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Tight Binding Model - solid state physics
Physics
Kendriya Vidyalaya Hebbal
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- 1 1 Tight-Binding
Approximation
239
Thus, theoreticallv. the cyclotron frequency is the same for all the orbits. This, however,
is not true and the resonance frequency is different for different orbits. A cyclotron mass rn*
for the orbit is defined by Eq. (8) a;d is given by
**
: A
(!!\
2n
####### \dE I
(8)
(8)
Thus, with the knowledge of the resonance frequency and from the knowledge of the mag-
netic field B, the effective mass rn* can be determined.
lo'r,
2ff
ra, -
w(,)Ir,
:
o]
8.
####### Tight-Binding
####### Approximation
In the band theory, it is assumed that a crystal is composed of
positive
ions which are perme-
ated by
quasi-free
electrons. This assumption leads to the formation
of bands in solids which
helps us to differentiate between conducLrs, insurators and semiconductors.
A completely different method of treating this problem is the tight-binding approximation.
In this method, it is assumed that a crystaiis formed by bringirrg the individual atoms close
together' when the atoms are far away from one another,
"uJh
uto,,. behaves as an isolated
system and hence, their energy levels are identical. But, when the atoms ur" t t
"tor",
to form a crystal, there is interaction
between the neighbouring atoms. In such a case, the
different atoms must be considered as a single system.- The orilinal energy states now shift
relative to one another so that there is no violation of Pauli exllusio, principle.
The closer
the atoms, the greater will be this shift. This energy shift wilt be more pronounced
for the
electrons in the outer orbits. If there are l/ atoms-in the system, then each electron state
for a single atom wiLl spread in to lr' levels. The breadth of the bands and the energy gaps
between them will depend on the
particular
energy state of the atom as well as on the atomic
separation.
Let us first consider an electron of a free atom. The potentiar
energy v^(r) otthis electron
will be due to the nucleus and the other electrons of the atom, where r represents
the distance
of the electron from the nucleus. The potential energy of tLe electron has the form shown
in tr'igure 8. If E6 be the energy of tie electro, u,rr".i ,p61rl u"-it, wave function, then the
corresponding Schrodinger wave equation will be given by"'
'
Here, we make the following assumptions:
1' The energy level ,o6 is nondegenerate,
that is, there is only one wave function corre-
sponding to Es,
- that the wave functions are normalized,
and
3' that the electron in the vicinity of aparticular nucleus is only slightly influenced by the
presence of other atoms.
Since a crystal const,qts of a large number of such atoms, the potential will be modified. Let
the modified potential
bev(r) hi"h h the periodicity
of the lattice. we take a particular
_1:"*
*.'nu of our coordinate system. The positioo or urr Jom, say jth, may then be
represented by Ri, where E is a rattice vector. This is shown in Figure
g.
240
Band Theory
of Solids
I
!
V
####### 8 atl*t
Equatir
and
The
F-t
expressil
where If b d
the proper r
Since
-l
where rY Ld
Equrrtir
E(k)
####### :
Figure
g: Schematic reqresentation
of
potential energ of an electron
in an atom aqd
in a solid'
####### oo
r?-4t
####### aa
jth atom
Two-dimensional space
showing
atomic
positions'
Since the electron
in the vicinity
of the nucleus
j is only slightly influenced
b{he
presence
of other
atoms, i. when
the end
point of vector ?"t
in the vicinity of
Ri, the wave
function
for the electron is
approximately
given by ,lto(?
E
i)
and
the energy of the
electron
wiII still be very
close to .Eo. The
total wave function
of the electron
with a wave vector
k
will then be a
linear combination
of the form
N
,l,x(r)
:Lc,+r(a
-1,)
j:
Since
we are dealing
with periodic
potential, the wave functions
are Bloch functions
which
restrict
the choice of ii
u,
n,)
(8)
a
o
E
,
o
a
Figure
8"6:
(8) +
Ci
: expi (l
Band TheorY
of Solids
For the
first term involving
-E6,
we neglect
all the terms
except rn:
j'
The first term
then
(8)
(8)
(8)
(8)
8, TtEl
8.11 a
C
In a simpbt
at
Thereforer',
Wheu &
level .Eo
hu
The width d
At the b!
At tbe tq
The
wiI
Ftom
Erl'
wave frrnctil
Thus, the ir
the oute *
and more c
model ard I
Let us u
values of t, d
17qp(il)
+r
rtFm
mlr
becomes
*;
I
o;e
-7'i)Eo't'^1?
-l
'1a':
Next,
we
.--+
v,(
-
R j)
we have
Second
term
First
term
:
-Es
(8)
consider
the second
term
of Eq.
(105) containing
the
perturbing
potential
and take
in to account
only
the
jth atom and
its nearest neighbours
rn'
Then,
Dfr--+
+)- l+;t"
-
E),bo(l
-
Ri)dr
r\
-J
:
fi
|
rAe -l',1v'1'
-
Ri)'h'(l
-7.'11a'
$ot
i
:m)
*+
t [
"*pt7,
(A,
-fi
*),t
;t" -
fi, *1v' 1'
-
R)'ho(?
-3,1a'
N;J
and
Second term
:
-a -
tf
exPl,k(i,,
-f,,*)
at:
-
|
,t;f;
-R,1v'1,
-
Ri),hr(?
