Transcript (1) Valance band

Band Theory of Solids
Bloch theorem
• A crystalline solid consists of a lattice which is
composed of a large number of positive ion cores at
regular intervals and the conduction electrons move
freely throughout the lattice.
• The variation of potential inside the metallic crystal
with the periodicity of the lattice is explained by Bloch
theorem.
Periodic positive ion cores Inside metallic crystals.
+
+
+
+
+
+
+
+
+
+
+
+
V0
+
a
+
+
+
One dimensional periodic potential in crystal.
• The periodic potential V (x) may be defined by
means of the lattice constant ‘a’ as
V (x) = V ( x + a )
• From Schr odinger wav e equation
d 2 8 2 m
 2 [ E  V ]  0
2
dx
h
d 2 8 2 m
 2 [ E  V ( x  a )]  0
2
dx
h
• Bloch has shown that the one dimensiona l
solution of the Schr odinger equation
 ( x)  U k ( x) exp( ikx)
Where U k (x) is a periodic with periodicit y of a crystal lattice.
U k (x)  U k (x  a)
where k represents the state of the motion of the electron
Kroning – Penney Model
• According to Kroning - Penney model the electrons move in a
periodic potential field provided by the lattice.
• Consider Schrödinger equation for this case, we can find the
existence of the energy gap between the allowed values of energy
of electron.
•
For one dimensional periodic potential field….
d  8 m

[ E  V ( x)]  0
2
2
dx
h
2
2
V0
1
2
+
+
+
-b
0
+
a
(1)  v( x)  0........o  x  a
(2)  v( x)  v0 .......  b  x  0
+
for
region., (1)
d 2
8 2 m

[ E ]  0
2
2
dx
h
for
region., (2)
d 2 8 2 m

[ E  V0 ]  0
2
2
dx
h
d 2
2


  0.....(1)
dx 2
d 2
  2  0.....( 2)
2
dx
where
8 2 mE
 
h2
2
8 2 m
 
[V0  E ]
h2
2
According to Bloch, the solution of a schrodinger equation
 ( x)  U k ( x)eikx .....(a)
Where Uk(x) is the periodicity of the lattice i.e,.
U k ( x  a)  U k ( x).....(b)
According to Bloch theorem
 k ( x  Na)   k ( x)e
ikNa
.....(c)
By using above a, b, and c Bloch conditions, the
solutions of equations (1) & (2) becomes
sin a
cos ka  P
 cos a
a
2
4 ma
where..P 
V0b
2
h
p is the scattering power of the potential barrier
V0b is called Barrier strength
2

h
2mE
P
+1
sin a
 cos a
a
+1
a
-1
-1
Conclusions
1. The motion of electrons in a periodic lattice is characterized
by the bands of allowed energy separated by forbidden
regions.
2. As the value of άa increases, the width of allowed energy
bands also increases and the width of the forbidden bands
decreases. i.e., the first term of equation deceases on the
average with increasing άa .
3. Let us now consider the effect of varying barrier strength P. if
V0b is large ,i.e. if p is large ,the function described by the left
hand side of the equation crosses +1 and -1 region as shown
in figure. thus the allowed bands are narrower and the
forbidden bands are wider.
P
sin a
 cos a
a
+1
0
-1
a
If P tends to infinite the allowed band reduces to one
single energy level :
p
0
a
4. If P tends to zero
cos a  cos ka
No energy levels exist:
 k
all energies are allowed to the electrons. 2
  k2
p0
2mE
k2 2  2

