atoms, molecule s, nanoclusters, and nanocolloids [10,140]. Furthermore, CDM has been used in the calculations of the polarizability of different molecules, fullerenes, and carbon nanotubes [1 41,142]. On the basis of the past success of CDM, we apply it to the study of the VDW interactions in the adsorption processes of
40 finite lengths of carbon nanotubes and gra phene nanoribbons on the surface of other finite nanotubes. The CDM is considered to be a strai ghtforward approach for the many-body VDW interactions. Within this method, each CNT or GNR structure is modeled as an arrangement of discrete N -dipoles positioned at the location of each atom. The corresponding VDW energy is calculated from the zero-point ground-state fluctuations of all atomic dipoles comprising the involved system. The polarizability of each atom i is described by the isot ropic Drude model as 2 2 0 2 0 0/ i i i i, where 0i is the static polarizability and 0i is the characteristic frequency of the ith atom. Then, the induced dipole moment pi on ith atom due to an external electric field Ei is N i j j j ij i i i 1) ( p T E p where the interaction tensor 5 2/ ) 3 (ij ij ij ij ijr r I r r T or 0 ijT if i j and i = j respectively, with I the unit tensor, rij the radius vector between i and j atoms. Since the VDW forces exist w ithout an external electric field Ei, that leads to selfconsistent coupled dipole equation in the form 0 /, 1 N i j j j ij i ip T p. This equation can be further written in the matrix form after making the substitution for i: N N N N N N N N p p p I/ 0 0 I/ 0 0 0 I/ p p p I/ T T I/ T T T I/ 2 1 2 0 0 2 02 02 2 01 01 2 2 1 0 1 2 02 21 1 12 01) ( ) ( ) ( (4.1) or, in the short notation, it can be re written as a genera l eigenvalue problem AP = 2BP where A and B are 3N3N matrixes, and P and 2 are 3N1 eigenvector and eigenvalue respectively. Here ij i ij ijT I A 0/ describes how each atom ic dipole couples to the
41 rest while diagonal matrix ) /(2 0 0 i i ij ij I B This general eigenvalue problem can be easily solved by the standard algorithm. However, we are especially interested in eigenvalues since each of them corresponds to the square of a coupled mode frequency (2) of the entire system. Then many-body VDW energy U of N-coupled dipoles is equal to the energy contributions from all coupled modes of the entire system N n nU3 12 /. The total many-body VDW interaction energy between two clusters A and B with atoms NA and NB, respectively, is found as follows: B A B AN n n N n n N N n n TNU3 1 3 1 3 3 12 / 2 / 2 / (4.2) where the self-energies of the two clusters are subtracted. UTN represents the total manybody VDW interaction energy between the two nanostructures (CNTs or GNRs or either one) which accounts for all many-body effects, th e atomic corrugation, and geometry of each nanostructure. 4.3 Studied Systems The structures of a finite length singl e wall carbon nanotube and a finite length graphene nanoribbon is shown at Figure 4.1. Each CNT is characterized by a chirality index (n,m) specifying the direction of rolling a graphene sheet into cylinder and its metallic or semiconducting properties [1,2]. The GNR is obtained by cutting a stripe from a graphene sheet. Since they resemble car bon nanotubes, it is expe cted that GNRs have similar electronic properties as CNTs. Consider ing the fact that there are many ways to cut the graphene sheet, we focus on the GNR obtained by unfoldi ng appropriate fine
42 length CNT with desired edge patterns. Hence, the GNRs can also be described by chiral index (n,m) which characterizes GNR properties as well [143,144]. Calculations of the VDW interactions report ed in this work have been carried out for the following finite graphitic structures two (6,0) CNTs, a (6,0) GNR and (6,0) CNT, and two (6,0) GNRs (shown in Figure 4. 1a-c). The nanostructures are of different lengths, and their free bonds are passivated by H atoms. In particular, we consider CNTs and GNRs with 7(CNT7, GNR7) and 16 (CNT16, GNR16) translational unit cells corresponding to 30.58 and 68.92 along the z-axis, respectively. The molecular structure of the CNTs is C168H12 (7 unit cells) and C384H12 (16 unit cells) and for the GNRs C168H40 (7 unit cells) and C384H76 (16 unit cells). The CNT diameter and GNR width are 4.70 and 15.42 respectively. Here we study the interactions between two CNTs, two GNRs, and a CNT and GNR of finite lengths. For all studied configurations, the initial surface-to-surface separation betw een the two structures is taken as D0 = 3.4 (Figure 4.1), which is the approximate equilib rium distance between graphene sheets in graphite and between carbon nanotub es in nanotube bundles .
43 Figure 4.1 a) Two (6,0) nanotubes (CNT16-CNT7); b) a (6,0) nanoribbon and (6,0) nanotube (GNR16-CNT7); c) two (6,0) nanoribbons (GNR16-GNR7); d) stacking symmetry of the hexagonal rings T-top, Bbridge, and H-hollow. The subscript indexes 16 and 7 denote the number of translational unit cells along the axial direction in each structure. The surface-to-surface distance is D0 = 3.4 .
44 4.4 High Symmetry Stacking of Graphitic Nanostructures For the structures such as carbon na notubes, nanoribbons, and graphene, the VDW force strongly depends on the mutual orientation betw een carbon rings. There are three high-symmetric orientations of those hexagonal rings with respect to each other, such as Top (T), Bridge (B), and Hollow (H) which are shown at Figure 4.1d. The T and H stackings are referred to well known AB and AA stackings in graphite structures, respectively, where one hexagon ring is loca ted on the top of a carbon atom of the other ring, and one hexagon ring is placed on the top of the other ring respectively . The B staking corresponds to the orientation of a hexagonal ring on the top of the C-C bond. Moreover, it has been demonstrated that AB stacking is more preferable than AA stacking in the graphitic structures [145, 146]. 4.5 Static Polarizability and Characteristic Frequency Most studies evaluating VDW interactions in structures usually take the atomic values for 0 i and 0 i [10,139,147]. This means that the atomic characteristics stay the same as they form a molecule or a cluster, and the molecular static polarizability is obtained by simply summing the atomic polarizabilities. However, it has been shown that such additive approach is not corr ect, and in many instances choosing 0 i and 0 i does not properly describe the experiment al data [148,149]. Since the chemical environment of the atoms changes when the molecules or clusters are formed, the atomic polarizability and characteristic frequency of atoms are altere d as well. This issue has been extensively studied in the past using the interacti on dipole models with various degrees of sophistication and comparing th e results to available experi mental and quantum chemical
45 data . Here, we take the following parame ters for the CNT static polarizabilities and characteristic frequencies H = 0.19 3, C = 1.25 3, H = 1.411016 rad/s, C = 1.851016 rad/s, and for the GNR st atic polarizabilities and char acteristic frequencies H = 0.25 3, C = 0.85 3, H = 1.411016 rad/s, C = 1.851016 rad/s [142,148]. These CNT parameters have been shown to be successful in the calculation of CNT polarizability [141, 150]. 4.6 Many-Body Van der Waals Energy We now proceed with th e calculations of the VDW interaction energy UTN for various graphitic nanostructures and configur ations using CDM and following Eq. (4.2). In Figure 4.2, we show the tota l many-body VDW interaction energy UTN for a short and long CNTs, a short CNT and a long GNR, and a short and long GNRs as a function of surface-to-surface distance D0 when the two structures have a common axial direction and are moved away from each other. For such a situation, the relative orientation of hexagon rings (H stacking for two CNTs a nd CNT/GNR, and B stacking for two GNRs) stays the same for all values of D0. One sees that the interaction energy decreases monotonically as the dist ance is increased and |UTN(CNT16-CNT7)| < |UTN(GNR16-CNT7)| < |UTN(CNR16-GNR7)| for all D0. The fact that the inter action between the two GNRs UTN(GNR16-GNR7) is the strongest is due to the larger aspect ratio si nce the overlap between the nanoribbons is the largest. Consequently, due to the CNT cylindrical curvature fewer atoms are f ound at closer distance, and UTN(CNT16-CNT7) is the weakest.
46 Figure 4.2 Total many-body VDW energy as a function of surface-to -surface separation distance D0 for two nanotubes (CNT16-CNT7), nanoribbon and nanotube (GNR16-CNT7), and two nanoribbons (GNR16-GNR7). The studied configurations and the direction of displacement are also shown.
47 4.7 Sliding Many-Body Van der Waals Interactions Besides the separation of graphitic stru ctures by moving them away from one another (Figure 4.2), other possi ble configurations can be co nsidered. Thus, we study the sliding of one nanos tructure along the xand z-axis with respect to th e other Figure 4.3. The interaction energy for the displacement along the axial direction is of particular importance for modeling of carbon nanotube and graphitic nanoribbon bearing devices, which have been proposed and/or dem onstrated recently . In addition, UTN for the relative displacement for all directions can be used to understand the corrugation of graphitic surfaces with differe nt curvature or even find mechanisms to distinguish nanotubes or nanoribbons by helical angles via interaction with larg er graphitic surfaces [152-154]. Figure 4.3 shows that there are oscillatorylike features for the axial sliding. These are much more pronounced for the CNT/ CNT and GNR/GNR relative motions. For example, the difference between the mi nimum and maximum energies for GNRs is | |max min TN TNU U =334 meV, while| |max min TN TNU U =3 meV for the CNT/GNR system. These features are attributed to a purely geomet rical effect arising from the preferential orientation of the carbon rings. It is interes ting to note that the strongest interaction min TNU occurs when the carbon rings are in H stacking (Figure 4.1d), while the maxTNUis associated with the B stacking for the CNT/GNR and GNR/GNR z-sliding. For the CNT/CNT, the oscillations in UTN are rather irregular with altering T and H preferred stackings. This is unlike the case of graphene/graphene VDW in teraction which is strongest for T stacking of the hexagonal rings (AB stacking). The oscillatory-like features are still present for
48 Figure 4.3 Total many-body VDW energy as a function of displacement for: a) two nanotubes (CNT16-CNT7), b) a nanoribbon and a nanotube (GNR16-CNT7), and c) two nanoribbons (GNR16-GNR7) in three sliding geometries. Th e displacement corresponds to the relative separation between the centers of mass of the considered nanostructures along the relevant axis. Parallel refers to a co mmon axial direction for the two nanostructures, and perpendicular referrers to the case when the nanostructure axes are perpendicular. The sliding geometries are depicted in the inserts. The surface-to-surface distance along y-axis is kept constant 3.4 .
