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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. - PowerPoint PPT Presentation
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Majed AbdELSalam Nashaat, Department Of Physics – Cairo University
Abbas Hussein Abbas, Department Of Physics – Cairo University
Loay Elalfy AbdelHafiz, Center Of Nanotechnology – Nile University
Team work
Supervisor
V.L. Katkov
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia.
BLTPBLTP
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
C-
--
-
Carbon Graphene
4 valence electrons
1 pz orbital
3 sp2 orbitals
Hexagonal lattice;1 pz orbital at each site
Step 1: Bloch sum (discrete Fourier Transform) of each localized wave function.
Step 2: Write wave function as linear combination of Bloch sums.
11 12
21 22
H HH k
H H
Step 3: Expand the Hamiltonian in terms of the Bloch sums. Eg. For two atoms per unit cell
11 11 12 12
21 21 22 22
B
E V k V k V kH k
V k V k E V k
3NN
21V k
11 11 12
21 22 22
B
E V k V kH k
V k E V k
2NN
22V k
Nearest neighbors only
Nearest + Distant neighbors
Tight-binding Models
11E22E
11 12
21 22
B
E V kH k
V k E
NN
21V k
Interaction sub-matrices
Dresselhause methodTight binding method
2a
1a
Two identical atoms in unit cell: A B
Band Structure of Graphene
Tight-binding model: P. R. Wallace, (1947) (nearest neighbor overlap = γ0)
2cos4
2cos
2
3cos41)( 2
0
akakakEE yyxF k
For A-A bilayer
For A-B bilayer
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
2cos4
2cos
2
3cos41)( 2
0
akakakEE yyxF k
Dresselhause method Tight binding method
For 10 - 10
For 5 - 5
1st brillouin zone
1st brillouin zone2ndzone 1st bril zone 2ndzone
2ndzone 1st bril zone 2ndzone
F0R 10-0
F0R 9-0
F0R 11-0
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