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Home , Electronic structure, Tight binding

Electronic Structure of Calculations Based on Method

Mehmet Ergin 11.01.2010

Abstract: Different methods using to calculate electronic band structure, however tight binding method is used widely and it works in more different cases. In this report, introductory knowledge is given about band structure and tight binding method. Tight Binding‟s 1D and 2D calculations also shown in this report.

Introduction: One another important subject is . Pauili exculision principle said that maximum only one allowed per Electronic band structure is the ranges , so that number of bands may be of , which is forbidden or partially field. “In momentum space, these allowed to have. Band structure determines particles fill up a sphere of radius p , the optical and electronic properties of materials. F surface of which is called the Fermi surface.” is important topic in electronic band [3] structure; it means, “the difference between the top of the highest occupied band and the bottom of the lowest unoccupied band” [1] In Various techniques used for to solids too many came together and the calculate the band structure energy. Some of number of orbitals becomes large, because of these methods are; Pseudopotentials, Green‟s that energy differences small and energy bands Function, Independent Electron is discrete. Although, when some of the Approximation, Muffin-Tin Potentials and intervals of energy contain no orbital, it Tight Binding. produced the band gaps.[2] have so many band which is partly filled, so metals Tight binding method is different from have high conductivity. Fully occupied band in other methods, since it is suitable for more insulator or called the valance cases and directly suitable for any real band. Valance band is the highest range of solid.[4] Tight binding method is the easiest electron , the highest point of the method for calculating band structures in not valance band is called the valance band edge only conceptually but also computationally. This method also referred to as Linear . Combination of Atomic Orbitals. [5] Tight binding method is based on Linear Combination of (LCAO) approach used in molecular physics. Because of this situation, tight binding method is not only applied to crystalline but also applied to non-crystalline materials. “The approximation is quite good for the inner electron of , but it is not often a good description of the conduction themselves.” [6]

One of the simplest and general case of in tight binding is an isolated s band. To calculate this case, we write Hamiltonian „H‟ Figure 1: Simplified diagram of electronic as; band structure

Hat ψ(n)= En ψ(n) (1) H = Hat + ΔU(r) (2) Limits: ka  0; cos(푘푎)  1 − 1 2 2(푘 푎) (11) Where Hat containing the potential and kinetic 2 energy of one ion. ΔU(r) is the potential 휀(푘 )= 퐸푠- 훽-2훾-훾(푘푎) (12) : Linear generated by other ions in the lattice. function having no dependence on k. Schrodinger equation also satisfied by the of the atomic problem Tight Binding Method for 2D Square Hat φn(r) = En φn(r) (3) Lattices In here n means, collectively a full set of Normally the energy structure of quantum numbers. Then if we will expand crystals depends on the interactions between formulation (3) we get, orbitals in the lattice. However, the tight binding approximation neglects interactions φ(r) = n bn φn(r) (4) between atoms separated by large distances, an approximation which greatly simplifies the Than we make a combination of localized analysis. In solid state physics, the tight states with the symmetry of the lattice: binding model calculates the electronic band structure using an approximate set of wave i k·R ψ(r) = R 푒 φ(r-R) (5) [1] functions based upon superposition of orbitals located at each individual atomic site. [7] Bloch function can be written like Formulation In a tight binding approximation including (5), φ in this formulation known as Wannier only first nearest neighbor s-orbital, the band formulation. structure is given by: 푖푘 푘 휀(푘 )=퐸푠+ 푘 푒 훾(푅 ) (13)

1D Band Structure of a Periodic Linear Carbon Atom’s Chain

. . . o o o o o o o o . . . → x

Distances between two atom is “a”, periodic chain consist of same atom.

Figure2: 2D square lattice fragment. Considering an atom inside the chain; there are

2 nearest neighbors. Where 훾 is the overlap integral between s- orbitals, R is the translation vector of the 휀(k)= 퐸 - 훽- 훾(푅 )cos⁡(푘 . 푅 ) (6) 푠 푛.푛 lattice, and ħ푘 is the "crystal momentum" the 훾(푅 )= - 푑푟 ∅ 푟 ∆푢 푟 ∅(푟 − 푅 ) (7) quantum number for periodic systems. The band structure for a simple cubic lattice can 훽 = − 푑푟 ∆푢(푟 )|∅(푟 )|2 (8) now be readily calculated. Assuming that the bond integrals couple only to the first four + R= −푎푥 -a푥 ← o→ +a푥 nearest neighbors with position vectors R + + equal to(−푎, 0) and(0, −푎), gives + + 푘 . 푅 = −푎푥 .푘 =−푎 k

휀(푘 )= Es- 훽- 훾cos⁡(푎푘) - 훾cos⁡(−푎푘) (9) 휀(푘 )= 퐸 +2 훾(cos(푘 푎)+cos(푘 a)) (14) 푠 푥 푦 휀(푘)= Es- 훽- 2훾cos⁡(푎푘) (10)

Es: energy of the atomis s-level.

Figure 3: Four nearest neighbors included in transfer, the first Brillouin zone.

Figure 6: Lattice of .

