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# Tight binding

Tight Binding method for graphene

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### Tight binding

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 10. HBB=HAA=ε2p 10 Energy diagram of Polyacetylene is the transfer integral Since HAB is real, so
11. 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. 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. 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. 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. 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. 16. By Sloter-Koster approximation (s=0): 16 Energy diagram of single-layer graphene
17. 17. 17 Energy diagram of single-layer graphene
18. 18. 18 Energy diagram of single-layer graphene
19. 19. 19 Energy diagram of bilayer graphene
20. 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. 21 Energy diagram of bilayer graphene
22. 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. 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. 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. 25 Energy diagram of multi-layer graphene
26. 26. 26 Energy diagram of multi-layer graphene Dirac points exist in odd number of layers.
27. 27. 27 Energy diagram of two and four layer graphene in constant electric field
28. 28. Energy diagram of single-layer graphene in modulated electric field 28
29. 29. Energy diagram of single-layer graphene in modulated electric field The rectangular primitive cell has 4RE carbon atoms. b=ac-c 29
30. 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. 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. 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. 33. Thanks for your attention