Tight Binding Method for Carbon Nanotubes - Essay Example

Page 12 October, 2006 Tight Binding Method For Carbon Nanotubes Discovery of nanotubes in the year 1991 by S. Iijima, opened a completely new field in the science of carbon related materials, especially carbon nanotubes, single-wall and multiwall structures. Nanotubes are cylindrical structures based on the hexagonal lattice of carbon atoms that forms crystalline graphite. They can be thought of as a sheet of graphite rolled into a cylinder. Nanotubes are of three types armchair, zigzag and chiral nanotubes. Besides having a single cylindrical wall (SWNTs), nanotubes can have multiple walls (MWNTs).Single-wall nanotubes can be either conducting or semi-conducting, depending on their structure. Carbon nanotubes are long, thin cylinders of carbon and have a very broad range of electronic, thermal, and structural properties that change depending on the different kinds of nanotube. The chiral vector of the nanotube, B'= nR1 + mR2 where R1 and R2 are unit vectors in the two-dimensional hexagonal lattice, and n and m are integers. Another important parameter is the chiral angle, which is the angle between Band R1. Chiral angle , = arc tan (3m/[2n + m]) and Diameter D = a3 (n2 + nm + m2)/ p ,Where, ac is the distance between neighboring carbon atoms in the flat sheet. The different values of n and m lead to different types of nanotube. They are armchair, zigzag and chiral nanotubes. Armchair nanotubes are Page 2 formed when n = m and the chiral angle is 30. Zigzag nanotubes are formed when either n =0 or m==0 and the chiral angle is 0. Other nanotubes, with chiral angles between 0 and 30, are known as chiral nanotubes. The properties of nanotubes are determined by their diameter and chiral angle, both of which depend on n and m. The electronic characteristics of the nanotubes have been done by numerical band structure, the structure of the chemical bonds. is given by the local spatial structure of the orbital. The electronic structure of the nanotube fragments are calculated by SCF-MO-LCAOVmethods. In this method, only valence electrons are taken into account and the three- and four-center integrals are omitted and the repulsion of lone electron pairs can be explained. The SCF convergence criterion was 10-8for total-energy changes and 10-5 for charge-density changes between two subsequent cycles. Band structure calculations of [n, 0] (n = 6, 7, 8, 9)tubes were performed using the tight-binding Hamiltonian, with a universal set of first and second nearest-neighbor hopping integrals that reproduce various carbon structures, including graphite. The 2s, 2px, 2py, 2pz, and s* orbital of each carbon atom are used as the basis set for expressing the tight binding model. The Hamiltonian matrix elements and related parameters are obtained by adjusting the model to fit photoemission band-structure data. The (6, 0) carbon tube seems to have the lowest diameter and are thermodynamically unstable. The bonds at the ends of the nanotube fragments get saturated by hydrogen atoms. The structural unit of the tube is the distorted carbon hexagon. All c-c bonds were assumed to be of the same length: 1.4 . Page 3 The distance between third-neighbor carbon atoms along the tube circumference is 2.39 . The point group symmetry of the (6, 0) nanotube fragment is determined by the number N of carbon hexagons along the tube axis. There is a difference between heat of formation of the nanotube fragments, caused by the boundary atoms affect, strongly at the central part of the nanotube fragment. In the above Figure, the dispersion curves of the (n, 0) tubes with n = 6... 11 are shown. This tube family splits into three groups. The (3n, 0) tubes have vanishing energy gaps. The gap increases in (3n + 1, 0) and in (3n + 2, 0) tubes. Consequently, (6, 0) and (9, 0) tubes will likely show metallic conductivity, similar to graph. In graphite, orbital are represented in carbon nanotubes, the radial orbital are analogous to the lone orbital of graphite .This changes the character of the frontier orbital of carbon nanotubes in comparison with those of graphite, and in particular for nanotubes the changed s-hybridization resulted in the different the reactivity of inner and outer surfaces of single tubes. The charge-density distribution of frontier orbital in Page 4 (n, 0) nanotube fragments shifted from the inside to the outside of the tube. , resulting in nonzero electric field in the radial direction .These dispersion relations show how the electronic energy in three types of nanotube varies with wave vector. Each curve corresponds to a single quantum sub band. Fermi level is at E = 0: states of lower energy are fully occupied, while higher energy states are completely empty. In an armchair (5,5) nanotube (left) and a zigzag (9, 0) nanotube (middle), an infinitesimally small amount of energy is needed to excite an electron into an empty excited state, and such nanotubes are metallic. For a zigzag (10, 0) nanotube (right) there is a finite band gap between the occupied and empty states, so this nanotube is a semiconductor. A small increase in diameter has a major impact on the conduction properties of carbon nanotubes. The density of electronic states as a function of energy has been calculated for a variety of nanotubes. As an example, consider the density of states for metallic Page 5 (8, 8), (9, 9), (10, 10) and (11, 11) armchair nanotubes .The graph below shows energy on X-axis and Density on Y- axis. While conventional metals have a smooth density of states, these nanotubes are characterized by a number of singularities, where each peak corresponds to a single quantum sub band. .At the Fermi Energy, the highest occupied energy level, the density of states is finite for a metallic tube and zero for a semi-conducting tube. As energy is increased, sharp peaks in the density of states, called Van Hove singularities, appear and specific energy levels, where, E is the energy difference between occupied and unoccupied states, especially near the peaks. In addition, the electronic mean free path exhibits a downscaling law with a lower dependence on the coverage density of grafted molecules than for conventional substitutional doping or homogeneous disorder. Page 6 Works cited Bulusheva L G, Okotrub A V, Romanov D A and Tomanek D Electronic. "Structure of (n, 0) Zigzag Carbon Nanotubes: Cluster and Crystal." pa.msu.edu. 12November1997. 10October2006 [http://www.pa.msu.edu/cmp/csc/eprint/DT114.pdf]. "Carbon nanotubes." physicsweb.org .01January 1998.11 October 2006 [http://physicsweb.org/articles/world/11/1/9/1#further]. "Carbon nanotube science and technology."A Carbon Nanotube page: personal.rdg.ac.uk 11 October 2006 [http://www.personal.rdg.ac.uk/scsharip/tubes.htm]. "Discrete Variational Method for the Energy-Band Problem with General Crystal Potentials."APS Physics. Physical Review Online Archieve.01 May 1970. 10 October 2006[http://prola.aps.org/abstract/PRB/v2/i8/p2887_1]. "Tight-binding modeling of materials." iop.org: Electronic Journals. 28 April 1997. 10 October 2006 [http://www.iop.org/EJ/abstract/0034-4885/60/12/001] ] SaitoR, DresselhausG, and DresselhausM S . "Physical Properties of Carbon Nanotubes" World Scientific Publishing Company Read More