-R',1a'
^t
- -
l,t
f,e -R'*1v'
1, -
R)'ho(?
-l'
;a'
--+
The vector
R- corresponds
to the
location
of one of
the nearest neighbours
of the
atom
7'
All tlre terms
in the summation
over
m, each
of which
is evaluated over
all
j (either in the
same
atom or
in the *"ur"rt
atom),
are equal
in magnitude
on the demand
of periodicity'
Since
the
summation
over rn
runs over all
the atoms
in ihe crystal,
the sum
is simply l[
times the
value
for a single
atom.
This factor
of l[ cancels
with the
factor of l[ in
the denominator'
Using Eqs.
(8) to
(8), Eq'
(8'105)
gives
E(k)
:
Eo
-
a
-
"r*exPik(R'
1 -
I
*)
Since V'(r
-
-R )
is negative,
o and 'y,
defined
by Eqs'
(S' 108) and
(8' 109) respectively'
are
positive.
. ir- _a
.
Equation(8)showsthattheenergyofanelectroninacrystaldiffersflom|hltlna
free
atom by a
constant factor
o
plus a term that
contains the
vector /c
'
ttre
last factor
on
ihe
right-hbnd
side of Eq.
(8) is
responsible
for the broadening
of discrete
atomic energy
levels in to an
energy band
in a solid'
- 11 Tight-Binding Approximation
243
8.
Application of Tight-Binding
Approximation
to a simple
Cubic
Lattice
In a simple cubic crystal with
a lattice constant
a, the nearest neighbour
atoms are situated
at
ThereforelT,
(8)
l,
-fr*:
(*o,0,0),
(0,fa,0),
(0,0,t4,)
E(k)
:
Eo
-
a
-
21(cosk*a *
coskoa *
cos k,a)
When atoms are
brought together so as
to form a crystal,
the single discrete atomic
level .E6 broadens in
to an erieigy band whose
component
levels are defined by Eq'
(8'111)'
The width of the energy
band dan be determined
as follows:
At the bottom of the
band, k
:
- Equation
(8), therefore,
gives
Ebotto-
-
Eo-a-6'Y
At the top of the band, k:
ltrla. Equation
(8), therefore,
gives
Et'oP:Eo-a*6'l
The width of
the energy band is, therefore,
Etop- Ebotto*:L2'Y
(8)
Flom Eq.
(8), it therefore follows that
as
7
increases, that
is, as the overlap of
the
wave functions on the
neighbouring atoms increases,
the width of the
energy band increases.
Thus, the inner electronic
levels of the free atoms
appeax almost discrete,
but as one moves to
the outer shells, the width
of the respective energy
bands
goes on increasing
because of more
and more overlap.
This result is consistent with
the conclusion drawn
from Kronig-Penney
model and also confirmed
by experiments.
Let us now try to
determine the effective mass
of an electron in an
energy band. For small
values of k, the cosine
term in Eq. (8) may
be expandedl8 to
yield
(8)
(8)
(8)
+
E(k)
:
Eo
-
a
-
zrlr
-
ry
*,
-
ry
.,
-
ry)
I k2a
E(k):Eo-a-rrl3-i)
E(k):Eo-a-61+1k2a
E(k)
:
Eo
-
o
-
r,
l,
-
t
AZ
ni +
n?))
r
exp (i0) +
exp (-i0)
: 2 cos 0
18For
small values of 0, cosl
=
L
e:
2
since we are investigating
the region near the zone boundary,
kt*a <
Tand the cosine term
in Eq. (8)
can be expanded to y"ield
E(k)
:
Eo
-
a
-
rrl-,
ry -,
*
rL'=o'
_,
*
r?"r
L
2
''
2
'--,
J
=+
E(k):
Ea
_
a*6t
_yaz
(tci
+n] +nir)
where
'(r/o)
E(k):Eo-a*h-B1a2k
-(tr/o)
(8)
k
:
k? + ki
+ tei
(8)
,rr"";T,:11H:fl
ilT?ffi"liagain
spherical, but their centres
are at the corners
of the
(u)
(b)
Figure
8'17: (a) constant
energ-y curves for the TB-A for a simple cubic lattice. constant
energ
;:f1}?SH:ii}l|}::Ilj#ff"""
;;;i;r
sma, '.,ur,,u, oi,ri
ib)
constant enersy surraces ror
Thus' the constantenergy
surfaces
are again spherical, but have their centres at the corners
of the zone, as shown in Figure
g.
i;;;
nearly free electron approximation,
the constant
;l"1flrfii[X1rT*lll"""'l
ror a much larger
"ur,,"
oi ii"
"'Li"
,""to. than in the tight-
QUESTTONS
- State and prove Bloch theorem.
2' Describe
Kronig-penney
model of erectrons
*orrrrg in a periodic
potentiar.
How does it
lead to the formation
oi forbidd*
"""rgi
SrprZ
- What do you mean by the effective
**"-of
an electron in an en
account for the negative
mass of u.,
"t""t,
in an energy buod?"'s
band?
How do you
Tight Binding Model - solid state physics
Course: Physics
University: Kendriya Vidyalaya Hebbal
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