Download - Snistnote

Free electron theory of Metals
Introduction
• The electrons in the outermost orbital's of the atoms
determine its electrical properties.
• The electron theory of solids aims to explain the
electrical, thermal and magnetic properties of solids.
• This theory has been developed in three main stages.
1.Classical free electron theory
2.Quantum free electron theory
3.Band theory of solids.
• Classical free electron theory:
Drude and Lorentz developed this theory in 1900. According to
this theory the metals containing free electrons obey the laws of
Classical Mechanics.
• Quantum free electron theory:
Somerfield developed this theory during 1928. According to this
theory free electrons obey the Quantum laws.
• Band theory of solids or Zone theory:
Bloch stated this theory in 1928. According to this theory, the
free electrons move in a Periodic field provided by the lattice
and the theory is also called Band theory of solids.
Classical free electron theory
o In an atom electrons revolve around the nucleus and every metallic
substance is composed of such like atoms.
o The valance electrons of atoms are free to move about the whole
volume of metals like the molecules of a perfect gas in a container.
o These free electrons moves in random directions and collide with
either positive ions fixed to the lattice or other free electrons.
o These all collisions are elastic that is there is no loss of energy.
o The electron velocities in a metal obey the laws of Classical
Maxwell - Boltzmann distribution.
• The free electrons move in a completely uniform
potential field due to ions fixed in the lattice.
• When an electric field is applied to the metal, the free
electrons are accelerated in the direction opposite the
direction of applied electric field.
• When an electric field E is applied on an electron of
charge e then it moves in opposite the direction of
applied electric field with a velocity Vd.
Electric field
Fixed Positive metallic ions
The Lorentz force acting on the electron FL  eE
The frictional force or opposing force can be expressed as
Fr  
mvd
r
Where vd is a drift velo city &  r is a relaxation time
When the system is in a steady state total force F{FL  Fr }  0
m
mv
dv
 eE  d
dt
r
 eE 
vd  (
mvd
r
 e r
m
0
)E
• If n is the number of conduction electrons per unit
volume, then the charge per unit volume is –ne.
• The amount of charge crossing a unit area per unit
time is given by the current density J.
i
J
A
 (ne)vd
 (ne)(
 e
E)
m
ne 2

E  E
m
ne 2
where 
 Electrical Conductivi ty
m
Drift velocity: The amount of velocity gained by the
electron by the application of unit electric field is
known drift velocity.
•
•
•
•
Relaxation time:
The duration of a time required for an electron to
decay its drift velocity to 1/e times of its initial
velocity.
When electric field is applied on an electron then it
drifts in opposite direction to the field applied.
After removal of electric field the drift velocity decays
exponentially.
Let us assume that the applied field is cut off after the
drift velocity of the electron has reached its steady
value.
dvd
v
 m d
dt

dvd
v
 d
dt

dvd
dt

vd

m
t
t
dvd
dt
0 vd  0  
{log vd }t0  
t

log vd (t )  log vd (0)  
Relaxation time can be defined as the
v (t )
t
log d

time taken for the drift velocity to decay
v d ( 0)

to 1/e of its initial value.
t
vd (t )  vd (0) exp(  )

ift    vd (t ) 
v d ( 0)
e
t

Mean free Path λ:
The average distance travelled by an electron between two
successive collisions in the presence of applied field is known as
mean free path.
Mobility µ:
Mobility of the electron is defined as the steady state drift velocity
per unit electric field.
e
E
vd
e
m



E
E
m
ne 2
e
conductivity 
 ne.
   ne
m
m
1
m
1
resistivit y   2 

ne 
ne

Drawbacks of Classical free electron theory
1. The phenomena such as photoelectric effect, Compton effect and
black body radiation couldn't be explained by classical free electron
theory.
2. According to the classical free electron theory the value of specific
heat and electronic specific heat of metals is given by 4.5Ru and
3/2Ru while the actual values are about 3Ru and 0.015Ru.
3. Electrical conductivity of semiconductor or insulators couldn’t be
explained using this model.
4. Ferromagnetism couldn’t be explained by this theory.
5. Wiedemann-franz law ( K / σT =L ) according to the classical free
electron theory, it is not constant at low temperature.
•
•
•
•
•
Quantum theory of free electrons
Somerfield proposed the quantum free electron theory
and he assumed that the valance electron are free in a
metal piece and they obey quantum laws .
According to quantum theory the free electrons occupy
different energy levels present in the metal.
According to this theory only Fermi level electrons are
responsible for electrical conductivity and other
properties of metals.
By this theory Somerfield exactly predicted the values of
specific heat of metals ,electrical conductivity and
thermal conductivity of metals.
If free electrons are moving with very higher velocities,
the potential energy becomes zero.
Total energy of outer most electron
E= KE + PE= p2/2m + V → p2/2m
From de Broglie Matter wave equation

