This document describes the tight-binding method for calculating the energy diagram of nanoelectronic systems. It introduces the tight-binding method and its application to calculating the energy diagrams of polyacetylene, single-layer graphene, bilayer graphene, and multi-layer graphene. It also discusses using the tight-binding method to calculate the energy diagrams of two and four layer graphene in a constant electric field and single-layer graphene in a modulated electric field. The document provides the theoretical framework and mathematical equations for applying the tight-binding method to these different nanoscale systems.

1. ‫نانوالکترونيک‬ ‫در‬ ‫انرژی‬ ‫دياگرام‬ ‫محاسبه‬ ‫برای‬ ‫بست‬ ‫تنگ‬ ‫روش‬
Tight-Binding Method for Evaluation of Energy
Diagram in Nanoelectronic
By:
Hojjatollah Sarvari
E-mail: hojjatsarvari@gmail.com
Supervised by:
Professor Dr. Rahim Ghayour
School of Electrical and Computer Engineering
Shiraz University, Shiraz, Iran
Sep 22, 2010

2. Outline
• Introduction
• Tight-Binding method
• Energy diagram of Polyacetylene
• Energy diagram of single-layer graphene
• Energy diagram of bilayer graphene
• Energy diagram of multi-layer graphene
• Energy diagram of two and four layer graphene in constant electric field
• Energy diagram of single-layer graphene in modulated electric field
• Other applications of Tight-Binding method
2

3. Introduction
• The electronic band structure of a solid describes ranges of energy
that an electron is "forbidden" or "allowed" to have.
• The band structure of a material determines several characteristics,
in particular its electronic and optical properties.
• Different method for calculation of energy diagram:
– Ab initio
– Nearly Free Electron Approximation
– K.P Approximation
– Tight-Binding Method
3

4. • In 1928, Bloch provided the initial framework to Calculate the band
structure.
• Bloch’s approach is based on the fact that the crystalline potential
Vc(r) is periodic in a lattice.
• In fact, it was Bloch who first modeled electrons as being tightly
bound to particular atoms, overlapping only weakly with neighbors.
• This tight-binding model was further developed and established by
Wannier in 1937, who showed how Bloch eigenfunctions could
always be summed together to obtain a complete set of
wavefunctions.
Introduction
4

5. • R is the position of the atom
• Ѱj denotes the atomic wavefunction in state j
• n is the number of atomic wavefunctions in the unit cell
Tight-Binding method
5

6. • The eigenfunctions in the solid (based on orbital atomic) are expressed
by a linear combination of Bloch functions or atomic orbital (LCAO)
as follows:
Tight-Binding method
6
The j-th eigenvalue as a function of k:
H is the Hamiltonian of the solid.

7. Tight-Binding method
Secular equation:
( ) 0i iH E k S C    
1
( )iH E k S

  
7
If exist, thus Ci=0 and Ѱ=0
A much simpler interpolation scheme for approximating the electronic band structure
is the parameterized tight-binding method conceived in 1954 by John Clarke Slater
and George Fred Koster. Hence, it is sometimes referred to as the SK tight-binding
method.

8. Procedure for obtaining the energy dispersion
• Specify the unit cell and the unit vectors.
• Specify the Brillouin zone and the reciprocal lattice vectors.
• For the selected k points, calculate the transfer and the overlap
matrix element.
• For the selected k points, solve the secular equation.
 The transfer and the overlap matrix elements are often treated
as parameters selected to reproduce the band structure of the
solid obtained either experimentally or from first principles
calculations.
8

9. Energy diagram of Polyacetylene
Ra is the atom site coordinate and j is the number of Bloch wavefunctions
k
a a
 
  
9
Polyacetylene (HC=CH-)n
Lattice unit vector:
Reciprocal lattice vector:
The Brillouin zone:
The Bloch orbitals consisting of A and B atoms:
1
2
,0,0b
a
 
  
 
 1 ,0,0a a

10. HBB=HAA=ε2p
10
Energy diagram of Polyacetylene
is the transfer integral
Since HAB is real, so

11. Secular equation:
-4 -3 -2 -1 0 1 2 3 4
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
k.a
E(k)(eV)
11
Energy diagram of Polyacetylene

12. • Graphene is the Mother of all nano-Graphitic forms. It can be wrapped up into
0D fullerenes, rolled into 1D nanotubes, cut into 1D graphene nanoribbons or
stacked into 3D graphite.
12
Energy diagram of single-layer graphene

13. • Graphene is made up of carbon atoms bonded in a hexagonal 2D plane.
Graphite is 3D structure that is made up of weakly coupled Graphene
sheets.
• The first calculations of the energy band structure of graphite appears to
have been made by Hund and Mrowska in 1937 as an academic exercise.
• In 1947, Wallace considered both the two- and three dimensional
approximations in graphite using the tight-binding model.
• A graphene sheet is one million times thinner than a sheet of paper.
Energy diagram of single-layer graphene
13

14. ac-c=0.142 nm
Unit vectors in real space:
a1 & a2
Reciprocal lattice vectors:
b1 & b2
High symmetry points:
Г, K , M
AA AB
BA BB
H H
H
H H
 
  
 
14
Energy diagram of single-layer graphene
Because we have two atom in the
unit cell, so H is a 2*2 matrix

15. 1 1 1
2 2 1
3 3 1
( , 0)
3
( , )
22 3
( , )
22 3
B A
B A
B A
a
R R R
a a
R R R
a a
R R R
  
