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Classical free electron theory of metals

Learning objectives
 Postulates of classical free electron theory
 Electrical conductivity in metals

 Expression for thermal conductivity in metals
 Differentiate thermal and electrical conductivity

Introduction
• Classical free electron theory:
Drude and Lorentz developed this theory in the year 1900,
which assumed that metals contain free electrons obeying
the law of classical mechanics
...

1
...
These electrons are called free electrons or conduction
electrons, as they contribute to conduction in the metal
...
The free electrons move about identical with the motion of gas molecules and
hence assumed to obey the kinetic theory of gases
...

3
...

4
...


The drift velocity ‘vd’ is the average velocity acquired by the free
electron of a metal in a particular direction by the application of the electric field
...


The collision time ‘τc’ is the average time taken by a free electron
between two successive collisions
...

From this model, it is assumed that collision time is equal to relaxation time and the
thermal velocity is equal to drift velocity
...

 The directions of their velocities are random and on an average the
net velocity is zero as shown in the fig1(a)
...

 When an potential difference V is applied to a metal piece of length
L and cross- sectional area A as shown in fig 2, the electron paths

curve slightly because of the acceleration caused be the curvilinear
as shown in fig 1(b)

Fig 1 (a) Random motion of electron when no external electric field is applied
(b) Motion of electron with drift caused by external electric filed forces
𝑉

E= 𝐿

Cross-sectional area, A

E
Fig 2 Electric field applied in metals

The current I in a conductor is proportional to the voltage drop across the segment; this
can mathematically expressed as
V α I
V=I R
𝑉

I=𝑅

---------1

Where R is a constant of proportionality called the resistance
...
The unit of resistance is ohm that is represented by Ω
...

The electric field strength developed across the metal is E=
V=EL ---------3

𝑉
𝐿

Substitute eq 2 and eq 3 in eq 1

𝑉
I=
𝑅

𝐸𝐿 𝐸𝐴
=
=
𝜌𝐿/𝐴 𝜌

𝐼 𝐸
=
𝐴 𝜌
𝐼 𝐼
𝐴 𝜌

J= = × 𝐸 ------4
Where J is the electric current density and is defined as the ratio of current I flowing per
unit area A
...
The SI unit of conductivity is the reciprocal of

the unit of resistivity, that is (Ωm)-1 OR ohm per meter
Therefore eq 4 can be rewritten as

J=σE

------5

Eq 5 is called the point form or microscopic form of Ohm’s law
...


ϻ = 𝑉𝐸𝑑=−𝑒τ
𝑚
and hence the conductivity is given by

σ= ne ϻ
Mean Free Path:
The average distance travelled by an electron between any two successive collision is
known as mean free path and is given by

λ= vrms τ
Vrms is the root mean square velocity of the electron

Differentiate electrical and thermal conductivity
Electrical conductivity

Thermal conductivity

Electrical conductivity (σ) is the ratio Thermal conductivity (K) is the amount of
between the current density (J) and the heat conducted per unit volume per unit time ,
applied electric field strength (E)

per unit thermal gradient
...

 But it could not account for specific heat of metals,
 temperature dependence of σ=1/T and
 the dependence of electrical conductivity on free electron
concentration 𝝈 =

𝒏𝒆𝟐 𝝉
𝒎

Quantum Free Electron Theory:
Partial failure of classical free electron theory led Somerfield (1928)
to propose the quantum free electron theory
...

Following are the assumptions of quantum free electron theory
...
The valence electrons are free to move about inside the metal
...
The potential inside the metal is constant
...
The force of attraction between the free electrons and the ionic
core and the force of repulsion between electrons is negligible
...
The energies of electrons are quantized
...


Similarities between Classical and Quantum Free Electron
Theories:
1
...


2
...

3
...


Problems on Electrical
conductivity

Problem 1
The mean free collision time of copper at 300K is equal to
2×10-14 s
...
Given that
free electron density,
n = 8
...

Collision time, τ = 2×10-14 s
Free electron density n = 8
...
789×107 ohm-1m-1

8
...
6×10−19
9
...
92 × 103 kgm – 3 ,
resistivity = 1
...
5 a
...
u
...


d = 8
...
5 a
...
u
...
73 × 10-8 ohm m
Carrier concentration n =
𝐴𝑣𝑎𝑔𝑎𝑑𝑟𝑜′𝑠 𝑁𝑜
...
free electrons per atom
Atomic weight
6
...
92 × 103 × 1
n=
= 8
...
5
𝑚
ρ = 1\ σ = 2
ne τ
τ =

𝑚

ne2ρ

=

9
...
5 ×

1028

× (1
...
73 ×

10−8

=

2
...


1
ρn

e

=

1

1
...
5×1028 ×1
...
245 × 10−3

Problem 3 :
A metallic wire has a resistivity of 1
...
14 Vm-1
...

E = 0
...
42 × 10−8 ohm m n = 6 × 1028
Electrons/m3
...
1×10– 31

6×

1028

× (1
...
42 ×

10−8

=

4
...


Fermi energy (EF)
Electrons are fermions (s = ±1/2) and obey Pauli exclusion principle; each state
can accommodate only two electron
...
It is denoted by ‘EF‘
...

Features :
1
...

2
...

3
...

4
...

5
...


Fermi distribution function f(E)
• At a temperature T, the probability of occupation of an electron
state of energy E, given by Fermi distribution function f(E)
...
38 1023 J/K
= absolute temperature in K

• The Fermi distribution function determines the probability of
finding an electron at energy E
...

• Amount of energy required to move an electron from Fermi energy into
vacuum
...

• Expressed in electron volts (eV)
• Energy difference between Fermi energy and vacuum level corresponds to
work function
• Impurities and chemical changes on surfaces can change the work function
The energy required to make an electron to escape from a metal surface can
be

• thermionic emission: heat energy, emitted electrons are
thermions
• photoelectric emission: light energy
• Schottky effect: electrical energy
• Secondary emission: energy of highly energetic charged particles

Thermionic Emission
• When a metal is heated to a suitable temperature, electrons are emitted
from its surface is called “thermionic emission”
...
The metallic cathode is heated in order to
supply the electrons
...

• Temperature of the metal is increased, the electrons below EF gain
energy to move above EF
...
These electrons are responsible for thermionic
emission
...

KBT = ф +1/2mv2

Photoelectric effect
Photoelectric effect observed by Heinrich Hertz in
1887 during experiments with radio receiver
...

Emitted electrons are called photoelectrons
...


Photoelectric Emission



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