Download Band Theory of Solids

Band Theory of Solids
• This theory has been developed in three main stages.
• 1. The Classical free electron theory:
Drude and Lorentz developed this theory in the year
1900.According to this theory, the metals contains free
electrons and obey the laws of classical mechanics.
• 2.The Quantum free electron theory:
Sommerfeld developed this theory during
1928.According to this theory, free electrons obey
quantum laws.
• 3.The Zone theory:
Bloch stated this theory in 1928. According to this
theory, the free electrons move in a periodic field
provided by the lattice. This theory is also called Band
theory of solids.
• The energy band theory of solids is the basic
principle of semiconductor physics and it is
used to explain the differences in electrical
properties between metals, insulators and
semiconductors.
• The concept of discrete allowed electron
energies that occur in a single atom and large
number of allowed energies of electrons are
explained by band theory of solids
Electron in a periodic potential – Bloch
theorem:
• According to Bloch a one dimensional lattice
consists of large number of positive ion cores at
regular intervals and the conduction electrons
moves freely throughout the lattice.
• The variation of potential inside the metallic
crystal with the periodicity of the lattice is
explained by Bloch theorem.
It experiences
zero Potential at centre of the ion core and maximum
potential between the two ion cores. The motion of the
electron through periodic lattice can be explained by
Schrödinger's time independent equation.
• 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

[ E  V ]  0
2
2
dx
h
d 2 8 2 m

[ E  V ( x  a )]  0
2
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 periodicit y of crystal lattice and can be written as
U k (x)  U k (x  a)
where k represents the state of motion of the electron
Kroning – penney model
• According to Kroning - Penney , the model consists of
a series of potential wells and potential barriers alternately
as follows.
Potential barriers
Square wells
V0
+
+
-b
+
0
+
a
+
• The electrons move in a periodic potential field
provided by the lattice between V = 0 and Vmax=V0.
• Note: The potential of the solid varies periodically
with the periodicity of space lattice.i.e. V(x) =
V(x+a)……=V(x+Na).
• For one dimensional periodic potential field , to
describe the motion of electron, we have ….
d 2 8 2 m
 2 [ E  V ( x)]  0
2
dx
h
We know that the potential experienced by the electron is
Zero In square wells and becomes maximum in potential
barriers.
i.e,
V ( x)  o.... for,.0  x  a
and
V ( x)  V0 .. for,.  b  x  0
For Square wells the schrodinge r
equation is reduced to
d 2 8 2 m

[ E ]  0
2
2
dx
h
where as for potential barriers, the
equation t hat can be applied is
d 2 8 2 m
 2 [ E  V0 ]  0
2
dx
h
d 2
2


  0.....(1)
2
dx
d 2
2


  0.....(2)
2
dx
where
8 2 mE
 
h2
2
8 2 m
 
[V0  E ]
2
h
2
The solutions of wave functions Ψ for equations (1) and (2)
are of the form , according to Bloch
 ( x)  U k ( x)eikx ....(3)
Where Uk(x) is the periodicity of the lattice and can be written as
U k ( x)  U k ( x  a)  ..........U k ( x  Na)
Where N is an integer and
K is wave propagation constant.
2
p
k



Differentiating equation (3) and substituting in equation
(1) & (2) and
And further solving we get
 2  2
sinh b sin a  cosh b cos a  cos k (a  b)...(4)
2
Kroning penney suggested that when V0 tends to infinity,
b approaches in such a way that the product V0b remains
finite and it is called barrier strength. Under such situations
sinhβb  βb & cosh βb =1
Equation (4) can be written as
 2  2
b sin a  cos a  cos ka...(5)
2
Substituat ing for
8 2 m(V0  E )
8 2 mE
2
 
and  
2
2
In equation (5)
2
8 2 m(V0  E )
8 2 mE
(
)(
)
2
2


 b sin a  cos a  cos ka
2 
4 2 mV0 ab sin a
(
)
 cos a  cos ka
2

a
sin a
P
 cos a  cos ka......(6)
a
4 2 ma
where..P 
V0b
2
h
p is called scattering power of the
potential barrier wh ich is the measure
of strength w ith which electrons are
attracted by the positive ions.
P
sin a  cos a
a
+1
+1
a
-1
-1
Un allowed bands
Allowed bands
• 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.
3. 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..
As P  infinite, the allowed energy bands are compressed
into energy levels and a line spectrum is resulted.
p
sin a  0
p
0
a   n
a
 
 n
a
8 2 mE
As we know that  
2
 n 2 8 2 mE
(
) 
a
2
n2h2
E
8ma 2
2
It clearly explains that the particle (electron) is confined to
One dimensional infinitely deep square well potential box ,
Where we can find desecrate energies ( En = n2E1).
cos a  cos ka
4. If P tends to zero
n
No energy levels exist:
 k 
all energies are allowed for the electrons.
a
2  k2
p0
2mE
k   2

