⚛️ Unit 4 – Part A (2-Mark Q&A)

1) Drawbacks of classical free-electron theory

• Overestimates electronic specific heat (predicts ~3/2 k_B per e⁻; experiment ≪).
• Cannot explain superconductivity, photoelectric & Compton effects.
• Fails to classify conductors/semiconductors/insulators (no bands).
• No account of quantum statistics or periodic potential.

2) Postulates of classical free-electron theory

• Metals contain many free (conduction) electrons moving randomly like gas.
• Electrons obey Newtonian mechanics & Maxwell–Boltzmann statistics.
• Ion cores form a uniform positive background; electron–ion/electron–electron forces neglected (except during collisions).
• Scattering gives a mean free time τ and path λ.

3) Mobility of electrons (definition & unit)

Mobility μ is the drift velocity per unit electric field: μ = v_d/E.

Units: m² V⁻¹ s⁻¹. Also, v_d = μE and σ = nqμ (n-type single carrier).

4) Wiedemann–Franz law & Lorenz number

For metals, the ratio of thermal to electrical conductivity is proportional to absolute temperature:

K/σ = L T, where L is the Lorenz (Lorentz) number.

Theoretical L: L = \pi^2 k_B^2/(3e^2) ≈ 2.44×10⁻⁸ W Ω K⁻².

5) Effective mass of electron (m*)

m* is the apparent inertial mass of an electron in a crystal responding to a force, determined by band curvature:

1/m* = (1/\hbar^2)\, d²E/dk² (from E–k diagram). It can be anisotropic, positive/negative, and differs from free mass m_e.

6) Mean free path (λ) & collision time (τ)

• Mean free path: average distance between two successive collisions. λ = v τ (use average speed).
• Collision/relaxation time: average time between collisions; sets damping of carrier motion.

Electrical conductivity: σ = n q² τ / m*.

7) Rest mass vs Effective mass

Aspect	Rest mass (me)	Effective mass (m*)
Definition	Mass of free electron in vacuum	Band-structure dependent inertial mass in crystal
Value	9.11×10⁻³¹ kg (constant)	Varies with E–k; can be ±, tensor
Origin	Intrinsic property	From periodic potential curvature

8) Statistics for electron energy distribution in metals

Fermi–Dirac statistics applies (quantum, indistinguishable fermions, Pauli principle).

Occupation probability: f(E)=1/[1+e^{(E−E_F)/(k_B T)}]. At T=0, all E<E_F filled.

9) Electrical & thermal conductivity (expressions)

• Electrical: σ = n q² τ / m*
• Thermal (free electrons): K = (π²/3)\, (k_B² T)\, (n q² τ / m*) / q² = L σ T

Here n = carrier density, q = e, τ = relaxation time, m* = effective mass.

10) Concept of a hole in semiconductors

A hole is an empty state in the valence band behaving like a mobile particle of charge +e and effective mass m*_h.

Under an electric field it drifts opposite to electron motion; current density J_p = q p μ_p E. Holes dominate in p-type material.