49 larger displacements (|D0|>20 ), but they are much less pronounced as the overlap area between the structures decreases. Also, this behavior is lost fo r graphitic structure slidings along the x-axis, for which UTN decreases monotonically fo r larger displacement. Comparing the strength of the VDW intera ction between the di fferent structures shows that the strongest attraction occurs between two GNRs regardless of their orientation. In fact, UTN for GNRs with parallel z-axis is ~3.4 times stronger than UTN when there is at least one CNT involved. Thus the planar geometry is the key to much more stable configurations due to the great er surface overlap. This indicates that it is much more difficult to separate two GNRs th an two CNTs or a CNT from a GNR. Thus, the adsorption is stronger in GNRs than in CNTs. 4.8 Van der Waals Energy Contributions To deepen our understanding of the VD W interaction between carbon nanotubes, we further investigate the que stion of the important cont ributions to the VDW energy originating from the inherently collectiv e nature of this phenomenon. Many of the reported studies dealing with CNT and graphe ne do not take many-body interactions into account, but rather use the continuous pair wise Hamaker approach [36,130,131]. While this may be appropriate for materials which are weakly polarizable with isotropic polarizability and relatively sma ll screening effects, it is not clear if such two-body VDW interaction is applicable in graphitic nanostructures. It is well known that graphitic nanostructures do not possess these features they have anisotropic and chirality dependent polarizabilities and chirality depende nt screening effects . At the same
50 time, their VDW interaction is registry depend ent, which is not realistically described by the pairwise model [153,154]. The standard approach to improve the intera ction in finite structures has been to simply add the three-body VDW force usin g the Axilrod-Teller-Muto expression [156,157]. Usually this is su fficient for noble gasses for which the three-body terms become prominent at small distances . Fo r nanoclusters made out of other materials, such as Na, Au, Ag or Si, studies indicate that the in clusion of terms beyond the twobody and three-body contributions is necessary to describe the VDW interaction correctly [34,159,160]. To calculate the contribution from different n-body terms and determine how each one affects the VDW energy U, we use the fact that U can be represented as an infinite series of power expansion in term s of the atomic polarizabilities ...4 3 2 U U U U where Un represents the n-body contribution to the VDW energy U . Each term in U can be obtained using the binomial theorem af ter expanding the eigenvalue matrix from the solution of Eq. (4.1) with respect to the atomic static polarizability. Here we consider the two-body, three-body, and four-body terms explicitly fo r the graphitic structures using the following: 1 1 6 ) 2 ( 22 3N i N i j ij ijr K U (4.3) 2 1 1 3 3 3) ( )) )( )( ( 3 1 ( 2 3N i N i j N j k ki jk ij kj ki jk ji ik ij ijkr r r K U u u u u u u (4.4) 1 1 12 ) 42 ( ) 2 ( 48 9N i N i j ij ijr K U (4.5)
51 N i N i j N j k j k ik ij ik ij ijkr r K U1 1 6 2 ) 43 ( ) 3 ( 4) ( ) ) ( 1 ( 8 9 u u (4.6) 3 1 21 3 ) 44 ( ) 4 ( 4) ( )) ( ) ( ) ( ( 2N i N i j N j k N k l li kl jk ij ijklr r r r ikjl f ijlk f ijkl f K U (4.7) where uij = rij/rij is a unit vector along the line connecting the i and j atoms and uij ukj=cos(i). The coefficients) 2 ( ijK, ) 3 ( ijkK, ) 42 ( ijK, ) 43 ( ijkK, ) 44 ( ijkK, and f(ijkl) are given as: j i j i j i ijK0 0 0 0 0 0 ) 2 ( (4.8) ) )( )( ( ) (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) 3 ( k j k i j i k j i k j i k j i ijkK (4.9) ) ) ( ( ) (2 0 0 0 0 3 0 0 0 0 2 0 2 0 ) 42 ( j i j i j i j i j i ijK (4.10) )] 2 )( ( 2 ) ( 2 [ ) ( ) ( ) (0 0 0 0 0 0 0 0 0 0 3 0 0 0 2 0 0 2 0 0 0 0 0 0 0 2 0 ) 43 ( k j i k j i k j k j i k j k i j i k j i k j i ijkK (4.11) )] )( )( ( ) ( ) [( ) )( )( )( )( )( (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) 44 ( l k j i l j k i l j l j k i k i l k l j k j l j k i j i l k j i l k j i ijklK (4.12) and
52 )] )( )( )( ( 9 ) )( )( ( 3 ) )( )( ( 3 ) )( )( ( 3 ) )( )( ( 3 ) ( ) ( ) ( ) ( ) ( ) ( 1 [ ) (2 2 2 2 2 2 ij li li kl kl jk jk ij jk li li kl kl jk ij li li kl kl ij ij li li jk jk ij ij kl kl jk jk ij li kl li jk kl jk li ij kl ij jk ijijkl f u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u (4.13) The interaction originating fr om four-body contributions is ) 4 ( 4 ) 3 ( 4 ) 2 ( 4 4U U U U The binomial theorem shows that each Un contribution contains / ) 1 )...( 1 ( n n N N N terms. Thus, the number of terms for each successive n-body interaction is ~ N times larger than the previous one. At the same time, Un ~ 1/r3 n. For large separations, the contribution for three-body, four-body, and higher order interactions diminishes greatly due to th at distance dependence. However, for close separations, like the equilibrium distance s in graphitic nanostructures, the ~Nn dependence of each Un can overcome the 1/r3 n functionality, and give substantial contributions to the total VDW energy. Th is competing behavior between the large number of terms for Un interactions and relatively small distances between graphitic materials is the key factor in the inhe rent many-body nature of their VDW forces. 4.8.1 Van der Waals Energies To give a quantitative description of th is issue, we evaluate total two-body (UT2), three-body (UT3), four-body (UT4), and the total many-body UTN using Eq.4.2 for two (6,0) CNTs, (6,0) CNT and (6,0) GNR, and two (6,0) GNR with 5 unit cells along the zaxis (22.06 ), when the nanostr uctures are moved apart along the y-direction. The results are given in Figure 4.4. It is evident that for two CNTs (Fi gure 4.4a), the two-body interaction overestimates the total UTN. Adding UT3 decreases UT2, while adding UT4
53 Figure 4.4 VDW energies for UTN, UT2, UT2+UT3, and UT2+UT3+UT4 interactions as a function of separation distance D0 for two (6,0) nanotubes (CNT5-CNT5), (6,0) nanoribbon and (6,0) nanotube (GNR5-CNT5), and two (6,0) nanoribbons (GNR5-GNR5) with 5 translational unit cells. The re lative geometries are also shown.
54 increases UT2+UT3, as compared to the total interaction UTN. For the CNT and GNR interaction (Figure 4.4b), it seems that UT2 is almost the same as UTN, and the effect of UT3 and UT4 have similar trends as for the two CNTs. For the GNRs, however, the twobody interaction is smaller than the total UTN, and their deviation is larger than the UT2/ UTN deviation for the two CNTs. Our calculations show that for all considered systems UT3 is always positive (Eq.4.4). Thus when added to the negative UT2, UT2+UT3 decreases. UT4 is always negative and due to the large number of terms, it becomes significant at distances D0 < 20 . When the separation be tween the nanostructures becomes comparable or larger than their size, the largest contribution to UTN is mainly from the two-body interaction. 4.8.2 Effect of Structural Si ze on Van der Waals Energies The above results suggest that since Un~Nn for larger nanostructures at their equilibrium distances (~3.4 ), the contribution from each n-body interactions may be even larger than the previous Un-1 term. We further study this issue by calculating the VDW energy and the various many-body term s for the CNT and GNR systems with common axial direction by keeping their su rface-to-surface distance constant, while changing their size. We show results for changing the length (Figure 4.5a) and the diameter (Figure 4.5b) of two CNTs, and the length (Figure 4.5c) and the width (Figure 4.5d) of two GNRs. Both structures are orie nted with parallel axial directions and the surface-to-surface distance is kept constant D0 = 3.4 . We see that in all cases UT2 is in the close proximity of UTN. However, UT2 does not exhibit the non-linear features of the to tal energy for the CNT axial and radial size
55 dependence as well as for the GNR width depend ence. For all structures the deviations of the three-body and four-body terms increase as the size of the system increases. In addition, |UT3|<|UT2|<|UT4| with exception for the radial dependence of the CNT energy for which |UT2|<|UT3|<|UT4|. UT2 and UT4 are always negative while UT3 is always positive. Figure 5a and 5d show that UTN exhibits periodic-like behavi or for every 4 and 2 unit cell increments for the two CNTs and GNRs, resp ectively. These non-line ar features are not found in the two-body intera ctions or when the UT3 and UT4 are added. In fact, adding UT3 and/or UT4 yields to even bi gger deviations from UTN. Figure 4.5b and 4.5c indicate that UT2 is very close to UTN, although some non-linearities are missing from the radial dependence of the CNT two-body interactio ns. Investigating th e VDW interactions between graphitic systems as a function of size shows that in some situations approximating the total energy by the twobody contribution may be a relatively good approximation, while in other situations, the two-body interaction misses functionalities present in the total energy. Adding the first one or two corrections (UT3 and UT4) does not result in an improvement of UT2, but rather it gives even bigger deviations from the total UTN.
56 Figure 4.5 VDW energies for UTN, UT2, UT2+UT3, and UT2+UT3+UT4 interactions between the CNTs and GNRs nanostructures: a) two (6 ,0) CNTs as a function of length; b) (n,0) CNTs with 4 unit cells along the z-axis as a function of diameter; c) two (6,0) GNRs as a function axial length; d) (n,0) GNRs with 4 unit cell along z-axis as a function of width. The surface-to-surface distance is D0 = 3.4 for all cases and chiral index n = 615.
57 4.8.3 Sliding Van der Waals Energies Our results from Figure 4.3 reveal that the relative orientation and displacement have a profound effect on the total VDW interact ion. This is manifested in the magnitude and corrugation effects in UTN. Further investigations show the importance of the manybody nature of the VDW long-ranged interactions for the different sliding displacements. Due to the axial and circular symmetries of the carbon nanotubes, we expect that the anisotropic polarizability and the larger overlap between the surfaces, the many-body effects are going to be even more pronounced in these sliding configurations. In Figure 4.6, we show our results for the UT2 and UT3 terms when the shorter nanostructure slides along the longer one keeping a common axial dire ction. One sees that the deviations of UT2 from UTN are rather different for th e considered cases. When there is a CNT involved (Figure 4.6a, 4.6b), UT2 always overestimates UTN. However, for two CNTs, the registry dependence of the two-body VDW energy is mu ch less pronounced th an the one of the total energy. There are also non-linearities in U which are missing in UT2 the largest difference in UTN between H and T stacking is 0.21 eV while it is only 0.01 eV in UT2. This oscillatory-like behavior is at a much smaller scale for both UT2 and UTN when the CNT slides along a GNR. For the GNR slidin g along the axial-direction, however, our results show that UT2 underestimates UTN by ~20 % (Figure 4.6c). The registry dependent features of UT2 are also much less pronounced when th e two structures overlap, and they are lost when the GNRs do not overlap completely (|D0|> 20 ). In all cases, UT3 is always positive and registry independe nt, and it is reduced significantly UT2 upon addition, while UT4 (not shown) is always negative and registry independent, and it brings significant deviations to UT2+UT3+UT4 compared to the total UTN.