2 2
E  ( )k
2m
h 2 2 2
E  ( 2 )( )
8 m 
h2 1
E( ) 2
2m 
h2 p2 p 2 1 2
E  ( ) 2   mv
2m h 2m 2
2
a
All energies are possible for electron.
The energy spectrum is continuous
2
Brillouin zones:
• The Brilouin zone is a representation of
permissive values of k of the electrons.
 2k 2
p  k & E 
2m
p 2
n
k 



a
(or )
n  2a
which reflects Braggs condition for normal incidence
Therefore the electron suffers Bragg s reflection s at the
n
values k  
which results in discontinu ities in the E - K curve.
a
E-k diagram :
E
For free
electron
Energy gap
Allowed
bands
Energy gap
3

a
2

a



a
a
2
a
3
a
k
n
note : k  
is the boundary condition for
a
electron motion and at these values , the energy is discontinu ous
 The region between -

a
and 

a
is called first Brillouin zone

2
 2
 the region between - , 
and  ,
is called second
a
a
a
a
Brillouin zone.
llly the other Brillouin zones are explained for electron motion.
• 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 splits 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 are
(1) Valance band
(2) Conduction band
• The band corresponding to the outer most
orbit is called conduction band and the
inner band is known as 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 values of forbidden ( band ) gap,
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 below.
Due to this fact the electrons cannot move from
valance band to conduction band even at high
values of energies.
In insulators at 00k , the energy gap is of the
order of 5eV to 10eV.
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 band is very small as
shown in figure above.
• Ge and Si are the best examples of semiconductors.
• Forbidden ( band ) gap is of the order of 0.7ev &1.1ev.
CONDUCTOS:
• In conductors there is no forbidden gap and both valence
and conduction bands overlaps each other as shown in
figure above.
• The electrons moves freely from valance band to
conduction band and vice versa.
• The value of energy gap is zero.
• 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 the electric field, the mass of the electron
varies and
this mass is called effective mass of the electron ( m* ).
• Consider an electron of charge q and mass m acted upon
by electric field.
f  qE
• The acceleration
ma  qE
qE
a
m
• Acceleration ( a ) is not a constant in the periodic lattice of
the crystal so mass of the electron is 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”.
d
vg 
dk
where
• Consider the free
electron as a wave
packet moving with a
velocity Vg, called group
velocity and is defined as
change in angular
velocity with respect to
wave propagation
2

 angular . frequency
T
k  wave.vector
d
vg 
dk
d
v g  2
dk
where..  2.,  
2 dE
vg 
h dk
1 dE
vg 
 dk
E
h
a
dv g
dt
1 d 2E
a
 dkdt
1 d 2 E dk
a
 dk 2 dt
sin ce., k  p
and ..
dp
F
dt
p
)
1 d E
 )
a
(
 dk 2
dt
1 d 2 E dp
a 2
(
)
2
 dk
dt
1 d 2E
a 2
F
2
 dk
2
d(
F
a
2
d 2E 2
dk
F
a *
m
2

where., m*  2
d E 2
dk
is called effective mass of
an electron in periodic lattice
E
a. Variation of E with K
b. Variation of v with K
c. Variation of m* with K
(a )
(b )
Point of
inflection
0
V
0
m
d. Variation of fk with K
(c )
The degree of freedom of an electron
is generally defined by a factor.
fk
2
m
m d E
fk    2 { 2 }
m
 dk
(d )


