1

Tight binding approximation

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: March 22, 2013)

The tight-binding model is opposite limit to the nearly free electron model. The

potential is so large that the electrons spend most of their lives near ionic cores, only

occasionally shift to nearest core atom quantum mechanically.

The nearly free electron method looks at the wave-functions outside the atomic cores,

where they look very like plane waves. Within the cores they look like atomic orbitals.

This suggests an entirely different scheme for the construction of electron wave-functions:

we try to combine atomic orbitals, each localized on a particular atom, to represent a state

running throughout the crystal.

1. One dimensional case: Bloch theorem

Suppose that the electrons are tightly bound to the nuclei. Thewavefunction would

coincide within the n cell, with an atomic eigenfunction a (such as 1s, 2s, 2p, ). Then

the wavefunction of the system may be expressed by a linear combination of atomic orbital

(LCAO) function,

1

1( ) ( )

N

k n a

n

x C x naN

.

x

x

2

The coefficient Cn can be determined as

ikna

n eC ,

from the requirement that )(xk should be the Bloch wave function;

)()( xeax k

ika

k .

((Proof))

Since

1

1 1

1 1( ) ( ) ( )

N N

k n a j a

n j

x a C x a na C x jaN N

,

1

1( ) ( )

Nika ika

k j a

j

e x e C x jaN

,

we get the relation

j

ika

j CeC 1 .

When C1 = ikae , nC is obtained as

ikna

n eC .

Then we have the Bloch wave as

1

1( ) ( )

Nikna

k a

n

x e x naN

.

((Note)) It is also clear that this form of )(xk satisfies the Bloch theorem,

( )

1

1( ) ( ) ( )

Nikx ik x na ikx

k a k

n

x e e x na e u xN

,

where

( )

1

( ) ( )N

ik x na

k a

n

u x e x na

.

)(xuk is a periodic function with a lattice period a,

3

[ ( 1) ]

1

( ) [ ( 1) ] ( )N

ik x n a

k a k

n

u x a e x n a u x

.

It can be easily shown that )(xk possesses all the required properties of the Bloch waves,

1 1

( ) ( ) ( ) ( )N N

ikna iGna ikna

k G a a k

n n

x e e x na e x na x

.

((Note)) Validity of the choice of above wave function

From the Bloch theorem, we have

( ) ( )ikx

k kx e u x

We choose ( )ku x as

( ) ( )k a

n

u x x na

we note that ( )x na is a wave function localized at x na [like a Dirac delta function

( )x na ]. It is clear that ( )ku x is periodic such that

( ) ( ) [ ( 1) ] ( )k a a k

n n

u x a x a na x n a u x

We note that

( ) ( )

( )

( )

ikx

k k

ikx

a

n

ikna

a

n

x e u x

e x na

e x na

In general case (3D system)

( ) ( )

( )

( )l

i

i

a l

l

i

a l

l

e u

e

e

k r

k k

k r

k R

r r

r R

r R

4

2. Three dimensional case

According to the Bloch theorem, we consider the wavefunction given by

N

j

ja

i

kje

N 1

)(1

)( RrrRk ,

where we assume that there is one atom per unit cell. The number of atoms is N.

jllajad )()(*

RrRrr .

The wavefunction )(rk is normalized as follows.

1

1

)()(1

)()(

,

,

**

N

lj

jl

ii

N

lj

laja

ii

kk

lj

lj

eeN

deeN

d

RkRk

RkRk

RrRrrrrr

The Hamiltonian H is defined by

)(2

22

rVm

H ℏ

,

where V(r) is the periodic potential of the present system.

Rl-1 Rl Rl+1

VaHr-Rl L

VHrL

5

Fig. Potential in the tight binding approximation. The potential ( )a lV r R is the

potential from the isolated atom at Rl. 1

( ) ( )N

a j

j

V V

r r rɶ . V(r) is a periodic

potential of the system: V(r)< ( )V rɶ .

Here ( )a lV r R is the potential of the atom isolated at the position vector lR . The

wavefunction ( )a l r R satisfies the Schrödinger equation,

2

2 (0)[ ( )] ( ) ( )

2a l a l a a lV

m r R r R r R

ℏ,

where )0(

a is the energy eigenvalue of the isolated atom. The Hamiltonian of the system is

given by

)(2

22

rVm

H ℏ

where )(rV is the periodic potential of the system. It is different from the potential ( )V rɶ

which is defined by

1

( ) ( )N

a j

j

V V

r r rɶ

and

)()( rRr VV l , ( ) ( )lV V r R rɶ ɶ

We start with the eigenvalue problem with the Schrödinger equation for the Hamiltonian,

)()( rrkkk

H ,

Noting that

1

( ) ( ) ( ) ( )

( ) ( )

N

a l a j a l

j

a l a l

V V

V

r r R r r r R

r r r R

ɶ

we get

6

2 22 2

1

22

1

22

1

1

1[ ( )] ( ) [ ( )] ( )

2 2

1{ ( ) [ ( ) ( )]} ( )

2

1[ ( )] ( )