58 Figure 4.6 VDW energies for UTN, UT2, and UT2+UT3 energies as a function of center of mass displacement along the z-direction for a) two nanotubes (CNT16-CNT7), b) a nanoribbon and a nanotube (GNR16-CNT7), and c) two nanoribbons (GNR16-GNR7).The inserts show the sliding geometries with the surface-to-surface separation distance D0=3.4 .
59 CHAPTER 5 DENSITY FUNCTIONAL THEORY Density functional theory (DFT) is one of widely used quantum mechanical technique for the calculations of electroni c structures of many-body systems such as atoms, molecules, clusters, and condensed mate rials. Its successful application in physics and chemistry is clearly shown in the enor mous amount of carried out research and published articles over the last decades. DFT results are found in a satisfactory agreement with numerous experimental data. Therefore, it has been considered that DFT is a powerful method in the nanotechno logy research as the size of the system becomes of nanometer scale. Moreover, this approach a llows exploring the f unctional properties of prospective nanodevices, while it can be very difficult to recreate that in a real experiment at this time. For the rest of the calcula tions of the dissertation, we will apply DFT method to the study of the adsorption of DNA bases on the surface of nanotubes as well as to the modulation of nanotube properties under two external factors such as a radial deformation and mechanical defects, along with radial deformation and external electric fields. Density functional theory takes its origin from the work of Thomas  and Fermi . They show how general DF T approach can work. However, their approximation is not accurate enough for present day problems since they ignore the
60 exchange correlation electron interactions. Howe ver, it has been firs t successful attempt of DFT application. Then, Di rac  in 1930 made extension to Thomas and Fermi approach by including local approximation for exchange interactions. Further, Hohenberg and Kohn describe DFT method which is used up to today . The extensive review of DFT can be found in the recent book by Martin . 5.1 The Hohenberg-Kohn Theorems In 1964, Hohenberg and Kohn formulated de nsity functional theory as used today for many-body systems . This approach applie s to any system of interacted particles in an external potential Vext(r). The Hamiltonian of such system can be written as j i j i i i ex i i er r e r V m | | 2 1 ) ( 22 2 2 H (5.1) where the first sum is the kinetic energy of the system, the second sum provides all exchange interactions in the system, and the last sum is the Hartree repulsion between electrons. Therefore, the Hamilt onian is determined if the Vext(r) is specified. Instead of solving a complex Schrdinger equatio n with full set of wavefunctions (r) for manybody systems, Hohenberg and Kohn replaced it with simpler equation for density n(r). This statement has been formulated in the following theorem: Vext(r) is uniquely defined by the ground state density n0(r) up to some constant for any system of interacted particles. The solution of th at new Schrdinger equation defi nes functional of the total energy E[n(r)] in terms of the density n(r) of the system, which is the second HohenbergKohn theorem. So, the ground state energy of the system is represented by the minimization of this functi onal with global minimum at the ground state density n0(r).
61 The proof of both theorems is simple and it can be found origin al Hohenberg-Kohns work . 5.2 The Kohn-Sham Approach The further development in density functi onal theory came one year later in 1965, when Kohn and Sham showed that problem of many-interacted elec trons can be replaced by self-consistent one-particle problem . They assume that ground state density of the original many-body system is the same as the density of system of non-interacting particles. This leads to the self-consistent Kohn-Sham equation: )] ( [ ) ( ) ( | | ) ( ) ( 2 1 ) ( ) ( 2 1 )] ( [' ' 2 *r n E dr r V r n drdr r r r n r n dr r r r n ECX ex i i i (5.2) where the first term is the kinetic energy of a non-interacting electron gas with the density i i ir r r n ) ( ) ( ) (* (5.3) where index i runs over all particles of the system; the second term is the Hartree energy with next term of external fiel d energy, and the last unknown term EXC[n(r)] is the exchange-correlation energy, which incorp orates all many-body in teractions in the system. The minimization of the E[n(r)] functional provides the ground state density n0(r) and energy E of the system. Thus, the Kohn-Sham appr oach is a very useful technique for prediction of properties of ma ny macro, micro, and nano systems. Moreover, it is the basis for all current first principle calculations.
62 5.3 Exchange-Correlation Functional Approximations The solution of Kohn-Sham equation can be found if exchange-correlation energy EXC is known. However, EXC is unknown most of the time due to its complexity in description of many-body intera ctions. The exact form of EXC has been calculated only for several simple systems. Therefore, EXC is commonly approxima ted in the electronic structure calculations. The simplest method for the treatment of exchange-correlation energy is local density appr oximation (LDA)  where EXC is approximated by a mean-field expression: dr r n r n r n EXC LDA XC ) ( )] ( [ )] ( [ (5.4) where XC is the exchange-correlation energy de nsity of the homogenous electron gas. XC can be accurately obtained from quantum M onte Carlo calculations where the problem of exchange and correlation is solvable . It is remarkable that for such a long tim e, LDA is the widely used approximation in DFT calculations even nowadays. It provid es accurate results for the systems with the slowly varying charge density, such as systems with sp bonds or large band gap semiconductors. The calculation results of geom etry, bond lengths, and angles can be just a few percent different from experimental one s. The success of LDA in DFT calculations can be partially related to the correct sum rules for exchange correlation holes . Despite LDA accomplishments, there is a lim itation where it can be applied. LDA usually ignores the corrections of inhomoge neous electron density to the exchange correlation energy. It also does not consider van der Waals forces a nd gives a very poor description of hydrogen bonding which are crucia l for biochemical studies. Moreover, it
63 fails to correctly estimate prope rties of strongly correlated materials, which usually have incompletely filled d or f electron orbitals. The next step in improving of LDA has been achieved by including the gradient of the electron density, which leads to the appearance of generalized gradient approximations (GGA) : dr r n r n F dr r n r n r n EXC XC GGA XC |] ) ( | ), ( [ ) ( )] ( [ )] ( [ (5.5) where FXC correction functional. There have been extensive numbers of studies carried out on GGA functional, but they do not al ways provide methodical improvement of LDA. The obtained results have to be carefu lly compared with the experimental ones. The most popular GGA functional are PBE derived by Perdew, Burke, and Ernzerhof , and PW91proposed by Perdew and Wang . 5.4 Pseudopotentials The pseudopotentials have been introdu ced as the method to simplify the manyelectron Schrodinger equation. The idea behind pseudopotentials is to exclude the core electrons from the calculations and repl ace them along with atom nuclei by ionic (pseudopotential) potential that interacts with valence electr ons. As a result, the rapidly oscillations of valance electron wave functions are smoothed by pseudopotential. Moreover, it is a well known f act that main physical propertie s of solids are characterized mostly by valance electrons rather than core electrons. The application of pseudopotentials significantly simplify the DFT calculations and improve the computational time. This approach has b een first introduced by Fermi in 1934 . Therefore, it has been forgotten for more th an 15 years, and it has been revived in the
64 1950s by Antoncik  and Phillips and Kleinman . The development of pseudopotentials is dynamically growing due to the fact the pseudopotentials are not uniquely defined. It leaves the freedom in the choice of the pseudopotential form to simplify the calculations and the explanation of the electronic structure. There has been some progress made in the development pseudopotentials with many approaches. However, the commonly used pseudopotentials are those that are ba sed on the projector augmented wave (PAW) method and those deri ved by Vanderbild and Blchl [175-177]. 5.5 Calculation Tool Our main method of investigation is ba sed on self-consistent DFT calculations within the local density approximation (LDA) for the exchange-correlation functional implemented within the Vienna Ab Initio Simulation Package (VASP) . The code uses plane-wave basis set, a nd it is a periodic supe rcell approach. The core electrons of the atoms are treated by ultrasoft Vanderb ilt pseudopotentials [ 176, 177]. This DFT method is rescalable for large systems, and it requires relatively sm all plane-wave basis set for each atom in the calcu lation. It also contains the opportunities to in clude external electric field in the system . Over the last decay, DFTLDA has been a useful model in studying large graphitic structures. It provi des an accurate descrip tion of the electronic structure along the graphite planes and nanotube axis, where the electronic structure is characterized by sp2 orbitals of the carbon atoms . Based on its past success, we utilize DFT-LDA for our studies. The calcula tion criteria for rela xation of the atom positions have been 10-5 eV for the total energy and 0.005 eV/ for the total force. Spinpolarization effects are in cluded in the study for de fective carbon nanotubes.
65 CHAPTER 6 DNA BASES ADSORPTION ON CARBON NANOTUBES The interaction properties carbon nanotube s with other compounds are the main element in many applications such as gas storage, sensors, biocompatible agents, functionalized elements and more, which we emphasize in the b ackground information. In particular, the adsorption of polymers, organic molecules, proteins, and DNA on carbon nanotube surfaces has attracted consid erable attention recently due to their potential applications in physics, chemistry, biology, and materials science. It has been demonstrated that physisorption of aromatic and organic molecules on the surface of the nanotubes can be achieved [183,184]. In addi tion, significant attention has been paid to DNA-carbon nanotube system s due to their potentia l applications in many new bio-nanodevices [48-53]. Othe r applications of DNA-carbon nanotube complexes related to antitumor drug delivery systems, enzyme immobilization, and DNA transfection have also been envisioned [ 185]. However, the interaction of DNA molecules with nanotubes is complex, and it depends on many external factors, such as temperature, pH, concentrations, solvent, electronic properties of nanotubes etc. To understand such complex nature of DNA interactions with nanotubes, we carried the studies of the smallest building DNA blocks, such as thei r bases (adenine and thymine) with nanotubes to determine the dominant contribution in larger DNA/nanotubes complex. We examine in detail the different adsorption configurations,
66 equilibrium distances, and elect ronic structure for the DNA base s/nanotube complexes. In addition, we carried the same analysis for the adsorption of DNA base s radicals (adenine and thymine radicals) on the same nanotubes. Experimental work has shown that all ionizing radiation exposures induce muta genic and recombinogenic lesions in DNA which are linked to strand breaks, base da mage, and clustered damage . In particular, damage mechanisms of DNA base s have been related to creation of base radicals [187,188] as well as tautomers of bases . Thus, we investigate the adsorption of radicals, obtained by deprotona tion which is removing a hydrogen (H) atom from the base [190,191], in order to pr ovide a model of how damaged DNA might interact with nanotube surfaces To provide analysis of the factors influencing the adsorption process in CNT/DNA base CNT/DNA base radicals systems, we use the DFTLDA approach, which is a reasonable model fo r this goal as it has been indicated by the other studies [181,192,193]. 6.1 Studied Systems DNA is an important element of the living cell that contains genomic information. It is a large quasi-one dimensional structur e consisting of two polynuc leotide chains as it is shown on Figure 6.1. Each chain consists of random repetition of nucleotides which are connected to each other through hydrogen bonds Each nucleotide has a phosphate group linked to a sugar ring and a base molecule also connected to th e sugar ring . The bases are the purines comprised by adenine (A) and guanine (G), and the pyrimidines comprised by thymine (T) and cytosine (C) (Figure 6.1). The stability of the DNA molecule is governed by the hydrogen bonds between the purines and pyrimidines (two
67 hydrogen bonds for A-T and three for G-C), the stacking of the bases, and the screening of the negative charges carried by th e phosphate groups on its backbone . We examine in more detail the structure of the DNA bases. This is motivated by the fact that our study will be centered at understanding of how the bases interact with carbon nanotubes. The purines (A and G) are nitrogenous molecules consisting of one six-member pyrimidine ring and one fi ve-member imidazole ring with an NH2 functional group for A and an NH2 and CO groups for G (Figure 6. 1). The pyrimidines (T and C) are nitrogenous bases consisting of one six-member pyrimidine ring with a CH3 and CO groups for T and a NH2 and CO groups for C (Figure 6. 1). Thus, purines and pyrimidines have a common structural feature from the orbitals perpendicular to the molecular plane, but the functional groups provide their different electron affinities. Our focus is to investigate the adsorp tion of two DNA bases on single wall carbon nanotube surfaces. These are adenine as a model for purines/CNT interaction and thymine as a model for pyrimidines/CNT in teraction. In addition, to understand the response of both metallic and semiconducting nanotubes, we study the adsorption of adenine on the metallic (6,6) CNT and the adsorption of thymine on the semiconducting (8,0) CNT. The A-radical and T-radical obt ained by the deprotonation have also been studied on the same (6,6) and (8,0) CNTs respectively.