a
0
k
k0


a
BONDING IN SOLIDS
Introduction:
•A solid is composed of billions of atoms packed closely
together and solids have usually strong elastic structures.
•According to strength and directionality the inter atomic
forces or bonds in solids are grouped into primary bonds
and secondary bonds.
CLASSIFICATION OF BONDS IN SOLIDS
BONDS
Bond energy
Bond
range
energy rang
0.1-10ev 0.01-0.5ev
PRIMARY SECONDARY
nter atomic
Inter molec
VANDER-W
HYDROGEN
IONIC
COVALENT
METALLIC
H
o
2
Neon
Nacl
H2,clNa,Al
2
in the form of ice
• Primary bonds are the strongest bonds
which hold atoms together. The three
types of primary bonds are
1.Ionic
2.Covalent
3.Metallic
• Secondary bonds are much weaker
than primary bonds and they are
1.Hydrogen bond
2.Vander waal’s bond
Ionic bond:
• An ionic bond is an attractive force existing
between a positive ion and a negative ion when
they are brought into close proximity.
Example:1
Na
Na+ + eCl + eclNa+ + clNacl
Example:2
Mg
Mg2+ + 2e2cl + 2e2clMg2+ + 2clMgcl2
Mgcl2, KOH and Al2o3
Formation of ionic bond in Nacl
Na
Cl
Na+ and Cl- ions formed by ionic bonding mechanism
Na+
Cl-
Characteristics of ionic bond:
• It is a very strong bond and ionic crystals
are rigid.
• These solids having high melting and
boiling point values.
• Conductivity is very less and these solids
are soluble in water and liquid ammonia.
• Transparent to visible light.
• Non-directional because the charge
distribution is spherical in nature.
• Ionic crystals have close packed structure.
• Melting points of ionic solids:
1. NaF
2. Nacl
3. NaBr
4. NaI
1270k
1073k
1023k
924k
Covalent bond:
• In this type of bonding the valence electron
are not transferred from one atom to the other
atom but their neighboring atoms share their
valence electrons under the formation of a
covalent bond.
+
H
+
H
+ +
H H
• Characteristics of covalent bond:
• These crystals are very hard and brittle.
• Bonding energy is very high, so MP & BP values
are high.
• These crystals are soluble in non-polar solvents
like benzene.
• Conductivity increasing with increasing
temperature.
• Covalent bonds are highly directional in
character.
• These bonds have saturation property.
• These crystals are transparent to longer
wavelengths but opaque to shorter wavelengths.
• Melting points of covalent solids:
• Diamond 3280k
Comparison between ionic and covalent solids:
Ionic solids:
Covalent solids:
• The bonds are non
directional.
• Bonds are relatively
stronger.
• Soluble in polar solvents.
• Not very hard.
• Possess high melting and
boiling points.
• The bonds are directional
• The bonds are relatively
weaker.
• Soluble in non polar
solvents.
• Very hard.
• Comparatively lower MP
and BP values.
Metallic bonding:
Positive ion co
+ + +
+ + +
+ +
 The valence electrons from
all the metallic atoms
belonging to the crystal are
free to move through out
the crystal.
 The crystal may be
considered as an array of
positive metal ions
embedded in a “cloud” or
“sea” of free electrons as
shown in figure.
 The bond formed between
these positively charged
metal ions to the negatively
charged electron cloud is
called metallic bond.
Electron clouds
Characteristics of metallic bond:
• Metallic solids are malleable and ductile.
• They have high electrical and thermal conductivities.
• Metallic solids are not soluble in polar and non polar solvents.
• These metals have high optical reflection and absorption coefficients.
• Due to the symmetrical arrangement of the positive ions in space lattice,
metals are crystalline.
• The metals are opaque to all electromagnetic radiation from very low
frequency to the middle ultraviolet, where they become transparent.
• Examples: Na, Aluminum.
Hydrogen bonds:
• The electrostatic force of attraction between
the hydrogen atom of one molecule and
more electronegative atom of the same or
another molecule is called hydrogen bond.
Example: H2o molecule in the form of ice.
Hydrogen bonds in H2O molecule
H
O
H
H
Hydrogen bonds
O
H
H
O
H
Characteristics of hydrogen bond:
1. Weak binding
2. Low electrical and thermal conductivities.
3. Hydrogen bonded crystals are
transparent.
4. These crystal have peculiar directional
properties.
5. Lose structure.
Vander - waal’s bond:
• This type of bonding arises due to mutual
polarization of atoms due to each other or
fluctuating dipole bonds between hydrogen
atoms are known as vander – waal’s
bonding and they are non directional.
• These bonds especially takes place in
noble gases which are cooled to very low
temperature.
Example: solid Ne & Ar
When an electron cloud density occurs at one side of an
atom or molecule during the electron flight about the
nucleus, vander - waal’s forces are generated.
Atom-A
+
Atom-B
_
+
_
High density clouds
High density clouds
Attraction
Characteristics;
1. Weak bonding and soft crystals.
2. Low melting and boiling points.
3. Poor electrical conductors.
4. Closed packed structure usually transparent
to electromagnetic radiation.
5. Non directional.
Forces between atoms:
• Electrical forces are responsible in binding the atoms
giving different solid structures.
• The forces between atoms can be of two types
1. Attractive forces
2. Repulsive forces
• Attractive force which keeps the atoms together. And
repulsive force which comes into existence when the
distance between atoms is small.
• The resultant force is equal to sum of the both factors.
Variation
F (r)of F (r)
Attractive for
inter
Resultant
force
atomic
r
F
max
ospacing
(r
r0
B

