Electrical and Thermal Properties of Solids

What is the Free Electron Theory of Solids and how did it come about?

Aims to explain the strucutres and properties of solids through their electronic structure:

1. Drude and Lorentz developed the Classical Free Electron Theory (1900) which assumed that metals contain free electrons obeying the laws of classical mechanics

2. Arnold Sommerfeld developed the Quantum Free Electron Theory (1928) which assumed that the free electrons obey quantum laws

3. Bloch proposed the Zone theory (1928) acc to which the electrons movie in a periodic potential provided by the lattice. This theory was able to explain almost all the properties of materials. This theory was also called as Band Theory of Solids because it explained the mechanism of superconductivity based on the band model.

List the Postulates of CFET

1. Large number of atoms combines to form a metal; the boundary of the neighbouring atoms slightly overlap on each other. Due to this overlapping there will be continuity from one atom to atom, thus valence electrons can move easily from throughout the body of the metal.

2. The free moving electrons not belongs to particular atom but each of them belongs to the metal to which they are confined to. Thus, each such electrons are called a free electron.

3. The electrons in the closed shells are called core electrons and those in the outer incomplete shell are called valence electrons.

4. The core electrons are strongly attracted by the nucleus and the valence electrons are free electrons.

5. In 1900, Drude assumed that the electrons in a metal are free to move and form electron gas.

Lorentz predicted that the kinetic theory of gases could be applied to the free electron gas.

When atoms are brought closer to form metal, the valence electrons get detached and move freely through the metal. Hence they are called free or conduction electrons.

The concentration of free electrons is ≈ 10²⁸/m³

List the Postulates/Assumptions in CFET

1. A metal is imagined as a structure of a 3-dimensional array of ions, in between which there are freely moving valence electrons confined to the body of the material and are called conduction electrons or free electrons.

2. These free electrons move in random directions and make collisions with either the ions fixed in the lattice or other free electrons. All the collisions are elastic in nature, so there is no loss of energy.

3. The free electrons are treated as gas molecules; thus they are assumed to obey the laws of kinetic theory of gases.

4. 4. The electric field due to the ionic cores is taken to be constant throughout the metal and the effect of repulsion between electrons will be neglected.

5. The potential energy of electrons moving through metals is constant (considered as zero). Therefore, the total energy of electrons is equal to their kinetic energy.

6. The electric current in a metal due to an applied field is a consequence of the drift velocity in a direction opposite to the direction of the field.

List the Merits of CFET

1. It explains the electrical conductivity and thermal conductivity of metals.

2. It explains the Wiedemann–Franz law.

3. Successfully explains the validity of Ohm’s law.

4. It explains the optical properties of metals.

What are the Limitations of CFET?

1. Specific Heat

2. Dependence of Electrical Conductivity on Temperature

3. Dependence of Electrical Conductivity on electronic concentration

4. CFET predicts that resistivity varies as √T, but actually it varies linearly with temperature.

5. CFET states that at low temperatures, the ratio of electrical conductivity (σ) and thermal conductivity (k) is constant but it is not true.

6. CFET states that all free electrons absorb the total supplied energy, but quantum theory states only few electrons absorb the supplied energy.

7. CFET fails to explain the electrical conductivity of semiconductors and insulators.

8. CFET fails to explain Photoelectric effect, Compton effect, Black body radiation.

9. CFET fails to explain variation of paramagnetic susceptibility with temperature.

10. CFET also fails to explain ferromagnetism.

List the Postulates of QFET

The theory proposed by Arnold Sommerfeld by using Fermi-Dirac statistics is known as quantum free electron theory (QFET).

The QFET is based on the following assumptions:

1. The allowed energy levels of an electrons are quantized.

2. The distribution of electrons in the various allowed energy levels takes place according to Pauli’s exclusion principle.

3. The velocity and energy distribution of the free electrons are governed by the Fermi-Dirac distribution function.

4. The electrons are assumed to move in a constant potential field.

5. The mutual repulsion between the electrons are assumed to be negligible.