§5.6 Tight-binding

Tight-binding is first proposed in 1929 by Bloch. The primary idea is

to use a linear combination of atomic orbitals as a set of basis

functions, thereby, we can solve the solid Schrodinger equation. This

method is based on such physical image, there is little difference

between the electronic states in solids and the free atoms which are

composed by them . Tight-binding is very successful in the study of

the band structure of the insulator. Since the atomic orbitals are

locate on different grids, basis function composed of them are

generally non-orthogonal. So, inevitably, we would encounter the

computing problems of multi-center integrals, and the form of

eigenequation is not easy.

Potential field )(rV

is a periodic function of lattice, which can be

rKil

leKVrV

)()(

From the Bloch theorem， in the wave vector space, bloch wave

function is a periodic function of reciprocal lattice vectors. Similar to

the potential function, the Bloch wave function in the wave vector

space is expanded to a Fourier series:

kRin

n

nerRWN

rk

),(1

, ）（

In the above formula, W is called Wannier function,α is the band number.

developed into the real space Fourier series:

The above equation was multiplied by

mRkieN

1

All wave vector within the brillouin zone are summed up:

),(1

),( rkeN

rRWk

Rkin

n

Combined with (5.15):

kKKN

rkKKi

ll

ll rdeN

2,

)2(1

Get

nnnnNrdrRWrRW ,,

* ),(),(

The above formula shows that Wannier function of the different energy

bands or the same band with different grid points are orthogonal. Known

from the translational symmetry of the Bloch wave function:

),(1

),( nk

n RrkN

rRW

When the spacing of atoms in the crystal is large, the probability of electrons

are trapped by nearby atoms is much larger than it move away from atoms, the

behavior of electrons in the vicinity of a grid is similar to the behavior of

electrons in the isolated atoms; when the electrons obviously deviated from the

grids, the wave function is a small quantity.),( nRrk

Isolated atom wave functions can be used to describe the wave function

under tight-binding conditions.

Take )()(),( nat

n RrkRrk

Wannier function can be translated into:

k

nat

n kN

RrrRW

)(

1)(),(

Using the orthogonality of the Wannier function, we can get:

1)(1

)()()(1

),(),(

2

*

2

*

k

nat

nN

at

k

nnN

kN

rdRrRrkN

rdrRWrRW

We can get:

)(1

),( nat

n

Rki RreN

rk n

The formula is called the Bloch wave function， it is a linear combination of atomic orbital wave function, so the tight-binding method is often called linear combination of atomic orbital method.

The above equation is substituted into the Schrodinger equation, and is rewritten as：

0)()()()()(2

22

nat

nat

nat

n

Rki RrRrVrVkERrVm

e n

Among them, ( ),at

nV r R

nR

is the potential field formed by atoms with the grid point .

We discuss the s-state of those non-degenerated electrons. When the

principal quantum number is certain, the s-state wavefunction will be more

localized，more suitable to the tight-binding. Using the following

relationship：

)()(

)()()(2

22

natss

ats

natssn

at

RrkEE

RrkERrVm

The above equation is multiplied by *( ),ats r

integrated over the crystal volume, we can get:

0)()()()(

)()()(

*

*

rdRrRrVrVre

rdRrrekEE

natsn

at

N

ats

n

Rki

natsN

ats

n

Rkis

ats

n

n

then,

Using the tight-binding model, we ignore the quadratic terms, and only reserve the term 0,nR

then, the first part of the above equation is equal to：

)(kEE sats

When , the integral term of the second part of the above formula is written as .sC

rdrrVrVrC ats

at

N

atss

)()()()(*

The integral term is negative obtained from the above figure.

0nR

0nR

When ,since the overlap is small for the isolated atom wave function of the two adjacent grids, so only the overlap integral of the adjacent grids are considered. S-state is spherical symmetry, so the integral values of the nearest grids are same.

rdRrRrVrVrJ natsn

at

N

atss

)()()()(*

So, the second part can be simplified as :

, is the nearest lattice vectornik Rs s n

n

C J e R

.

In summary, the band of the s-state of the tight-binding electron is

( ) ,nik Rats s s s n

n

E k E C J e R

is the nearest lattice vector.

For example, for a simple cubic crystal, there are six of the nearest

neighbor atoms. Substitute coordinates of the six atoms into the above

formula, we can get:

ssatss JCEE 6min

The maximum of the Energy is:

ssatss JCEE 6max

The width of the band:

sJE 12

)coscos(cos2)( akakakJCEkE zyxssatss

The minimum of the Energy is:

The width of the band is determined by the size and the coefficient of Js,

and Js depends on the overlap integral, the coefficient depends on the

number of the nearest neighbor grid points, i.e., the coordination number of crystal. We can expect, the larger overlap degree of the wave function, and the more coordination numbers of the crystal, the wider band, on the contrary, the narrower band.

An energy level of an electron in the isolated atom becomes a band in solid.