푖푘푟 휑푘= 푅∈퐺 푒 ∅(푥 − 푅) (15) We have to think that there are two orbitals per unit-cells.

∅ 푥 = 푏1∅1(푥 ) +푏2∅2(푥 )= 푛 푏푛 ∅푛 (16) Figure 4: Tight Binding band structure of 2D square lattice Carbon atom, 3D view. Then, consider the Hamiltonian for a single electron in the atomic potential given by all the carbon atoms:

푝 2 퐻 = + (푉 (푥 − 푥 − 푅 ) + 푉 (푥 − 2푚 푅∈퐺 푎푡 1 푎푡 푥 2 − 푅 )) (17)

푥 1,2 denote the position of the two carbon atoms within the unit-cell.

H∅1 =∈1 ∅1 + ( 푅≠0 푉푎푡 푥 − 푥 1 − 푅 +

Figure 5: Tight Binding band structure of 2D 푉푎푡 푥 − 푥 2 − 푅 + 푉푎푡 푥 − 푥 2 )∅1 (18) square lattice Carbon atom, 2D view. ∈1 is the eigenvalue for the atomic pz state.

H∅1,2=∈1,2+∆푈1,2 ∅1,2 (19) Tight Binding Band Structure of Graphene Let ∈1=∈2 and choose ∈1,2=0,

Graphene is a single sheet pf carbon atoms, H∅1=∆푈1∅1 and H∅2=∆푈2∅2 which is arranged in the honeycomb structure. We should solve to Schrödinger equation: Referring to figure, the lattice vectors can be written as: 푎 1= 푎0 3 (1 2, 3 2), H휑푘 = 휀(푘)휑푘 , 휀(푘)<∅푗 |휑> = <∅푗 |∆푈푗 |휑> 푎 2= 푎0 3 (−1 2, 3 2) (20) We have two parameters; so two equations are required to solve this eigenvalue problem. Then assume that only nearest-neighbor overlap integrals have to be taken into account. ∗ For instance, ∅1 ∅2 is non-zero as is ∗ ∅1 푥 ∅2(푥 − 푎 1 ) .

∗ −푖푘 ,푎 1 −푖푘 ,푎 2 <∅1|휑>=푏1+푏2( ∅1 ∅2)(1+푒 +푒 ) (21)

∗ 푖푘 ,푎 1 푖푘 ,푎 2 <∅1|휑>=푏1+푏2( ∅2 ∅1)(1+푒 +푒 ) (22)

Overlap integral is real: ∗ (23) 훾0 = ∅1 ∅2 ∈ 푅 ∗ ∗ 훾1 = ∅1 ∆푈1∅2= ∅2 ∆푈2∅1 (24) Figure7: Tight Binding band structure of graphene, 3D view. −푖푘 ,푎 1 −푖푘 ,푎 2 <∅1|∆푈1|휑>=푏2 훾1 (1+푒 +푒 ) (25)

푖푘 ,푎 1 푖푘 ,푎 2 <∅1|∆푈1|휑>=푏1 훾1 (1+푒 +푒 ) (26) Eigenvalue problem is formulated as:

훼 푘 = 1 + 푒−푖푘,푎 1 + 푒−푖푘,푎 2 (27)

휀(k) 훼(훾0 휀(k)−훾1) 푏1 0 ( ∗ ) (푏 )= (0)| (28) 훼 (훾0휀(k )−훾1) 휀(k ) 2

휀(푘 )

3 + 2 cos 푘 . 푎 1 + 2 cos 푘 . 푎 2 + = +훾 (29) − 1 Figure8: Tight Binding band structure of

2 cos 푘. 푎 2 − 푎 1 graphene, 2D view.

If we express this result in different form by Conclusion: using the (x,y) components. Tight binding approximation is the most practical method to calculate electronic 휀 (푘 푘 )= 푥 푦 band structure. It used in theoretical solid state + 3 푎푘푥 2 푎푘푥 physics and material science. Tight binding −훾1 1 + 4 cos 푎푘푦 cos + 4푐표푠 ( ) 2 2 2 method is worked in many different cases and (30) easily to calculate band structure in conceptually and computationally. a is the lattice constant, a= 3 푎0 [8] 1D linear carbon atom chain investigated and band energy is a cosine function. In 2D square carbon lattice and graphene also investigated, these systems band energy is equal to two cosine functions summation. In 2D systems electronic band structures figure was plotted by Matlab and especially in plot, figure 7, band gap is easily seen. References:

[1] N. W. Ashcroft and N. D. Mermin, “Solid State Physics” (1976)

[2]http://en.wikipedia.org/wiki/Band_structure

[3] K. Huang, “Statistical Mechanics” (2000)

[4] F. Liu, Self-Consistent Tight-Binding Theory Phys. Rev. B (1995)

[5] E. Kaxiras, “Atomic and of Solids” p121, p130 (2003)

[6] C. Kittel, “Introduction to Solid State Physics” (1996)

[7] D.G Pettifor, “Bonding and Structure of and Solids” (1995)

[8]http://www.stanford.edu/~kghosh/CNTElec tronicBandStructure_KG.pdf