2

1[ ( ) ( )] ( )

l

l

l

l

Ni

a l

l

Ni

a l

l

Ni

a l a l

l

Ni

a l

l

V e Vm mN

e V V VmN

e VmN

e V VN

k R

k

k R

k R

k R

r r r r R

r r r r R

r R r R

r r r R

ℏ ℏ

ℏɶ ɶ

ℏ

ɶ

Thus we have

2

2 (0)

1

1

1

1[ ( )] ( ) ( )

2

1[ ( ) ( )] ( )

1( )

l

l

l

Ni

a a l

l

Ni

a l

l

Ni

a l

l

V em N

e V VN

eN

k R

k

k R

k R

k

r r r R

r r r R

r R

ℏ

ɶ

By multiplying both sides of this equation by )( '

*

la Rr and integrating over all the space,

we have

(0) *

'

1

*

'

1

( ) ( ) ( )

( )[ ( ) ( )] ( )

l

l

Ni

a a l a l

l

Ni

a l a l

l

e d

e d V V

k R

k

k R

r r R r R

r r R r r r Rɶ

We note that

','

*)()( lllalad RrRrr

(orthogonality of the wave function)

Thus we get

(0) *

, ' '

1 1

( ) ( )[ ( ) ( )] ( )l l

N Ni i

a l l a l a l

l l

e e d V V

k R k R

kr r R r r r Rɶ

or

7

'(0) *

'

1

( ) ( )[ ( ) ( )] ( )l l

Ni i

a a l a j

l

e e d V V

k R k R

kr r R r r r Rɶ

When we define the potential difference by

( ) ( ) ( )V V V r r rɶ

which is negative, we get

'( )(0) *

'

1

( ) ( ) ( )l l

Ni

a a l a l

l

e d V

k R R

kr r R r r R

Note that this integral remains unchanged when the variable of the integral is changed from

lRr to r, since

( ) ( )lV V r R r .

Here we put

hll RRR ' ,

Fig. Schematic form of the potential energy for tightly bound electrons, alng a given

direction in the crystal. (Rigamonti and Carretta, Structure of Matter, Springer,

2007).

Then we have

8

N

h

aha

i

a Vde h

1

*)0( )()()( rrRrrRk

k

When the matrix element ht is defined by

)()()(*

rrRrr ahah Vdt ,

the energy eigenvalue is approximated by

...

..

)0(

...

..

)0(

)0(

nn

i

nna

nn

i

nna

h

i

hak

h

h

h

e

et

et

Rk

Rk

Rk

.

The sum over hR includes only values for which Rl denotes a nearest neighbor of Rj. In

the present case, both and are positive since V is negative. Note that the n.n denotes

the nearest neighbor pairs.

3. 3D systems: scc, fcc, and bcc

(a) Simple cubic

9

For the simple cubic lattice with the lattice constant a,

)coscos(cos2)0(

akakak zyxak ,

10

Fig. ContourPlot for the constant energy surface in the reciprocal lattice space.

75.0coscoscos akakak zyx .

When 1akx , 1aky , and 1akz , we get

)(62222)0(

zyxak kkka .

(b) fcc

There are twelve nearest neighbor atoms in the fcc lattice;

)1,1,0(2

1

aa , )1,0,1(

22

aa , )0,1,1(

23

aa , )1,0,1(

24

aa , )1,1,0(

25

aa ,

)0,1,1(2

6

aa , )0,1,1(

27

aa , )1,1,0(

28

aa , )1,0,1(

29

aa , )0,1,1(

210

aa ,

)1,0,1(2

11 a

a , )1,1,0(2

12 a

a

11

]2

cos2

cos2

cos2

cos2

cos2

[cos4)0(

akakakakakak xzzyyx

ak

.

Fig. ContourPlot for the Fermi surface of Copper (M. Suzuki and I.S. Suzuki).

https://www.researchgate.net/publication/322027719_Understanding_of_open_orbits_in_

copper_Fermi_surface_with_the_use_of_Mathematica

(c) bcc

There are eight nearest neighbor atoms in the fcc lattice;

)1,1,1(2

1

aa , )1,1,1(

22

aa , )1,1,1(

23

aa , )1,1,1(

24

aa , )1,1,1(

25

aa ,

)1,1,1(2

6 a

a , )1,1,1(2

7 a

a , )1,1,1(2

8 a

a ,

12

2

cos2

cos2

cos8)0( akakak zyx

ak .

((Mathematica program for the derivation of the energy dispersion in the Appendix)).

4. 2D system

For the 2D square lattice, we get

2

)cos(cos2

)0(

)0(

a

yxak akak

where changes between -2 and 2,

akak yx coscos .

Fig. ContourPlot of k = constant in the tight binding approximation for the 2D

square lattice (the first Brillouin zone). (kxa/2 - kya/2For the low energy,

the ContourPlot shows a circle. ContourPlot of cos(2x)+cos(2y )= ,

where = -2, to 2 ( = 0.1). The first Brillouin zone is denoted by the black thick line.

h=2

h=1

h=-1

h=-2

h=0

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

13

Near the origin of the k-space ( 1akx and 1aky ), we get

)(4222)0(

yxak kka . (electron-like)

The effective mass m* is defined as

2

2*

2 am

ℏ

.