r
)
Repulsive fo
• If two atoms are
separated by the
distance r, the net
force between
the atoms is
given as
A
f (r )  M
r
N
Where A,B,M and N are
constants
• The first term in the above equation represents attractive
force which according to inverse square law.
• The second term represents the repulsive force which is
very strong at small distances and the value of N≈9.
• When the separation between atoms is equal to zero. the
separation r0 is called equilibrium distance.
f (r ) 
A
ro M

B
rNo
equilibrium position
f (r )  0
A
ro
M

B
rNo
ro N
B

r0 M
A
B
r0

A
B N 1M
r0  [ ]
A
N M
• Cohesive energy:
• It is defined as the minimum amount of
energy is require to form bond in between
atoms or ions.
• It is also called as bond energy or bond
dissociation energy.
• Calculation of cohesive energy :
• To calculate the cohesive energy let us consider the general situation of two
identical atoms.
• The potential energy = decrease in potential energy due to attraction +
increase in potential energy due to repulsion.
• Work done on the system is stored as potential energy and work done
moving through small distance dr is given by
U (r ) 
du
(
r
)



f ( r ) dr
A
B
  [ M  N ]dr
r
r
  [ Ar
  Ar
M
M
 M 1
 Br
N
]dr
dr   Br
N
dr
 N 1
Ar
Br


c
M  1  N  1
1 M
1 N
Ar
Br


c
1 M
1 N
A
1
B
1

[ M 1 ] 
[ N 1 ]  c
M 1 r
N 1 r
a
b
 M 1  N 1  c
r
r
a
b
U (r )  m  n  c
r
r
Where a and b are new constants related
to A and B as a a = A/M-1
b = B/N-1
n = N-1
a
b
U (r )  m  n
and m = M-1
When r =∞,U=0 hence c=0
r
r
t a equilibrium position r=r0 the potential energy is minimum
URepulsive energy
U (r ) 
o
a b
 n
m
r
r
U (r ) 
b
rn
r0
r
Attractive energy
U (r ) 
a
rm
du
[ ]r  r0  0
dr
For
d a b
minimu
{ [
 ]}
dr r
r
m
am
bn

0
potentia
r
r
am
bn
l energy

m
n
m 1
0
r0
r0
n 1
0
m 1
nm
r0
bn

am
n 1
r  r0
0
bn
r0  [ ]
am
1
nm
a
b
U ( r0 )  m  n
ro
r0
This energy corresponding to the equilibrium position (r=r0)
is called the bonding energy or the energy of cohesion of the
molecule.
• This is also called the energy of dissociation
this can be calculated as follows.
bn
r0  [
]
am
1
nm
1
1 am
 m[
]
n
r0
r0 b n
a
b
U ( r0 )  m  n
ro
r0
a
1 am
 m  b{ m [
]}
ro
r0 b n
a
a m
 m { m  }
r0
r0
n
a
m
U (r0 )  m [1  ]
r0
n
Thus the minimum value of energy U (ro) is negative.
The positive quantity U (ro) is the dissociation energy
the molecule since m≠n the attractive and repulsive
energies are not equal though the attractive and
repulsive force’s are equal in equilibrium.
The total binding is essentially determined by the
energy of attraction.
2
d U
[ 2 ]r  r0  0
dr
Since U has a minimum at
r=r0
dU
am(m  1) bn(n  1)
[ 2 ]r r0 


