§5.6 §5.6 tight-binding tight-binding is first proposed in 1929 by bloch. the primary idea is to...
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§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.
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