2 2
E  ( )k
2m
a
h2
2 2
E  ( 2 )( )
8 m 
h2 1
E( ) 2
2m 
h2 p2 p 2 1 2
E( ) 2 
 mv
2m h
2m 2
Brillouin zones
• The Brilouin zone is a representation of permissive
values of k of the electrons in one, two or three
dimensions.
• Thus the energy spectrum of an electron moving in
the presence of a periodic potential fields is divided
into allowed zones and forbidden zones.
E-k diagram :
E
Energy gap
Allowed
bands
Energy gap
3

a
2

a



a
a
First
Brillouin zone
2
a
3
a
k
• Origin of energy band formation in solids:
• When we consider isolated atom, the electrons are tightly
bound and have discrete, sharp energy levels.
• When two identical atoms are brought closer the outer most
orbits of these atoms overlap and interact.
• If more atoms are brought together more levels are formed
and for a solid of N atoms , each of the energy levels of an
atom slipts into N levels of energy.
• The levels are so close together that they form an almost
continuous band.
• The width of this band depends on the degree of overlap of
electrons of adjacent atoms and is largest for outer most
atomic electrons.
E1
E1
E2
E1
E2
E3
N atoms
ΔE
N energy levels
• The energy bands in solids are important in
determining many of physical properties of
solids. The allowed energy bands
(1) Valance band
(2) Conduction band
• The band corresponding to the outer most
orbit is called conduction band and the next
inner band is called valence band. The gap
between these two allowed bands is called
forbidden energy gap.
•
•
•
•
Classifications of materials into Conductors,
Semiconductors & Insulators:On the basis of magnitude of forbidden band the
solids are classified into insulators, semiconductors
and conductors.
Insulators:In case of insulators, the forbidden energy band is
very wide as shown in figure.
Due to this fact the electrons cannot jump from
valance band to conduction band.
In insulators at 00k and the energy gap between
valance band and conduction band is of the order.
Conduction band
Forbidden gap
INSULATORS
Valance band
Conduction band
SEMI CONDUCTORS
Forbidden gap
Conduction band
Valance band
Valance band
CONDUCTORS
SEMI CONDUCTORS:
• In semi conductors the forbidden energy ( band ) gap is very small as
shown in a figure.
• Ge and Si are the best examples of semiconductors.
•
Forbidden ( band ) is of the order of 0.7ev & 1.1ev.
CONDUCTOS:
• In conductors there is no forbidden gap. Valence and conduction
bands overlap each other as shown in figure above.
• The electrons from valance band freely enter into conduction band.
Effective mass of an electron
• The effective mass of an electron arises due to periodic potential provided by the
lattice.
• When an electron in a periodic potential of lattice is accelerated by an electric field,
then the mass of the electron varies, mass is called effective mass of the electron m*.
• Consider an electron of charge q and mass m acted on by electric field.
f  qE
ma  qE
qE
a 
m
• The acceleration
• Acceleration is not a constant in the periodic lattice of the crystal so mass of the
electron replaced by its effective mass m* when it is moving in a periodic potential
or crystal lattice.
• Now we can find a relation
for m* in terms of ‘e’ and
wave vector “k”.
• Consider the free electron
as a wave packet moving
with a velocity Vg
d
vg 
dk
where
2

 angular . frequency
t
k  wave.vector
d
vg 
dk
v g  2
d
dk
E
where..  2.,  
h
2 dE
vg 
h dk
1 dE
vg 
 dk
a
dv g
dt
1 d 2E 1
a
 dk dt
1 d 2 E dk
a
 dk 2 dt
sin ce., k  p
and ..
dp
F
dt
p
d
(
)
1 d 2E
 )
a
(
 dk 2
dt
1 d 2 E dp
a 2
( )
2
 dk dt
1 d 2E
a 2
F
2
 dk
The effective mass of an electron
1 d 2E
F
a 2
2
 dk
2
F
 2
a d E 2
dk
2

m  2
d E 2
dk
E
a. Variation of E with K
(a )
0
b. Variation of v with K
c. Variation of
m*
with K
d. Variation of fk with K
V
(b )
0
m
(c )
fk
The degree of freedom of an electron
is generally defined by a factor.
2
m
m d E
fk    2 { 2 }
m
 dk
(d )


a
0
k
k0

a