0
m 2
n2
dr
r0
r0
2
 am(m  1)r0
bn(n  1)r0
n2
m 2
 bn(n  1)r0
m2
0
 am(m  1)r0
n2
Substituting for ro
b n
bn(n  1)  am(m  1)( )( )
a m
(n  1)  (m  1)
nm
Hence we understand that the forces acting
between the atoms are mostly electro static in
Calculation of Cohesive energy of Ionic
solids:
• The bond energy of diatomic ionic molecules
different from the cohesive energy of an ionic
crystal.
• The ionic molecules being stable because of
presenting the following balancing forces.
1. Electro static coulomb attractive forces
2. Negligible vander - waal’s forces of
attraction.
3. Inter atomic repulsive forces
• The resultant force of attractive and repulsive forces
brings the system to an equilibrium state where the
minimum potential energy is present.
• Let us consider ions of charges Z1e and Z2e
separated by a distance ‘r’
2
z1 z2e
Attractiveforce 
2
4 0 r
For complete crystal coulomb potential
energy
Where A is Mode lung's constant
Az1 z2e

4 0 r
Potential energy due to repulsive force
b
 n
r
The potential energy of a crystal can be expressed as
2
Az1 z2 e
b
U (r )  
 n
4 0 r
r
2
For univalent alkali halides Z1=Z2=Z3…there fore
 Ae 2
b
U (r ) 
 n
4 0 r r
The total energy of on kile mole could be written as
b
Ae2
U (r )  N A [ n 
]      (1)
r
4 0 r
r  r0 andU (r )  U min
dU
[
]r r0  0
dr
Ae
2
bn

2
n 1
4 0 r0
r0
2
n 1
0
Ae r
b
     (2)
4 0 n
Substituting equation 2 in 1
[U (r )]r  r0  U min
2 n 1
0
n
0
0
2
Ae r
Ae
 N A{

}
4 nr
4 0 r0
2
U min
Ae N A 1

[  1]
4 0 r0 n
Where Umin represents cohesive energy of an ionic
compound.
• Made lung constant (A):
• The Made lung constant A is function of crystal
structure.
• It can be calculated from the geometrical
arrangement of ions in the crystal.
• Let us consider the equilibrium positions of ions in
Nacl structure as shown in a figure.
• Let us choose the central Na+ ion as the reference
ion having a single positive charge on it.
• Six cl- ions are surrounding this Na+ ion first
nearest neighbors.
• Let us consider them at unit distance.12 Na+ ions
are the second nearest neighbors at a distance
√2.
+
Na
Cl
1 unit
+
Na
1 unit
√2 units
Cl
+
Na
1 unit
√3 units
√2 units
Cl
• Eight cl- ions are the third nearest neighbors at a
distance √3 and so on.
• The Made lung constant for the Nacl structure can be
written as a summation series.
6 12 8
6 24 24
A 




 .............
1
2
3
4
5
6
This converges to a value 1.74756
Made lung constants for some typical
Ionic crystals:
•
•
•
•
Nacl
1.74756
Cscl
1.76267
Fluorite 2.51939
Zinc
1.638
Bond energy:
Bond energy is defined as the energy of the formation of
one kilo mole of a substance from its atoms or ions.
Bond energy of Nacl molecule:
Consider Na and Cl atoms are infinitely separated.
•
The ionization energy of a Na atom is 5.1ev.
Na + 5.1ev = Na+ + e-------------1
•
The electron affinity of a chlorine is 3.6ev
cl + e- = cl- + 3.6ev--------2
• Thus the net energy required to form one sodium ion and
one cl ion is equal to 5.1ev-3.6=1.5ev
• Electrostatic attraction between Na+ ion and cl- ion
brings them together to the equilibrium spacing r0 =
0.24nm and forms an ionic bond resulting Nacl molecule.
• Potential energy of Na+ and cl- ions at equilibrium
position
e 2

4 0 r0
(1.602 1019 ) 2

4  8.85 1012  0.24 1091.602 1019
 6ev
• The negative sign indicates that there is
electrostatic attraction between the ions.
• The potential energy of resultant Nacl
molecule => 1.5 - 6 = - 4.5ev
• Bond energy of Nacl molecule = -potential
energy of Nacl molecule = 4.5ev
• Thus the formation of sodium chloride
molecule from their ions can be written as
Na+ + cl-
Na+ + cl- + 4.5ev
Nacl