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