68 Figure 6.1 Chemical structure of DNA with its purines (Adenine and Guanine) and pyrimidines (Thymine and Cytosine) bases. Hydrogen bonds between DNA bases are shown by dotted lines .
69 6.2 Calculation Parameters The adsorption of adenine on the metallic (6,6) CNT and the adsorption of thymine on the semiconducting (8,0) CNT is considered with non-spin-polarized DFTLDA which has been describe earlier. These ar e closed valence shell structures. Hence, it does not require performing spin-polarized calculation. However, spin-polarized calculations are done for the ad sorption of the partially fill ed valence shell A-radical on (6,6) CNT and T-radical on (8,0) CNT. The bare nanotubes are fully relaxed ini tially obtaining an optimized length of 12.25 for the (6,6) CNT, which corresponds to 5 unit cells along the tube axis, and an optimized length of 17.03 for the (8,0) CN T, which corresponds to 4 unit cells along the tube axis. The lateral sizes of both s upercells are 22.04 and 22.12 for the (6,6) and (8,0) CNTs, respectively, in order to a void interactions between tubes from adjacent unit cells. A 1x1x7 Monhkorst-Pack k-grid for sampling the Brillouin zone is taken. During the adsorption calculations, the entire system let completely relax in each case. For all molecules, the adsorption energy is calculated using the relation: CNT mol CNT mol adsE E E E/ (6.1) where Emol is the total energy for the relaxed molecule only, ECNT is the total energy for the relaxed pristine CNT, and Emol/CNT is the total energy of the relaxed molecule/CNT structure. 6.3 Adsorption Configurations We have already seen that carbon nanotubes structure ha s several symmetric sites for interactions with other structures relate d to the relative position of the hexagonal ring
70 in each molecule with respect to the nanot ube carbon rings. They are top Figure 6.2a, bridge Figure 6.2b, hollow Figure 6.2c and they have been determined and investigated by us in the Van der Waals interactions in gra phitic nanostructures (Figure 4.1d). Thus, we investigate several positions of the adenine and thymine molecules on the CNT surfaces. In addition, adenine is an exte nded molecule due to the five-member ring attached to its hexagonal ring. As a result, th ere are many more possible configurations as compared to those for a molecule with just a hexagonal ring [182,183,196]. 6.3.1 Adenine and Thymine Configurations We considered fifteen locations of aden ine on the (6,6) CNT, and they are shown on Figure 6.2 after relaxation of the entire system. Figure 6.2d shows the most stable configuration after ad sorption Top_60b, where the adenin e is in the top symmetry site and the axis of the molecule, is at about 60o with respect to the nanotube axis. The N atom from the NH2 group is in the top position while the two H atoms are above the two C atoms from the nanotube. Thymine has only one hexagonal ring with a methyl functional group. We consider five configur ations on the (8,0) CN T surface Figure 6.3 (shown after relaxation of the structure). The most stable one is near the top symmetry site Figure 6.3a, where the hexagonal ring and the C atom from the CH3 group are also in top location. Thymine is slightly tilted with the oxygen opposite to the CH3 group closer to the nanotube axis.
71 Figure 6.2 The symmetry adsorption sites of a hexagonal ring struct ure: a) top symmetry site, b) bridge symmetry site, and c) hollo w symmetry site. Adsorpti on configurations of adenine on the (6,6) CNT surface: d) Top_60b configuration with front and top views (the most stable configuration), e) Top_60a f) Top_90a, g) Top_90b, h) Top_a, i) Top_b, j) Bridge_a, k) Bridge_b, l) Bridge_30, m) Bridge_60, n) Bridge_90, o) Hollow, p) Hollow_30, r) Hollow_60, s) Hollow_90. The nu mbers 30, 60, and 90 represent relative angles between the longitudinal axis of adenine (dashed line) and the z-direction of the nanotube.
72 Figure 6.3 Equilibrium configurations of thymine adsorbed on (8,0) CNT: a) Top_a configuration with front and t op views (most stable configur ation), b) Top_b, c) Hollow, d) Bridge_a, e) Bridge_b.
73 6.3.2 Adenine and Thymine Radical Configurations For the A-radical, we take the most stable configuration for A/CNT (Top_60b). There are several A-radicals th at one can obtain if a H atom is removed . We take only this deprotonated radical fo r which the H atom from the NH2 (N6 atom) group has been removed Figure 6.4. This radical ha s the second largest most stable energy as compared to the one for which H from N9 is removed, but since N9 in adenine is connected to a DNA sugar ring, such radical is much more diffic ult to create when adenine is part of the whole DNA molecule . Figure 6.4a shows the A-radical/CNT configuration after relaxati on. For the T-radical, we al so take the most stable configuration from T/CNT Top_a. The depr otonated one obtained by the removal of a H atom from N3 is considered here. Such radical has the most stable energy as compared to other deprotonated struct ures, and it can occur in DNA . The T-radical/CNT is shown on Figure 6.4b after relaxation. The T-radical is somewhat more tilted as compared to T/CNT with the oxygen opposite to the CH3 group being closer to the nanotube axis as a result of charge redi stribution in the ab sence of H atoms.
74 Figure 6.4 Equilibrium configurations of a) A-radical on (6,6) CNT and b) T-radical on (8,0) CNT with front and top views respective ly after relaxation. Each complex is in Top configuration.
75 6.4 Structural Changes of DNA Bases and Their Radicals To understand better the structural change s in the DNA bases and their radicals involved in the adsorption process, we also calculated their bonds and relative angles before and after adsorption Figure 6.5. Ther e is a little change in bond lengths for all molecules. For thymine, adenine, and A-radica l, there is also a little change in the dihedral angles. Therefore, these compounds st ay flat before and after adsorption. Some noticeable changes were found for thymine a nd T-radical themselves in the dihedral angles. All dihedral angles (the ir absolute value) for thymine are larger than the dihedral ones for the T-radical with the largest difference of 170 being in the angle N3-C4-C5-C7. This indicates that the T-radical is not flat and it has a bent form. After adsorption, however, the dihedral angles for the T-radical become very close to the ones for thymine, and the molecule stays essentiall y flat above the nanotube surface. 6.5 Adsorption Energies The adsorption energies and equilibrium distances after relaxation have been calculated according to the Eq. 6.1, and they ar e given in Table 6.1. The data indicate that many local minima for the adsorption of thymine and adenine are possible on the carbon nanotube surface. This is in agreement with pr evious results for the adsorption of benzene aromatics on nanotubes which showed that the adhesion energy is very shallow, and there are several local equilibrium positions [ 181,182,196,197]. The values of the adsorption energies and distances for adenine and t hymine are consistent with a physisorption process where no chemical bond is formed between the two species.
76 Figure 6.5 DNA base structures before and af ter adsorption: a) bond distance for adenine and A-radical before and after adsorption; b) bond distance for thymine and T-radical before and after adsorption; c) angles fo r thymine and T-radical before and after
77 adsorption; d) angles for adenine and A-radi cal before and after adsorption. The dashed circles show the hydrogen atoms removed to form the deprotonated radicals. The largest adsorption energy has been obtained for Adenine/CNT 0.354 eV with a distance 3.14 (Top_60b) although the T op_b configuration is just 1 meV below. For Thymine/CNT the largest adsorption ener gy is 0.316 eV with the distance 3.10 (Top_a). We compare these values to the ones reported for the nucleoside/CNT adsorption (Eads=0.420.46 eV) . One notes that the adsorption energies calculated here are only about ~0.1 eV smaller than the ones given in Ref. . The equilibrium distances, however, were found to be of similar magnitude ~3.10 for A and T calculated here, and ~3.3 for the nucleosid e in Ref. . This indicates that the interaction process is dominated mainly by th e base, and the rest of components in the larger DNA fragment play a secondary effect. Our adsorption energy and equilibrium distan ce values are of similar order as the ones reported for the DFT-LDA values for ben zene and other simple benzene derivatives, where similar relaxation and convergence criteria were applied . Thus the nitrogenous bases adsorb in a similar manne r as the carbon benzenes. For the radicals, after the relaxation with spin-polarization was done, we find that the adsorption energies are larger than the ones for the full molecules Table 6.1. The distances are of the same order.