h
p
2
2
p
p
k



h
h


(
)
p
2
p  K
p2
fromeq (1) E 
2m
 2k 2
E
2m
n
k
L
n 2
2 (
)
2 2
2
2 2

n

h
n
L
E
E

E

2m
2mL2
8mL2
n
When an external electric field E is applied the force
exerted on the electron is
F  eE
dp
 eE
dt
d (k )
 eE
dt
dk
 eE

 eE  dk 
dt
dt

dv dp
F m

dt
dt
dv d (k )

m

 dv 
dk
dt
dt
m
  eE
 eE
dv 
(
dt ) 
dt
m

m
If the no of electrons per unit volume is n, then the current
density J is given by….
J   ne( dv )
 eEdt
J   ne(
)
m
ne 2 dt
J 
E
m
J  E
ne 2 dt
where 
m
Draw backs
•It does not explain temperature variation of electrical conductivity
•It does not explain why metals prefer only certain structures.
•Distribution means how the electrons in materials
distribute among the different possible energy states.
•A small metal piece contains very large number of
electrons
•Each electron posses quantized energy states .
•The electrons obey Pouli’s exclusion principle.
•Hence the suitable statistics to find the distribution of
electrons in a metal is Fermi Dirac distribution.
 According to this law, the probability F(E) or P(E) of an
electron occupying that a given energy state ‘ E ’ at
temperature ‘ T ’ is given by
F(E) = P(E) =
1
--------- (1)
1 + e (E –EF) / KBT
Where,
F(E) = P(E) = Fermi – Dirac probability function
KB = Boltzmann constant
Explanation(F –D statistics)
E
E
At
T=0K
EF
Where,
E3 > E 2 > E1 > E F
T3 > T 2 > T 1 > T
E3
E2
E1
At
T=0K
EF
O
1.0
T1
T2
T3
P(E)
Figure-1
O
1.0
Figure-2
P(E)
From equation (1) we may discuss the following 2 cases.
CASE:1
(1)
Let T=0K
for E < EF , e(E-E F/KT) = 0
F(E)=1
This means all energy states below EF are completely
filled.
(2)
Let T=0K
for E > EF , e(E-E F/KT) = 
F(E)=0
then the exponential term becomes infinite and F(E) = 0
i.e. there is no probability of finding an occupied state of
energy greater than EF at absolute zero.
Hence, This means all energy states EF are empty.
That means EF is the Fermi energy level is the energy level
of the Highest occupied states at 0k.
CASE:2
Let T> 0K and E = EF ,
Then F(E) = P(E) = ½ at any temperature.
Thus, Fermi level is that energy level for which the
probability of occupation is
(½) at any
temperature above 0K.
The effect of temperature on Fermi Dirac Distribution
 As the temperature increases, thermal agitations grows
up.
 As a result more and more electrons jump up to the
levels above EF from the levels below EF leaving
vacancies as shown
EF
EF
T=0
T>0
When T > 0 K , some of levels below EF are not
completely filled up and some of the levels above EF
are not completely empty.
They are partially filled and partially empty.
Both the electrons in the levels above EF and
vacancies in the levels below EF contribute for
conduction in semi conductors.
But the metals are good conductors even at
room temperatures.
If the temperature raised, the resistance of the
metals increases due to increase of the collisions
or decrease of the mobility.
Use :
By using this we can explain specific heat of
metals.
Free electron theory of
Metals
Introduction
• The electrons in the outermost orbital's of the
atoms determine its electrical properties.
• The electron theory of solids aims to explain the
electrical, thermal and magnetic properties of
solids.
• This theory has been developed in three main
stages.
1.Classical free electron theory
2.Quantum free electron theory
3.Band theory of solids.
• Classical free electron theory
Drude and Lorentz developed this theory in 1900.
According to this theory the metals containing free
electrons obey the laws of Classical mechanics.
• Quantum free electron theory
Somerfield developed this theory during 1928. According
to this theory free electrons obey the Quantum laws.
• Band theory of solids or Zone theory
Bloch stated this theory in 1928. According to this
theory, the free electrons move in a Periodic field
provided by the lattice and the theory is also called Band
theory of solids.
Classical free electron theory
o In an atom electrons revolve around the nucleus and every metallic
substance is composed of such like atoms.
o The valance electrons of atoms are free to move about the whole
volume of metals like the molecules of a perfect gas in a container.
o These free electrons moves in random directions and collide with
either positive ions fixed to the lattice or other free electrons.
o These all collisions are elastic that is there is no loss of energy.
o The electron velocities in a metal obey the laws of Classical Maxwell -
Boltzmann distribution.
• The free electrons move in a completely uniform
potential field due to ions fixed in the lattice.
• When an electric field is applied to the metal, the
free electrons are accelerated in the direction
opposite the direction of applied electric field.
• When an electric field E is applied on an electron
of charge e then it moves in opposite the direction
of applied electric field with a velocity Vd.
Positive ion cores
_
_
_
_
_
_
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Electron clouds
_
_
_
_
_
_
The Lorentz force acting on the electron FL  eE
The frictional force or opposing force can be expressed as
Fr  
mvd
r
Where vd is a drift velo city &  r is a relaxation time
When the system is in a steady state total force F{FL  Fr }  0
m
mv
dv
 eE  d
dt
r
 eE 
vd  (
mvd
r
 e r
m
0
)E
• If n is the number of conduction electrons per unit
volume, then the charge per unit volume is –ne.
• The amount of charge crossing a unit area per
unit time is given by the current density J.
i
J
A
 (ne)vd
 (ne)(
 e
E)
m
ne 2