Near the zone edge ( akx and ak y ), we get

])()[(22)0( akak yxak . (hole-lik)

For the 1D lattice, we get

akxak cos2)0( .

When 1ka , we get

22)0(

2 kaak .

The effective mass m* is defined as

2

2*

2 am

ℏ

.

Fig. Formation of the energy band in the tight binding approximation

14

3. Summary

The following interesting conclusions may be derived from the above figures.

(a) The energy spectrum of the electrons consists of a number of allowed energy bands separated by energy gaps.

(b) The discontinuities in the energy spectrum, occur for the zone boundaries.

______________________________________________________________________

REFERENCES

1. M. Suzuki and I.S. Suzuki, Lecture note on solid state Physics, Bloch theorem and

energy band. http://www2.binghamton.edu/physics/docs/note-energy-band.pdf 2. See Out of the Crystal Maze, Chapters from the history of solid state physics edited

by L. Hoddeson, E. Braun, J. Teichmann, and S. Wert (Oxford University Press, New York, 1992).

3. S.L. Altmann, Band Theory of Metals (Pergamon Press, Oxford 1970). S.L. Altmann, Band Theory of Solids An Introduction from the point of view of symmetry

(Clarendon Press, Oxford 1991). 4. J.M. Ziman, Principle of the Theory of Solids (Cambridge University Press 1964).

5. C. Kittel, Introduction to Solid State Physics, seventh edition (John Wiley & Sons, New York, 1996).

6. C. Kittel, Quantum Theory of Solids (John Wiley & Sons, 1963). 7. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinheart and Winston,

New York, 1976). 8. E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics Part2 Landau and Lifshitz

Course of Theoretical Physics volume 9 (Pergamon Press, Oxford 1980). 9. J. Callaway, Quantum Theory of the Solid State, second edition (Academic Press,

New York, 1991). 10. A.A. Abrikosov, Introduction to the Theory of Normal Metals (Academic Press,

New York, 1972). 11. M. Kuno, IntroductoryNanoscience Physical and Chemical Concepts

(Garland Science, Taylor & Francis Group, London and New York, 2012). 12. R.L. Liboff Introductory Quantum Mechanics (Addison-Wesley Publishing

Company, New York, 1980). 13. C. Kittel Introduction to Solid State Physics eighth edition (John Wiley &

Sons, New York, 2005). 14. A. Rigamonti and P. Caretta, Structure of Matter: An Introduction Course with

Problems and Splutions (Springer, 2007)

________________________________________________________________________ APPENDIX

((Mathematica))

Program to find the form of energy dispersion for the tight binding approximation for the fcc and bcc

(a) fcc lattice

15

(b) bcc lattice

tight binding approximation for fcc

Clear@"Global`∗"D; a1 = 2; A1 =a1

280, 1, 1<;

A2 =a1

281, 0, 1<; A3 =

a1

281, 1, 0<;

R1@h1_, h2_, h3_D := h1 A1 + h2 A2 + h3 A3;

Rmag = R1@h1, h2, h3D.R1@h1, h2, h3D;list1 =

Table@8Rmag, R1@h1, h2, h3D<, 8h1, −1, 1, 1<,8h2, −1, 1, 1<, 8h3, −1, 1, 1<D êê

Flatten@�, 2D &;

list2 = Sort@list1, �1@@1DD < �2@@1DD &D;eq1 = Select@list2, H1 < �@@1DD < 3L &D;L1 = Length@eq1D; X@i_D := eq1@@i, 2DD;k = 8kx, ky, kz<;f1 = SumBExpB�

a

2Hk. X@iDLF , 8i, 1, L1<F êê

ExpToTrig êê FullSimplify

4 JCosAa ky2

E CosAa kz2

E +

CosAa kx2

E JCosAa ky2

E + CosA a kz2

ENN

16

Tight binding effect for bcc

Clear@"Global`∗"D; a1 = 2; A1 =a1

28−1, 1, 1<;

A2 =a1

281, −1, 1<; A3 =

a1

281, 1, −1<;

R1@h1_, h2_, h3_D := h1 A1 + h2 A2 + h3 A3;

Rmag = R1@h1, h2, h3D.R1@h1, h2, h3D;list1 =

Table@8Rmag, R1@h1, h2, h3D<, 8h1, −2, 2, 1<,8h2, −2, 2, 1<, 8h3, −2, 2, 1<D êêFlatten@�, 2D &;

list2 = Sort@list1, �1@@1DD < �2@@1DD &D;eq1 = Select@list2, H2 < �@@1DD < 4L &D;L1 = Length@eq1D; X@i_D := eq1@@i, 2DD;k = 8kx, ky, kz<;f1 = SumBExpB�

a

2Hk. X@iDLF , 8i, 1, L1<F êê

ExpToTrig êê FullSimplify

8 CosAa kx2

E CosAa ky2

E CosAa kz2

E