78 Table 6.1 Adsorption energies and equilibrium dist ances for adenine and adenine-radical (A-radical) and (6,6) CNT, and thymine and thymine-radical (T-radical) on (8,0) CNT. The equilibrium distances are measured from the surface of nanotube to the center of the hexagonal ring of each molecule. Configuration Eads (eV) d ( ) Adenine/(6,6) CNT Top_a 0.324 3.06 Top_b 0.353 3.05 Top_60a 0.296 3.11 Top_60b 0.354 3.14 Top_90a 0.317 3.10 Top_90b 0.319 3.16 Bridge_a 0.338 3.13 Bridge_b 0.321 3.14 Bridge_30 0.301 3.13 Bridge_60 0.320 3.14 Bridge_90 0.319 3.13 Hollow 0.281 3.25 Hollow_30 0.297 3.24 Hollow_60 0.324 3.21 Hollow_90 0.324 3.19 Thymine/(8,0) CNT Top_a 0.316 3.10 Top_b 0.291 3.16 Bridge_a 0.303 3.11 Bridge_b 0.312 3.22 Hollow 0.247 3.31 Radicals A-radical/(6,6) CNT 0.517 3.14 T-radical/(8,0) CNT 0.661 2.96
79 6.6 Electronic Structure Calculations Afterwards, we calculate and analyze the electronic structure of the DNA bases and their radicals adsorbed on CNT in order to further understand the adsorption process. Figure 6.6 shows the total elec tronic density of states ( DOS) for the DNA bases and their radicals on the CNT surfaces. The DOS for th e pristine nanotubes is also given as a reference. First, we analyze the adsorption of th e DNA bases on CNT. The six-member rings have enhanced stability attributed to the delocalized electronic structure. The Highest Occupied Molecular Orbital (HOMO) for bot h adenine and thymine is deep in the energy, and it is completely filled while the Lowest Unoccupied Molecular Orbital (LUMO) for both molecules is completely em pty. The band gap of the (8,0) nanotube is not changed, and there are no modifications around the Fermi level of the metallic (6,6) nanotube. Therefore, the electronic transpor t properties of CNT are not expected to change upon adsorption of these DNA bases. The DOS is perturbe d at around 1.5 eV under EF for the (8,0) tube and around 2 eV under EF for the (6,6) tube (Figure 6.6b,e). These results are very similar to the tota l DOS for benzene/CNT and simple benzene derivative/CNT upon adsorption [181,196] wher e no significant cha nges in the carbon nanotube electronic structure were found due to the benzenes adsorption. Mulliken analysis shows that in both cases there is little charge tran sfer in the system 0.01e. The total DOS for the radicals is also given in Figure 6.6c and 6.6f. The spin polarized calculations indicate that the DOS for both spin species for the A-radical is very similar except for a peak for spin down which appears at the Fermi level of the (6,6) nanotube. This state mainly of p character is primarily compos ed of molecular states, and
80 Figure 6.6 Total density of states for a) pristine (8,0) CNT, b) Thymine/(8,0) CNT, c) Tradical/(8,0) CNT, spin-up, and spin-down, d) pristine (6,6) CNT, e) Adenine/(6,6) CNT f) A-radical/(6,6) CNT, spin-up and spin-down. For the radicals spinpolarization effects were in cluded in the calculations.
81 there is very little contribution from the C orbitals from the nanotube. Mulliken analysis shows charge transfer ~0.05 e in the system. For the T -radical the spin polarized calculations show that the band gap of the (8,0) nanotube is changed very little, and no peaks appear in it. For the spin down a p eak appears below the top of the valence band composed mainly of p molecular states, and it does not affect the Fermi level of the nanotube. No such peaks are found for the spin up around the Ferm i level. Mulliken analysis gives charge transfer ~0.03 e in the adsorption process. 6.7 Charge Densit y Distributions To gain further insight into the adsorpti on electronic structure of the radicals for the new features in and around the Fermi leve l, we give plots of the charge density isosurfaces of four states be low the Fermi level Figure 6.7. For T-radical/CNT, we take the first four states under the Fermi level, and for the A-radical, we take the state at the Fermi level and the three states right beneath it. For both cases two of the states are due to the molecule, and the other two are due to the nanotube. We find practically no hybridization between the states of the radicals and the highest valence nanotube levels. The adsorption causes some charge redistribution, but mainly within the molecule. This is due to the fact that delocalized nanotube electronic states ar e not easily polarized as it was found for the adenine/graphite system . The rehybridization occurs mainly within the surfaces of the radicals.
82 Figure 6.7 Isodensity surfaces of the charge de nsity for four valence states below Fermi level with isovalue 0.023 -3 for a) A-radical/(6,6) CNT and 0.022 -3 for b) Tradical/(8,0) CNT (created with VMD ).
83 CHAPTER 7 DEFECTIVE CARBON NANOTUBES U NDER RADIAL DEFORMATION Controlling carbon nanotubes properties by more than one external factor simultaneously is considered a new research fi eld. However, as we have discussed it in the background information, there are not so ma ny studies which have been carried out in that direction. To extend this new area, we perform the calculations of the structural changes, characteristic energies, electron ic, and magnetic properties of nanotubes under combined effect of radial deformation and m echanical defects, such as a Stone Wales, nitrogen substitutional impurity, and mono-vacancy. These calculations and results are presented in the following sections. 7.1 Studied Systems As a model for investigating the combin ed effect of radial deformation and defects, we consider a zigzag (8,0) single wall carbon nanotube. To avoid the interaction between nearest-neighbor de fects, we construct a supe rcell consisting of four translational unit cells along the z-axis of the nanotube whic h has the length of 17.03 after relaxation. The radial deformation is simulated by applying stress to the cross section of the nanotube along the y-direction from both sides. Consequently, the nanotube is compressed along the y-direction and extended along the x-direction (Figure 3.4a,b). The nanotubes are squeezed by two types of strain. The first one is deforming the
84 nanotube between two hard surfaces with a cr oss section smaller than the nanotube one, which we refer to as Peanut deformation (Figure 3.4c). The second one represents squashing the nanotube between two hard surfa ces with a cross section larger than the nanotube one Flat deformation (Figure 3.4d). Such deformations are modeled by fixing the y-coordinates of the top and bottom rows at low curvature in Peanut and Flat structures while the rest of the atoms are free to relax. We show th at the (8,0) nanotubes can undergo large radial deformations up to yy=75% and yy=65% for Peanut and Flat deformation respectively. After the radial deformation, defects, such as a Stone Wales defect, nitrogen impurity, or mono-vacancy are introduced on the high curvature of the nanotube. Experimentally this can be achieved, for ex ample, by deforming the tube using an atom force microscope and then intentionally introducing defects us ing electron or ion irradiation methods. The radially deformed carbon nanotube with or without defects has an elastic response even under high degree of squeezing. When the applied stress is removed, the nanotube comes back to its origin al circular cross-sectional form regardless of the defect presence. 7.2 Structural Changes In this section we analyze the changes in the carbon nanotube structure under radial deformation and various defects under tw o types of deformati on Peanut and Flat. The main focus is on the bonds on the high curvature and the bonds comprising the defects (Figure 3.6). When the perfect nanotub e is radially deformed, the bonds along the
85 Figure 7.1 (8,0) nanotube bond length evolu tion as a function of applied strain yy for a) no defect, b) with a Stone Wales defect, c) with a nitrogen impurity, d) and e) with a mono-vacancy.
86 nanotube axis (C-C1) decrease while th e curvature bonds (C-C2) increase to reduce the strain on the high curvature (Figure 7.1a). Under moderate radial deformation (yy<50%) there is no significant difference between Peanut and Flat deformation. However, at high deformation (yy>50%), the difference becomes more evident since the Flat deformation applies much more distortion on the high curv ature rather than the Peanut does. As a result, the bonds on the high curvature change mo re in Flat than in Peanut deformation, while there are no changes along nanotube axis (F igure 7.1a). It is worth mentioning that in the region of (yy~2025%), the (8,0) nanotube s hows interesting changes in the structure, which can be seen in the jump of the bond length profile. Later, there will be more discussion of this jump due to the admixt ure of different orbitals from the C atoms. Energetically favorable configuration of Stone Wales defect happens when two heptagon rings align with the nanotube ax is (Figure 3.6b). Figure 7.1b) shows the evolution of various bond changes as a function of yy. Both Peanut and Flat deformations produce almost the same changes in the def ect bond lengths with the exception of the C1C2 bond. In fact, the bonds in the heptagon rings (C1-C2, C2-C3, and C3-C6) undergo more changes than the bonds in pentagon ones (C3-C4, C4-C5, and C6-C7). Such changes are explained by the location of the heptagon rings on the high curvature because the pentagon rings are dist ant from it. Moreover, the C1-C2 functional behavior looks similar to the one for the C-C2 in the defect-free nanotube. The presence of a nitrogen impurity on the high curvature of the nanotube does not create much disturbance in its structure (Figure 3.6c). Si nce the atomic size of the nitrogen atom is larger than the ca rbon one, it leads to the decrease in N-C1 and N-C2 bond lengths by 0.01 and 0.03 as compared to the C-C1 and C-C2 in defect-free
87 nanotube for all yy respectively (Figure 7.1c). Anothe r evidence of small disturbance due to the nitrogen atom can be seen from the functional behavior of the N-C1 and N-C2 bonds, which exhibits similar dependence on yy as the corresponding C-C bonds in the defect-free nanotube (Figure 7.1a). Removing a carbon atom from the carbon network creates a vacancy leaving three dangling bonds behind. Such configuration is not stable, and it undergoes reconstruction with the formation pentagon and nonagon alon g the axis of zigzag nanotube (Figure 3.6d). On the contrary to the previous ca ses (defect-free, Stone Wales, and nitrogen impurity) where the bond lengths show conti nuous behavior, vacancy has more dramatic dependence as a function of deformation yy (Figure 7.1d,e). In pa rticular, when the distance between the two low curvatures beco mes comparable with graphite interplanar distance, there are some rela tively sudden jumps and di ps in bond lengths for Flat deformation. We again observe that most changes in the bond le ngths happen on the high curvature. For instance, the bond lengths of C1-C2 and C6-C7 increase by 0.06 and 0.08 , respectively, while the C5-C8 decreases by 0.11 for the Flat deformation. As a result, the presence of mechanical defects causes the structural changes in nanotubes where nitrogen impurity creates th e lowest perturbation in nanotube network whiles the mono-vacancy the largest one. 7.3 Characteristic Energies of Deformation and Defect Formation The structural changes described above lead to changes in deformation and defect formation energies. We introduce the character istic energies of radial deformation and defect formations. The energy required to perform radial deform ation is defined as:
88 E E Eyy yy f (7.1) where yyE is the total energy for the deformed nanotube, and E is the total energy for the undeformed perfect nanotube. The defect form ation energies for the Stone Wales defect, nitrogen impurity, and vacancy at the high cu rvature of the radially deformed nanotube are calculated, respectively as: yy yy yy SW f SW E E E / / (7.2) C N N f N yy yy yyE E E / / (7.3) yy yy yy MV f MV E E E/ / (7.4) where MV N SW yyE, / are the total energies for the defo rmed nanotube with the appropriate defect (SW Stone Wales, N nitr ogen impurity, and MV mono-vacancy), N,C are the chemical potentials for free nitrogen and carbon atoms, respectively, and is the chemical potential of a carbon atom in the na notube (energy per C atom in the supercell). Applied radial deformation of up to yy<35% transforms the circular (8,0) nanotube to an elliptical-like one (Figure 3.4b). Further increase of yy>35% results in Peanut or Flat structures. As it can be seen on Figure 7.2a, the deformation energy becomes more distinctive between Peanut and Fl at deformations for higher radial stress yy>45%. What actually happens for high deform ation is that for the Peanut deformation only one row set on the top and the bottom of low curvatures are brought together. Consequently, there are no significant ch anges on the high cu rvature and elongation along a-axis (Figure 3.4c). The Flat deformation, however, has five rows of atoms from top and bottom on the low curvatures brought toge ther (Figure 3.4d). As a result, it leads
89 Figure 7.2 The energy of a) deformation a nd b) defect formation as a function of yy for Peanut and Flat deformations in (8,0) nanot ube. Cross-sectional vi ews of the deformed and defective (8,0) nanot ubes are also provided.
