Tight Binding Method for Calculating Band Structure Of Carbon Nanostructures

Team work Majed AbdELSalam Nashaat, Department Of Physics – Cairo University Abbas Hussein Abbas, Department Of Physics – Cairo University Loay Elalfy AbdelHafiz, Center Of Nanotechnology – Nile University

Supervisor V.L. Katkov Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia.

Aim Of Practice Calculate band structure for different carbon Nanostructure and investigate their characteristics ( metallic – semiconductor ) Using tight binding method and Dresselhause method For Graphene – bilayer ( A-A & A-B) Carbon nanotube – graphene Nano ribbon The effect of electric field on Gb ( A-A & A-B)

Outlines Tight – binding method Graphene band structure Bilayer graphene Carbon nanotube Graphene Nano ribbon infinite thermal conductivity :meaning that no temperature differential can exist between two superfluids or two parts of the same superfluid.

Carbon Graphene C - Hexagonal lattice; 1 pz orbital at each site 4 valence electrons 1 pz orbital 3 sp2 orbitals

Tight – binding method   Step 1: Bloch sum (discrete Fourier Transform) of each localized wave function.   Step 2: Write wave function as linear combination of Bloch sums. Step 3: Expand the Hamiltonian in terms of the Bloch sums. Eg. For two atoms per unit cell

Interaction Range 2NN 3NN NN Tight-binding Models Nearest neighbors only Nearest + Distant neighbors Tight-binding Models 2NN NN Interaction sub-matrices 3NN

Band structure calculation Tight binding method Dresselhause method 1- Eigen value equ. In matrix form:   2- Non trivial sol. is given by:   3- Solving the Det w.r.t 𝜀 we get the band structure

Graphene Band Structure of Graphene Two identical atoms in unit cell: A B Band Structure of Graphene Tight-binding model: P. R. Wallace, (1947) (nearest neighbor overlap = γ0)

Graphene & Graphite

Bilayer graphene

For A-A bilayer

For A-B bilayer

A tunable graphene bandgap opens the way to nanoelectronics and nanophotonics Wang: Department of Physics at the University of California at Berkeley Generate a bandgap in bilayer graphene that can be precisely controlled from 0 to 250 milli-electron volts (250 meV, or .25 eV). For A-A bilayer For A-B bilayer

Carbon nanotube

Band structure for carbon nanotube Dresselhause method Tight binding method  

Band structure for armchair carbon nanotube 1st brillouin zone 2ndzone 1st bril zone 2ndzone For 5 - 5 1st brillouin zone 2ndzone 1st bril zone 2ndzone

Band structure for zigzag carbon nanotube F0R 9-0 F0R 10-0 F0R 11-0

Graphene Nanoribbon a) Nz: no zigzag chains (Nz-zGNR) Narrow rectangle made from graphene sheet , Has width in order of nm up to tens of nm. Considered as quasi-1D nanomaterials. Has metallic or semiconducting character. a) Nz: no zigzag chains (Nz-zGNR) b) Na :no of armchair chains (Na-aGNR) width of the GNRs can be expressed in terms of the no of lateral chains The red lines are the zigzag or armchair chains that are used to determine Nz or Na respectively.

For A-A bilayer ribbon with ү1 = 0 For A-A bilayer ribbon with ү1 = .4 eV

For A-A bilayer ribbon with doped Hydrogen atom Eg=0.3 eV

Conclusions

Refrences Tight binding approach to incorporate accurate bandstructure in nanoscale device simulation (Anisur Rahman and Mark Lundstrom School of Electrical and Computer Engineering Purdue University, West Lafayette) Carbon Nanotube and Graphene Device Physics, H.-S. P H I L I P WONG