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. BLTP 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 Carbon - - C 4 valence electrons 1 pz orbital 3 sp2 orbitals Graphene Hexagonal lattice; 1 pz orbital at each site 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 H11 H k H 21 H12 H 22 Interaction Range Tight-binding Models Nearest + Distant neighbors Nearest neighbors only HB E11 k V k 21 HB V12 k E22 HB NN Interaction sub-matrices E22 E11 E11 V11 k k V k V k 21 21 V21 k E11 V11 k k V k 21 V22 k V21 k V k V12 k E22 2NN 22 V12 k V12 k E22 V22 k 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 Two identical atoms in unit cell: A a1 B a2 Band Structure of Graphene Tight-binding model: P. R. Wallace, (1947) (nearest neighbor overlap = γ0) E (k ) EF 0 1 4 cos k a k a 3k x a cos y 4 cos2 y 2 2 2 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 E (k ) EF 0 1 4 cos Tight binding method k a k a 3k x a cos y 4 cos2 y 2 2 2 Band structure for armchair carbon nanotube For 10 - 10 1st brillouin zone 2ndzone 1st bril zone 2ndzone For 5 - 5 2ndzone 1st bril zone 2ndzone 1st brillouin zone Band structure for zigzag carbon nanotube F0R 9-0 F0R 10-0 F0R 11-0 Graphene Nanoribbon 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 • 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