90 to major changes on high curvature and appropriate elongation along a-axis (Figure 3.4d). Meanwhile, radial deforma tion energy shows steeper increa se in Flat that in Peanut deformation. In addition, there is a repulsi on between two low curvatures of nanotube at yy>50%, where separation distance becomes sma ller than graphite interplanar distance. The repulsion is stronger for Flat rather than Peanut deformation because fewer atoms are brought together in Peanut as co mpared to Flat deformation. Figure 7.2b also shows varia tions in the defect forma tion energy as a function of the deformation. All energies f MV N SW yyE, / decrease for all types of defects as yy increases. Therefore, the defect formation occurs much easier in radially deformed carbon nanotubes than in the undeformed ones. This has b een shown for Si doping of carbon nanotubes . Analysis of defect fo rmation energies shows that the lowest amount of energy is required to produce Stone Wales defect, more energy is needed to make nitrogen impurity, and the most energy-expensive is the vacancy formation at high curvature of carbon nanotubes. There is a sm all difference between Peanut and Flat deformations for Stone Wales defect and nitrogen impurity, while this difference is significant for mono-vacancy. It relates to the f act that Stone Wales and nitrogen impurity creates small disturbance in nanotube st ructure for both deformations where the C-C bonds in mono-vacancy exhibit more drama tic changes for the Flat that Peanut deformations (Figure 7.1d,e). Thus, it is easi er to break the thr ee bonds to form a monovacancy after squashing between two wide surfaces as compared to the case of two narrow ones (Figure 3.4c-d). As a result, the higher the deformation yy is, the easier it is to make defects in car bon nanotube structure.
91 7.4 Electronic and Magnetic Properties of Carbon Nanotubes One of the main questions in development and design of new nanodevices is what electronic response one can expect from car bon nanotubes under the combined effect of radial deformation and different defects. We have already seen that the radial deformation can cause metal-semiconductor or semiconductor-metal transitions while the mechanical defects are able to have si milar effects, induce magnetism and change conductivity. 7.4.1 Band Gap Modulations To answer this question we analyze the modulation of the band gap Eg of defectfree and defective (8,0) nanotube under radial deformations yy (Figure 7.3). Initially, the defect-free (8,0) nanotube is a semiconductor with Eg=0.55 eV. Applying radial deformation leads to decreasing of Eg until its closure at yy =23% where semiconductormetal transition occurs. As we observed earlie r, this semiconductor-metal transition is reflected in disconti nuous change in the C-C1 and C-C2 bonds (Figure 7.1a). On the other hand, it corres ponds to strong hybridization on high cu rvature which causes structural and semiconductor-metal transformati ons in (8,0) nanotube. Interestingly, such transition arises before (8,0) nanotube under goes Peanut or Flat deformations. Further increase of radial deformation does not open second bang gap in the (8,0) nanotube. The presence of defects at high curvat ure tunes the value of band gap in nanotubes. For instance, Stone Wales defect increases Eg while Eg decreases for nitrogen impurity and mono-vacancy as compared to defect-free nanotube at yy=0%. Applying radial deformation to the defective nanotube has shown similar trends as in defect-free
92 Figure 7.3 Energy band gap Eg of perfect and defective (8 ,0) nanotubes as a function of radial deformation yy. Insert shows the semiconductor-met al transition in (8,0) nanotube in detail.
93 nanotubes (Figure 7.3). Moreover, the semi conductor-metal transition happens at the same degree of deformation yy=23% for nitrogen impurity and mono-vacancy. However, it occurs at yy=25% for Stone Wales defect. Furthe r increasing deformation opens the band gap in nanotube with mono-vacancy and Stone Wales defect again while nanotube with nitrogen impurity remains metallic for all deformations. In addition, the band gap can be modulated by Peanut or Flat deformatio ns at high deformation. Still, we can see that strong hybridization is the majo r factor in semiconductor-metal transition in defective nanotubes, but defect-deformation coup ling is significant for further opening of the band gap at high deformations. 7.4.2 Electronic Band Structures of Ra dially Deformed Carbon Nanotubes Without/With Stone Wales Defect To elucidate the mechanism of semiconduc tor-metal transformation in defect-free and defective nanotubes, we carry out comprehe nsive analysis of the electronic structure changes for various cases. Th e applied radial deformation removes the degeneracy in sub-bands of defect-free nanotub es (Figure 7.4a). Increasing yy causes the lowest conduction band and the highest valence band to move towards the Fermi level. They cross the Fermi level at the point different than for yy=23%. The further deformation leads to the shift of crossing point with EF towards the X point. For sufficiently large yy>50%, multiple crossings at the Fermi leve l from higher conduction and lower valence bands occur, which improve metallization of na notubes and increase the density of states at the Fermi level. The initial analysis reveals that all atoms of carbon network equally contribute to the energy band. However, this input changes with applied deformation.
94 Figure 7.4 The energy band structure of a) de fect-free (8,0) nanotube b) (8,0) nanotubes with Stone Wales defect at high curv ature site for several degrees of yy. The red level is associated with the Stone Wales defect.
95 The atoms from high curvature have the main contribution to the energy bands around the Fermi level. For yy>50% the atoms from the low curvature in the Peanut deformation (top and bottom rows) also contribute to the energy bands at the Fermi level. The band structure evolution of the defo rmed (8,0) nanotube with a Stone Wales defect shows similar behavior as compared to the defect-free one (Figure 7.4b). However, Stone Wales defect increases initial value of band gap and introduces flat defect level in the conduction band at E 2.9 eV (in red). Under appl ied radial deformations yy>0%, this flat level starts to move away from the Fermi level while increasing hybridizations brings the lowest conducti on and the highest valence bands closer together. Small band gap Eg0.01 eV opens up for the Peanut deformation at yy=55% and yy=65%. There is no such opening for Flat deformation, and the band gap remains closed. The analysis of conduc tion and valence bands reveal s that the lowest conduction bands consist of the orbital from defect atoms where hybridizations is significant. 7.4.3 Charge Distribution in Radially Deformed Carbon Nanotubes Without/With Stone Wales Defect The applied radial deformation induces nonuniform charge distribution on the nanotube surface (Figure 7.5). For a def ect-free nanotube, the charge accumulation appears on the high curvature. Even more ir regular allocation is caused by Peanut and Flat deformations (Figure 7.5a-c). Stone Wa les defect creates local disturbance in the nanotube structure, which reduces charge accumulation on the defect itself, except for the C6-C7 bond even though it is on the high curvatur e. Most of the charge is gathered around the atoms surrounding Stone Wale s defect (Figure 7.5d-f).
96 Figure 7.5 Total charge density plots with isosurface value is 0.0185 e/3 for a) defectfree (8,0) nanotube with yy=25%, b) Peanut structure with yy=65%, c) Flat structure with yy=65%, and d) (8,0) nanotube w ith Stone Wales defect with yy=25%, e) Peanut structure with yy=65%, f) Flat structure with yy=65%.
97 7.4.4 DOS of Radially Deformed Carbon Nanotubes With Nitrogen Impurity and Mono-Vacancy The effects of nitrogen impurity and mono-vacancy in radially deformed nanotubes have been investigated in terms of Density of States (DOS) for different yy. Since the nitrogen atom supplies one ex tra electron in nanotube, and mono-vacancy provides dangling bond, the calculations includ e spin-polarized effect to account for magnetic effect in nanotubes. The possibi lity of magnetism in nanotube has been predicted by other authors [99,200,201]. The density of states of nitrogen doped nanotubes is responsive to the concentrati on of impurities . In creasing the impurity concentration more than 1% provides a flat level in the band gap. In our calculation we do not observe any flat level because the nitrogen concentration is 0.78% (Figure 7.6a). As the radial deformation yy increases, the band gap decreases, and the conduction and valance bands move towards the Fermi level. For yy>45%, the DOS for Peanut and Flat deformation raises around the Fermi level wh ich suggests that the nanotube shows more metal-like behavior. The nitrogen impurity contributes significan tly to conduction band in the vicinity of the Fermi level which is partially occ upied. The highest valence band consists of contribution from atoms on the hi gh curvature. The extra nitrogen e induces spinpolarization effects (magnetism) in the nanotube electr onic structure at yy=0% and sharp features in the DOS for spin up around ~ 2.8 eV (Figure 7.6a). As yy increases, the spin-polarized effect is overtaken by hybridization. Eventu ally, it completely disappears at yy>25% except for at yy=65% for Flat deformati on. Thus, applied radial deformation provides the opportunity to cont rol not only electronic but also magnetic
98 Figure 7.6 a) Total Density of States for spin up and down carriers of (8,0) nanotube with a substitutional nitrogen impurity on its high curvature at different yy for Peanut and Flat structures. b) The magnetic moment of th e (8,0) nanotube with the nitrogen impurity as a function of yy. c) Total Density of States for spin up and down carriers of (8,0) nanotube with mono-vacancy defect on its high curvature at different yy for Peanut and Flat structures.
99 properties of nanotubes. To eluc idate this effect further, we investigate changes in local magnetic moment m under yy (Figyre 7.6b). For small deformations m increases with a maximum at yy=5%, and then it reduces to m=0 at yy=25% when the nanotube becomes almost a metal (yy=23%). However, complicated magne to-mechanical coupling produces large magnetic moment m=0.92 B (not shown on Figure 7.6b) at yy=65%. It gives evidence of new methods of tuning nanot ube properties and complex interrelation between defects and deformations. Finally, we consider a radially deform ed (8,0) nanotube with a mono-vacancy (Figure 7.6c). The presence of mono-vacancy is manifested in a sharp localized peak around 3.41 eV in the DOS conduction regi on for the spin up and down at yy=0%. This state mainly consists of contributi ons from the mono-vacancy atoms. As the deformation progresses, the peak intensity decreases and becomes broader, and peak location moves closer to the Fermi level due to hybridization from the nanotube atoms on high curvature along with the C atom s of mono-vacancy. Such process repeats again for yy>40%, where sharp peaks around the Fe rmi level occur and become wider and sharper with increasing yy. The major contribution to these peaks comes from dangling bond atom C1 and the C atoms of the pentagon ring of mono-vacancy (Figure 3.6d). DOS for spin up and down demonstrat es that there is no spin-polarized effect in radially deformed nanotubes with mono-vacancy for all yy except yy=65% in Peanut deformation where a small magnetic moment m=0.04 B has been found. These results are in partial agreement with previously re ported findings that zigzag nanotubes with mono-vacancy do not show magnetic response for given orientation of defect . However, the appearance of magnetism at yy=65% can originate from several reasons,
100 such as uncoordinated C atom w ith a localized unpaired spin, concentration of vacancies, nanotube chirality, sp3 pyramidization, and related bond length changes, as well as vacancy location with respect to the nanotube structure [99,126,202]. Another explanation of the magnetism in radially deformed nanotubes with mono-vacancy is the magnetic flat-band theory together with hybridization [203,204]. The presence of flat bands around the Fermi level leads to increa sed electron-electron in teraction and results in the occurrence of magnetic polarization whic h is common for edge states in zigzag nanoribbons [203,204]. We can consider a monovacancy on the high curvature and large deformations such edge states. 7.4.5 Spin Density Distribut ion in Radially Deformed Carbon Nanotubes With Nitrogen Impurity and Mono-Vacancy Further insight in the nanotube magnetis m can be gained by exploring the spin density distribution for various deformations yy (Figure 7.7). Initially extra nitrogen e is localized on the defect site but increasing defo rmation obliterates this localization. As an example, the unpaired e is much more delocalized at yy=20% in comparison to the yy=10% case, where the e spin density localized around the impurity site is much more intense. Similar behavior has been reported for semiconducting nanotubes, where the nitrogen e is localized around the impurity, while for metallic nanotube it is completely delocalized . However, the e spin localization around the impurity appears again at yy=65% in Flat deformation (m=0.92 B). Moreover, similar localization of the spin density happens around the dangling bond in the nanotube with mono-vacancy yy=65%.
