1. Electrical Engineering Materials
Dr. Md. Sherajul Islam
Associate Professor
Department of Electrical and Electronics Engineering Khulna University of Engineering & Technology
Khulna, Bangladesh

2. LECTURE - 2
Electrical and thermal conduction in solids

3. ELECTRICAL CONDUCTIVITY
In terms of electrical properties, the materials can be divided into three groups
(1) conductors (2) semi conductors and (3) dielectrics (or) insulators.
Electric current
The rate of flow of charge through a conductor is known as the current. If a charge ‘dq’ flows through the conductor for ‘dt’ second then
Ohm’s law
At constant temperature, the potential difference between the two ends of a conductor is directly proportional to the current that passes through it. where R = resistance of the conductor

4. ELECTRICAL CONDUCTIVITY
Resistance of a conductor
The resistance (R) of a conductor is the ratio of the potential difference (V) applied to the conductor to the current (I) that passes through it.
The specific resistance (or) resistivity of a conductor
The resistance (R) of conductor depends upon its length (L) and cross sectional area (A) i.e., or
where  is a proportional constant and is known as the specific resistance (or ) resistivity of the material.

5. ELECTRICAL CONDUCTIVITY
The electrical conductivity is also defined as” the charge that flows in unit time per unit area of cross section of the conductor per unit potential gradient”. The resistivity and conductivity of materials are pictured as shown below,
Conductivities and resistivities of materials

6. ELECTRICAL CONDUCTIVITY
Conductors
The materials that conduct electricity when an electrical potential difference is applied across them are conductors.
The resistivity of the material of a conductor is defined as the resistance of the material having unit length and unit cross sectional area.

7. ELECTRICAL CONDUCTIVITY
The electrical conductivity () of a conductor
The reciprocal of the electrical resistivity is known as electrical conductivity (σ) and is expressed in ohm1 metre1.
The conductivity ()
We Know that, R = V/I

8. Free Electron Theory
The electron theory explain the structure and properties of solids through their electronic structure.
It explains the binding in solids, behaviour of conductors and insulators, ferromagnetism, electrical and thermal conductivities of solids, elasticity, cohesive and repulsive forces in solids etc.
Development of Free Electron Theory
The classical free electron theory [Drude and Lorentz]
It is a macroscopic theory, through which free electrons in lattice and it obeys the laws of classical mechanics. Here the electrons are assumed to move in a constant potential.

9. Free Electron Theory
The quantum free electron theory [Sommerfeld Theory]
It is a microscopic theory, according to this theory the electrons in lattice moves in a constant potential and it obeys law of quantum mechanics.
Brillouin Zone Theory [Band Theory]
Bloch developed this theory in which the electrons move in a periodic potential provided by periodicity of crystal lattice.It explains the mechanisms of conductivity, semiconductivity on the basis of energy bands and hence band theory.
The Classical Free Electron Theory
According to kinetic theory of gases in a metal ,Drude assumed free electrons are as a gas of electrons.

10. The Classical Free Electron Theory
Kinetic theory treats the molecules of a gas as identical solid spheres, which move in straight lines until they collide with one another.
The time taken for single collision is assumed to be negligible, and except for the forces coming momentarily into play each collision, no other forces are assumed to act between the particles.
Here is only one kind of particle present in the simplest gases. However, in a metal, there must be at least two types of particles, for the electrons are negatively charged and the metal is electrically neutral.

11. The Independent Electron Approximation.
Drude Model
Drude’s Assumptions
Matter consists of light negatively charged electrons which are mobile, & heavy, static, positively charged ions.
The only interactions are electron-ion collisions, which take place in a very short time t.
The neglect of the electron-electron interactions is
The Independent Electron Approximation.
The neglect of the electron-ion interactions is
The Free Electron Approximation.

12. Drude’s Assumptions Continued
3. Electron-ion collisions are assumed to dominate. These will
abruptly alter the electron velocity & maintain thermal equilibrium
4. The probability of an electron suffering a collision in a
short time dt is dt/τ, where
1/τ  The Electron Scattering Rate.
Electrons emerge from each collision with both the direction &
magnitude of their velocity changed; the magnitude is changed
due to the local temperature at the collision point. 1/t is often an
adjustable parameter. See the figure.
The mean time between
collisions is t.
Trajectory of a mobile electron
Ion

13. Drude Model

14. Drude Model

15. Drude Model

16. Drude Model
Ohm’s law

17. Mean Free path

18. Temperature dependence of resistivity: pure metals:
If conduction electrons are only scattered by thermal vibrations of the metal ions, then in the mobility expression refers
to the mean time between scattering events by this process.
An electron moving with a mean speed u is scattered when its path crosses the cross sectional area s of a scattering center
To find the temperature dependence of conductivity, we first consider the temperature dependence of the mean free time, since this determines the drift mobility

19. Temperature dependence of resistivity: pure metals:

20. Temperature dependence of resistivity: pure metals:

21. Temperature dependence of resistivity: pure metals:

22. Success of classical free electron theory
It is used to verify ohm’s law.
It is used to explain the electrical and thermal conductivities of metals.
It is used to explain the optical properties of metals.
Ductility and malleability of metals can be explained by this model.

23. Drawbacks of classical free electron theory
From the classical free electron theory the value of specific heat of metals is given by 4.5R, where ‘R’ is called the universal gas constant. But the experimental value of specific heat is nearly equal to 3R.
With help of this model we can’t explain the electrical conductivity of semiconductors or insulators.
The theoretical value of paramagnetic susceptibility is greater than the experimental value.
Ferromagnetism cannot be explained by this theory.