E  E
m
ne 2
where 
 Electrical Conductivi ty
m
Drift velocity: The amount of velocity gained by
the electron by the application of unit electric
field is known drift velocity.
•
•
•
•
Relaxation time:
The duration of a time required for an electron
to decay its drift velocity to 1/e times of its
initial velocity.
When electric field is applied on an electron then
it drifts in opposite direction to the field applied.
After removal of electric field the drift velocity
decays exponentially.
Let us assume that the applied field is cut off
after the drift velocity of the electron has
reached its steady value.
dvd
v
 m d
dt

dvd
v
 d
dt

dvd
dt

vd

m
t
t
dvd
dt
0 vd  0  
{log vd }t0  
t

log vd (t )  log vd (0)  
Relaxation time can be defined as the
time taken for the drift velocity to decay
to 1/e of its initial value.
log
vd (t )
t

v d ( 0)

t
vd (t )  vd (0) exp(  )

ift    vd (t ) 
v d ( 0)
e
t

Mean free Path λ:
The average distance travelled by an electron between two successive
collisions in the presence of applied field is known as mean free path.
Mobility µ:
Mobility of the electron is defined as the steady state drift velocity per
unit electric field.
e
E
vd
e
m



E
E
m
ne 2
e
conductivity 
 ne.
   ne
m
m
1
m
1
resistivit y   2 

ne 
ne

Drawbacks of classical free electron theory
1. The phenomena such as photoelectric effect, Compton effect and
black body radiation couldn't be explained by classical free electron
theory.
2. According to the classical free electron theory the value of specific
heat and electronic specific heat of metals is given by 4.5Ru and 3/2Ru
while the actual values are about 3Ru and 0.015Ru.
3. Electrical conductivity of semiconductor or insulators couldn’t be
explained using this model.
4. Ferromagnetism couldn’t be explained by this theory.
5. Wiedemann-franz law ( K/œT =L ) according to the classical free
electron theory, it is not constant at low temperature.
Quantum theory of free electrons
• Somerfield proposed the quantum free electron theory
and he assumed that the valance electron are free in a
metal piece and they obey quantum laws .
• According to quantum theory the free electrons occupy
different energy levels present in the metal.
• According to this theory only Fermi level electrons are
responsible for electrical conductivity and other properties
of metals.
• By this theory Somerfield exactly predicted the values of
specific heat of metals ,electrical conductivity and thermal
conductivity of metals.
• If free electrons are moving with very higher velocities, the
potential energy becomes zero.
Total energy of outer most electron
E= KE + PE= p2/2m + V → p2/2m
From de Broglie matter wave equation….

h
p
2
2
p
p
k



h
h


(
)
p
2
p  K
p2
fromeq (1) E 
2m
 2k 2
E
2m
n
k
L
n 2
2 (
)
2 2
2
2 2

n

h
n
L
E
E

E

2m
2mL2
8mL2
n
• When an external electric field E is applied the force
exerted on the electron is
F  eE
dp
 eE
dt
d (k )
 eE
dt
dk
 eE

 eE  dk 
dt
dt

dv dp
F m

dt
dt
dv d (k )

m

 dv 
dk
dt
dt
m
  eE
 eE
dv 
(
dt ) 
dt
m

m
If the no of electrons per unit volume is n, then the current
density J is given by….
J   ne( dv )
 eEdt
)
m
ne 2 dt
J 
E
m
J  E
J   ne(
ne 2 dt
where 
m
Draw backs
•It does not explain temperature variation of electrical conductivity
•It does not explain why metals prefer only certain structures.