   
    
15
Energy diagram of single-layer graphene

16. By Sloter-Koster approximation (s=0):
16
Energy diagram of single-layer graphene

17. 17
Energy diagram of single-layer graphene

18. 18
Energy diagram of single-layer graphene

19. 19
Energy diagram of bilayer graphene

20. 1 1 1 1
1 1 1 1
2 2 2 2
2 2 2
1 2 1 2
1 2 1 2
2 1 2 1
2 1 2 1 2
A A A B
B A
A A A B
B A B B
A A A B
B A B
B B
A A A B
B A B B B
H H
H H
H
H H
H H
H H
H
H
H
HH H
 
 
 
 
 
 
11 22 33 44 2
12 43 0 ( )
pH H H H
H H t f k
   
 
13 1 2 1 2 1| |A A A AH H H t   
14 23 1 2 3| | ( )A BH H H t g k  
  
24 1 2 2| | ( )
6 6
( ) exp( ) 2cos( )exp( )
6 63
B B
x y x
H H t g k
a
g k ik k a ik a
  
  
Layer-layer separation: 0.335 nm
20
Energy diagram of bilayer graphene

21. 21
Energy diagram of bilayer graphene

22. 12 12
12 12
23 23
1 1
1 1
2
23
2
2 2
3 3
3
13 13
13
3
13
2
3
LL L
L L
L L
L
L L
L L
L L
L
H
L
L
L L
L
L L
L L
 
 
 
 
  
 
 
 
  22
Energy diagram of multi-layer graphene

23. '
5 0( ) ( 2 )i iA A A Ar r H r r c dr  
   
'
0 ( ) ( ) , 1,2,3.i i
j
A A B A ABr r H r r R dr j  
    
'
4 0( ) ( ) , 1,2,3.i i
j
A A B B ABr r H r r R c dr j  
     
23
Energy diagram of multi-layer graphene

24. -5 -4 -3 -2 -1 0 1 2 3 4 5
-10
-5
0
5
10
15
k.a
E(k)(eV)
24
Energy diagram of multi-layer graphene

25. 25
Energy diagram of multi-layer graphene

26. 26
Energy diagram of multi-layer graphene
Dirac points exist in odd number of layers.

27. 27
Energy diagram of two and four layer graphene
in constant electric field

28. Energy diagram of single-layer graphene in
modulated electric field
28

29. Energy diagram of single-layer graphene in
modulated electric field
The rectangular primitive cell has 4RE carbon atoms.
b=ac-c
29

30. (d) The low-energy bands are not sensitive to the direction of a modulated
electric potential.
Energy diagram of single-layer graphene in
modulated electric field
30
(a) Energy dispersion strongly deformed and induce several band-edge states.(b) When the strength increases, the energy bands are further deformed and
more band-edge states are created.
(c) The modulation period can also affect the numbers of band-edge states

31. Other applications of Tight-Binding method
• Electronic properties of carbon nanotube (CNT)
• Electronic properties of graphene nanoribbon (GNR)
• Effect of defect on the electronic properties of materials
• Effect of electric field and magnetic field
• Effect of doping on electronic properties of materials
• Description the Hamiltonian of a system for simulation of
nano-transistors
31

32. References
1) R. Saito, M .S. Dresselhaus, G. Dresselhaus, PHYSICAL PROPERTIES OF CARBON NANOTUBES,
(Imperial, London, 1998)
2) F. Bloch. The quantum mechanics of electrons in crystal lattices, Zeitschrift fur Physik, 52, 555-600 (1928)
3) G. H. Wannier. The structure or electronic exitation levels in insulating crystals, Phys Rev 52, 191-197
(1937)
4) F. Hund and B. Mrowka. Ber. verk. Akad. Wiss. Lpz. math. phys. Kl. 87, 185, 325 (1937)
5) Philip Richard Wallace. “The band theory of graphite”, Phy. Rev. 71, 622-634 (1947)
6) B. Partoens and F. M. Peeters, From graphene to graphite: Electronic structure around the K point, Phys. Rev.
B 74, 075404 (2006)
7) B. Partoens and F. M. Peeters, Normal and Dirac fermions in graphene multi-layers: Tight-binding
description of the electronic structure, Phys. Rev. B 75, 193402 (2007)
8) C. L. Lu, C. P. Chang, J. H. Ho, C. C. Tsai, M. F. Lin, Low-energy electronic properties of multilayer
graphite in an electric field, Physica E 32 585–588 (2006)
9) A. Gruneis, C. Attaccalite, L. Wirtz, H. Shiozawa, R. Saito, T. Pichler, A. Rubio, Tight-binding description
of the quasiparticle dispersion of graphite and few-layer graphene, Phys. Rev. B 78, 205425 (2008)
10) J. H. Ho, Y. H. Lai, C. L. Lu, J. S. Hwang, C. P. Chang, M. F. Lin, Electronic structure of a monolayer
graphite layer in a modulated electric field, Physics Letters A 359 70–75 (2006)
11) J. H. Ho, Y. H. Chiu, S. J. Tsai, and M. F. Lin, Semi metallic graphene in a modulated electric potential,
Phys. Rev. B 79, 115427 (2009)
32

33. Thanks for your attention