101 Figure 7.7 Spin density isosurface plots for a) (8,0) nanotube with nitrogen impurity for yy=10% at 0.004 B/3, b) (8,0) nanotube with nitrogen impurity for yy=20% at 0.004 B/3, c) Flat (8,0) nanotube with nitrogen impurity for yy=65% at 0.0122 B/3, d) Peanut (8,0) nanotube with mono-vacancy for yy=65% at 0.018 B/3.
102 CHAPTER 8 RADIALLY DEFORMED CA RBON NANOTUBES UNDER EXTERNAL ELECTRIC FIELDS In addition to the tuned properties of carbon nanotubes by radial deformation together with mechanical defects, they can also be altered by the simultaneous application of radial deformation and an ap plied external electric field. Since the application of the electric fiel d in nanotubes is crucial for carbon nanotube based devices such as field-effect-transistors , se nsors , and electron emitters , the combined effect of electric field with other external fact ors can lead even to richer diversity of nanodevices. Such opportunity ha s been demonstrated in nanotubes under the simultaneous effect of electric field and vacancy defects [205 ,206]. This effect leads to the tunable decrease of the nanotube conductivity, which is a crucial element of nanotube spintronics and electr ical switching devices Thus, to extend such opportunities, we present the theoretical calculation of band gap modulation, band st ructure changes, and charge density redistributions in nanotubes under combined effect of radial deformation and electric field in the following sections. 8.1 Studied Systems To study the changes in carbon nanotubes under combined effect of radial deformation and an external electric field, we consider zigzag semiconducting (8,0) and
103 metallic (9,0) single wall carbon nanotubes. The constructed supercell consists of one translational unit cell along z-axis with length of 4.26 fo r the (8,0) and 4.25 for the (9,0) nanotubes after relaxation. The external applied electric field is generated by a sawtooth electric potential  in th ree different directions such as Ex and Ey along x and y axes, respectively, and E45 0 along 450 from the xaxis. 8.2 Deformation Energies We have already seen how the deform ation energy changes in (8,0) nanotubes under radial deformation (Figure 7.2). Using the Eq. (7.1) we calcu late the deformation energy of (9,0) nanotubes as well. Figure 8.1 shows these deformation energy for (8,0) and (9,0) carbon nanotubes as a function of yy. The deformation energy for both nanotubes has non-linear dependence, and it rises with the increasing yy showing that higher deformation requires more energy to overcome strain energy on the high curvature. On the other hand, the deformati on energy for (8,0) nanotube is steeper than for (9,0) due to the larger curvature effect in (8,0) nanotube as compared to the (9,0) one since the (8,0) nanotubes has smaller diameter than (9,0) nanotube. This difference in the deformation energy becomes more evident as the deformation yy progress (Figure 8.1).
104 Figure 8.1 The deformation energy as a function of radial deformation yy for one unit cell of (8,0) and (9,0) nanotubes.
105 8.3 Band Gap Modulations Furthermore, we explore the electronic pr operties of radially deformed nanotubes in an external radial electr ic field. We focus on the modul ation of the band gap by an external electric field in na notubes with various degrees of deformation. As we have already seen from earlier discussion, appl ying radial deformation causes the metal semiconductor transition in semiconductor nanotubes and causes opening and closing of a band gap in metallic nanotubes because of the hybridization from the high curvature regions. Figure 8.2a,b show that th ese dependences in semiconductor (8,0) and semi-metal (9,0) carbon nanotubes as a function of yy. The same behavior can be achieved by the application of an external electric field to undeformed (8,0) and (9,0) nanotubes. Eg dependences in Figure 8.2c,d and Figure 8.2a,b look similar. However, the semiconductor-metal transition happens faster for the (8,0) nanotube over a shorter field strength interval as compared to the (9,0) nanotube. In this case, Eg for (8,0) nanotube is closed at E=0.4 V/ while for (9,0) tube the closure occurs at E=0.8 V/. Therefore, we can obtain the same electronic characteristics of nanotubes from radial deformation and the applied external electric field. In additi on, it provides the flex ibility of controlling nanotube properties at different circumstances.
106 Figure 8.2 Eg as a function of yy for a) (8,0) and b) (9,0) nanotubes; Eg as a function of E for c) (8,0) at yy=0%; d) (9,0) at yy=0%; e) (8,0) at yy=10%; f) (9,0) at yy=10%; g) (8,0) at yy=20%; h) (9,0) at yy=20%; i) (8,0) at yy=25%; j) (9,0) at yy=25%. The insert indicates electric field directions as follows: Ex is along xaxis, Eyalong yaxis, and E45 0 along 450 from xaxis in xy plane.
107 The symmetry of the nanotube is decrea sed under applied radial deformation. Consequently, the x and y directions of the cross secti on are not equivalent any more (Figure 3.4b). It has been shown before that electronic structure of the defective nanotube depends on the orientation of ex ternal electric field . To understand the electric field effect on our systems, we analyze the Eg evolution of the deformed (8,0) and (9,0) nanotubes under E along x, yaxes and the 450 direction in the xy plane (insert in Figure 8.2c,d). Radially deformed (8,0) nanotube at yy=10% still has a relatively large band gap Eg= 0.37 eV, which decreases under applie d electric field an d vanishes at E~0.45 V/ for all three directions. Each Eg dependence has distinct profile for E<0.4 V/. Eg initially increases and then starts to decrease for ydirection. However, it sharply decreases for x direction, and it shows intermediate, almost linear decline for 450 direction (Figure 8.2e). At yy=20%, the applied electric fi eld has a small effect on Eg over relatively long range of electric fiel d strength for all direct ions until it closes Eg at E~0.8 V/ (Figure 8.2g). The (8,0) nanotube becomes metallic at yy=25%, but applied electric field opens Eg with its maximum at E~0.2 V/ for all directions. The largest Eg is for the electric field along ydirection and the lowest for xdirection (Figure 8.2i). On the other hand, the radially deformed (9,0) nanotube at yy=10% has the largest Eg=0.08 eV, which continues to increase under applied electric field in all directions until E~0.60.7 V/, where Eg is sharply closed (Figure 8.2f). This behavior is similar for Eg in the (8,0) nanotube at yy=20%. More drastic behavi or of the band gap is observed at yy=20% for electric field along ydirection. The original value of the band gap increases by ~ 4 times at E=0.7 V/ which is followed by its rapid closure at E=0.9
108 V/. At the same time, the weak electric field E along x and 450 directions closes the band gap. The same dependence has been found at yy=25%. The band gap reaches maximum for E=0.8 V/ while there is no effect of applied electric fi eld on the band gap along xdirection. 8.4 Electronic Band Structure Changes Better understanding of band gap modulati ons comes from the analysis of the electronic band structure for (8,0) and (9,0) nanotubes at yy=10% and yy=25% respectively, for several orientations (Figur e 8.3). The electronic level of conduction and valance bands are associated with the quantum angular momentum J which defines the rule for admixture of allowed states . It has been shown that electronic states can mix according to the selection rule J=0, 2 for radial deformation of nanotubes . This admixture and increasing hybridization on high curv ature bring the lowest conduction and the highest valance bands cl oser to the Fermi level. However, the applied electric field mixes the el ectronic states in accordance with J =1 and also decreases the values of the band gap to its closure at EF [208,209]. The combination of radial deformation and external electric field causes the band structure modulation according to J=0, 1, 2 selection rule and hybridization on the high curvature of nanotubes. As we have seen from the band ga p behavior, the strengt h and orientation of applied electric field with respect to the na notube cross section are crucial for electronic
109 Figure 8.3 Energy band structure evolu tions for a) (8,0) nanotube at yy=10% and b) (9,0) nanotubes at yy=25% and electric field strength E=0.1, 0.3, 0.5 V/ along x, yaxis, and 450 from xaxis in xy plane, respectively. The Fermi level is shown as a dashed line.
110 properties of nanotubes. This ha s been reflected in the band structures for (8,0) and (9,0) nanotubes (Figure 8.3). Most changes happe n with the lowest unoccupied single degenerate conduction state for both (8,0) and (9,0) nanotubes. Therefore, application of electric field along xdirection (through the high curvat ure) has revealed the largest hybridization on the high curvat ure. Thus, depending on the in terplay between the radial deformation and electric field strength/orien tation, the electronic properties of nanotubes can reveal semiconductor-metal or metal semiconductor transformation (Figure 8.3). 8.5 Charge Density Distributions To gain further insight in the role of the applied electric field direction with respect to the radial CNT cross section, we calculate the total charge density difference for (8,0) and (9,0) CNTs as, (r) = (r) (r) (9.1) where (r) is the charge density of the radi ally deformed nanotubes for specific E, and (r) is the charge density of the deformed nanotube. The plots for (r) are given in Figure 8.4. It shows that the charge redistribution depends strongl y on the direction of the electric field. For the semiconducting (8,0) nanotube the applied Ey field causes charge accumulation inside and charge depletion outside around the nanotubes highest curvature sides Figure 8.4a. The charge redistribution on the flat regions is such that the charge accumulation and depletion outside and inside th e nanotube is almost equal. For Ex, however, there is well pronounced charge depletion at the hi ghest curvature sides, while the charge
111 Figure 8.4 The total charge density diffe rence plots for (8,0), and (9,0) CNT at E = 0.1 V/. a) Applied el ectric field along y axis for isosurface value 0.0022 e/3; and b) along x axis for isosurface value 0.008 e/3 for (8,0) CNT. c) Applied electric field along y axis; and d) along x axis for isosurface value 0.009 e/3 for (9,0) CNT. The electron accumulation/deplet ion regions are displayed in blue ()/red (+).
