Transcript Tight binding method for calculating band structure of carbon

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