Tight Binding Method for Calculating Band Structure Of Carbon Nanostructures

1. Tight Binding Method for Calculating Band Structure Of Carbon Nanostructures

2. 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

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

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

5. Outlines • Tight – binding method • Grapheneband structure • Bilayer graphene • Carbon nanotube • Graphene Nano ribbon

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

7. 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

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

9. Band structure calculation Dresselhause method Tight binding 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

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

11. Graphene & Graphite

12. Bilayer graphene

13. For A-A bilayer

14. For A-B bilayer

15. A tunable graphenebandgap 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

16. Carbon nanotube

17. Band structure for carbon nanotube Dresselhause method Tight binding method

18. Band structure for armchair carbon nanotube For 10 - 10 1stbrillouin zone 2ndzone 1stbrilzone 2ndzone For 5 - 5 1stbrillouin zone 2ndzone 1stbrilzone 2ndzone

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

20. GrapheneNanoribbon • Narrow rectangle made from graphenesheet , 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 redlines are the zigzag or armchair chains that are used to determine Nz or Na respectively.

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

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

23. Conclusions

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