112 accumulation is inside and outside of the inte rmediate curvature CNT sides Figure 8.4b. For the metallic radially defo rmed (9,0) CNT, the charge redistribution is mainly outside the nanotube Figure 8.4c,d, showing the scr eening of the electric field due to the metallic nature of the system. The Ey field causes charge accu mulation and depletion on the two opposite flat CNT sides, while the Ex field is responsible for the charge accumulation and depletion on the two oppos ite highest curvature CNT sides. The results from our calcula tions show that the appli cation of radial squashing and applied electric field at the same ti me can provide a vari ety of CNT properties modulations. If semiconducting me tallic tubes are slightly deformed, great variations in the electronic structure can be induced by exte rnal electric fields at different radial directions. For example, we showed that by changing the direction of the field from Ey to Ex for the slightly deformed (8,0) CNT, the energy gap can be reduced ~ 1.5 times for the same field strength Figure 8.2e). At the same time, for metallic CNT smaller fields along y axis can increase Eg significantly, while fields along x axis affect their band structure very little or not at all Figure 8.2i,h,j Moreover, for some CNT with relatively small Eg, smaller electric fields tend to not aff ect the energy gap mu ch regardless of the applied electric field orientation Figure 8.2f while for other CNT this is not the case Figure 8.2g. Thus, the combination of deformation, el ectric field strength, and direction may provide additional flexibility in accommodati ng greater capabilities and reduce the need of very large deformations or very large applied external el ectric fields for CNT property modulations.
113 CHAPTER 9 CONCLUSIONS In this dissertation, we provided a comprehensive study of carbon nanotube properties under several external factors such as adsorption, combined effect of radial deformation and mechanical defects and comb ined effect of radi al deformation and external electric field. In particular, the emphasis was given to physisorption mechanisms, which were described by Van der Waals dispersion forces, between carbon nanotubes and the other graphitic structures (nanotubes and nanoribbon s), and biological molecules (DNA bases and their radicals). In addition, this work showed new opportunities in the modulation of electronic and magnetic na notube properties under two external factors for their expanded applications in current and new nanodevices. 9.1 Many-Body Van der Waals Interaction Forces By applying the coupled dipole met hod, which takes into account many-body effects and discrete natu re of the materials, we were ab le to calculate the Van der Waals interactions between carbon nanotubes and othe r graphitic nanostructures. Our results demonstrated a strong relation between the inhe rently collective nature of Van der Waals forces together with the anisotropy, geometry and mutual orientation of these materials. We found that the total many-body Van der Waal s energy was several times stronger for planar systems because of the larger aspect ratio as compared to the Van der Waals
114 energy of cylindrically circular nanotubes wi th a similar size. The calculation results showed that the description of the inter actions between finite size nanotubes and nanoribbons by well-known pairwise approximati on at distances comparable to the size of the structures was not sufficient in a ll studied cases. Even inclusion of higher interaction orders (three-body and four-body) to enhance pairwise approach did not provide any improvements. Thus, all many-body in teractions were nece ssary in order to capture the correct magnitude and alignment of adsorbent with carbon nanotube surface. 9.2 Adsorption of DNA Bases and Their Radicals We used the DFT method to understand th e important factors in the adsorption process of DNA bases on carbon nanotubes. Our re sults revealed that the adsorption of the studied DNA bases and their radicals wa s of physisorption nature. The energies, equilibrium distances, and electronic structur e showed that the ma in contribution comes from the noncovalent interac tion between the delocalized orbitals from the molecules and the nanotubes. We also saw that even in the cases of the radicals which have larger electron charge transfer abiliti es (due to the removal of a hydrogen atom) than the full molecules, the interaction was still physisorp tion. The structural analysis of nanotubes did not show significant deformations in their geometry and the molecules upon adsorption. The exception was found for the thymine-radical which has some bending in its shape before adsorption, but after the adsorption it became flat. For the adenine and thymine cases, there wa s little charge mi gration between the molecules and the nanotubes. Small charge re distribution was found within the molecules themselves. Analysis of the molecular orbi tals showed that they are not changed
115 significantly. This analysis revealed the existence of a repulsive Pauli barrier between the orbitals of the molecule and the na notube leading to minimization of the interaction and determining the equilibrium position of the molecules on the nanotube surfaces in a similar manner as the AB stacking in graphite. A detailed examination of the contributions to the total energy also showed that the attraction was due mainly to the exchange-correlation interaction which overcomes the repulsive Pauli barrier. For the radicals, there was some charge transfer to the nanotube although it was still small. Larger charge redistribution within the radi cals was seen as compared to the whole molecules. The peaks that appear in the total DOS (right under the band gap for the Tradical/CNT (8,0) and right at the Fermi leve l for the A-radical/CNT (6,6) were mainly due to the molecular orbitals with very lit tle admixture from th e nanotube bands. Again the attraction was mainly from the exchange-c orrelation contribution to the total energy. Our study provided detailed information of the equilibrium distances, adsorption energies, bond changes, charge transfer, and density of stat es characteristics about the interaction of the smallest structure building blocks of DNA. They suggested that DNA bases and their radicals could be easily immobilized on nanotube surfaces; DNA bases and their radicals could also be used for noncovalent functionalization of carbon nanotubes. 9.3 Combined Effect of Radial De formation and Mechanical Defects Furthermore, we discussed electronic and magnetic carbon nanotube properties under combined effect of radial deform ation and mechanical defects applied simultaneously. Our analysis was based on DF T calculations. We showed that defects,
116 such as Stone Wales, nitrogen impurity, a nd mono-vacancy, create a local distortion in the carbon network. This dist urbance was lowest for nitrogen, larger for Stone Wales defect, and largest for mono-vacancy. The analys is of the characteristic energies revealed that it takes more energy to deform a sma ller diameter nanotube in comparison to larger one. The larger degree of radial deformati on made it easier to introduce a defect on the high curvature of the nanotube The Stone Wales defect re quired the lowest formation energy, while it was higher for nitrogen im purity, and it was the largest for monovacancy. We found that applied radial defo rmation causes semiconduc tor-metal transition in defective nanotubes at approximately the same value of applied strain (yy=2325%) regardless of the type of defect. The electronic structure calculations showed that the main factors responsible for semiconductor-metal transition in defective nanotubes were the hybridization from the high curvature and the presence of defect states around the Fermi level region. We observed that interplay between radial deform ation and type of defects such as Stone Wales defect and mono-vacancy can open the band gap again, after it had been closed due to the same factors. However, there wa s not such opening for the nitrogen impurity and defect-free nanotubes regardless of the type of deformation. On the other hand, mutual effect of radial de formation and defects led to magneto-mechanical coupling in the nanotube structure because of the presen ce of extra an electron from the nitrogen atom and dangling bond from the mono-vacan cy. The magnetic properties of nanotubes could be controlled by the degree of radi al deformation. The origin of magnetic properties of nanotubes was also discussed.
117 9.4 Combined Effect of Radial Defo rmation and External Electric Fields In addition, the combined effect of radial deformation and external electric field was given in terms of band gap modulations, electronic band structure changes, strength of applied electric field, electr ic field orientation, and select ion rules. We showed various responses in semiconductor and metallic car bon nanotubes. The electronic structure was sensitive to the strength and electric field or ientation with respect to the cross section of nanotubes. The interplay between selection rules of mixing diffe rent states in the radial deformation and applied el ectric field along with hybridization cont rolled and tuned the band gap of nanotubes. Finally, we di scussed semiconductor-metal transition in semiconducting nanotubes and opening of the band gap in metallic nanotubes.
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137 ABOUT THE AUTHOR Yaroslav Shtogun was born in Ivankivtsi village, Krements district, Ternopil region, Ukraine. He received his Bachelor of Science in Physics at the National Taras Shevchenko University of Kyiv in 2002, and his Master of Science at the National Taras Shevchenko University of Kyiv in 2004. Afte rwards, Yaroslav entered the Ph.D program in Applied Physics at the University of South Florida in Spring 2005. His research work focuses on the fiel d of nanoscience and nanomaterials, specifically, on properties of carbon nanostructu res under external fact ors. The results of his studies have been published in high impact journals such as Journal of Physical Chemistry C, Carbon, Journal of Applied P hysics, and Nano Letters. Yaroslav also has various conference proceedings and has given presentations at important conferences. He was invited to write chapters to the book Carbon Nanotubes published by IN-TECH in Fabruary 2010. He was certifie d in Materials Science and Engineering and received Summer 2007 THARP Scholarship Yaroslav is also a re viewer of ACS Nano, the Journal of Physical Chemistry, the Journal of Applied Physics, Chemical Physica, and Physica E. He did an internship at the Tampa General Hospital in Department of Information Technology and Standards. He wa s trained in different technologies of datamining, analytic analysis, software devel opment, and assisted in project management groups in various duties.
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Properties of carbon nanotubes under external factors :
b adsorption, mechanical deformations, defects, and external electric fields
h [electronic resource] /
by Yaroslav Shtogun.
[Tampa, Fla] :
University of South Florida,
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Dissertation (Ph.D.)--University of South Florida, 2010.
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ABSTRACT: Carbon nanotubes have unique electronic, optical, mechanical, and transport properties which make them an important element of nanoscience and nanotechnology. However, successful application and integration of carbon nanotubes into new nanodevices requires fundamental understanding of their property changes under the influence of many external factors. This dissertation presents qualitative and quantitative theoretical understanding of property changes, while carbon nanotubes are exposed to the deformations, defects, external electric fields, and adsorption. Adsorption mechanisms due to Van der Waals dispersion forces are analyzed first for the interactions of graphitic materials and biological molecules with carbon nanotubes. In particular, the calculations are performed for the carbon nanotubes and graphene nanoribbons, DNA bases, and their radicals on the surface of carbon nanotubes in terms of binding energies, structural changes, and electronic properties alterations. The results have shown the importance of many-body effects and discrete nature of system, which are commonly neglected in many calculations for Van der Waals forces in the nanotube interactions with other materials at nanoscale. Then, the effect of the simultaneous application of two external factors, such as radial deformation and different defects (a Stone Wales, nitrogen impurity, and mono-vacancy) on properties of carbon nanotubes is studied. The results reveal significant changes in mechanical, electrical, and magnetic characteristics of nanotubes. The complicated interplay between radial deformation and different kinds of defects leads to the appearance of magnetism in carbon nanotubes which does not exist in perfect ones. Moreover, the combined effect of radial deformation and external electric fields on their electronic properties is shown for the first time. As a result, metal-semiconductor or semiconductor-metal transitions occur and are strongly correlated with the strength and direction of external electric field and the degree of radial deformations.
Advisor: Lilia M. Woods, Ph.D.
Van der Waals interactions
Electronic band structure
t USF Electronic Theses and Dissertations.