💻 24PY112 Unit 4 – Part B (Essay Q&A)

Q1: Using the classical free electron theory, derive the mathematical expression for electrical conductivity and thermal conductivity of metals and hence deduce the Wiedemann-Franz law.

(i) Derivation of Electrical Conductivity (σ)

In the classical free electron theory, when an electric field 'E' is applied to a metallic rod, the electrons experience a force.

By equating (1) and (2), we get the acceleration 'a' of the electron:

This acceleration is related to the drift velocity (v_d) and relaxation time (τ):

Substituting (3) into (4):

The current density (J) is defined as J = nv_de, where 'n' is the free electron density. Substituting (5) into this equation:

According to Ohm's law, current density is also J = σE. By comparing this with equation (6), we get the expression for electrical conductivity (σ):

This is the expression for electrical conductivity based on classical free electron theory.

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(ii) Derivation of Thermal Conductivity (K)

Thermal conductivity (K) is defined as the amount of heat (Q) conducted per unit time (t) per unit area (A) per unit temperature gradient (dT/dx).

Consider a uniform rod with a hot end A at temperature T and a cold end B at T-dT, separated by a mean free path λ. The number of electrons crossing a unit area per unit time is (1/6)nv, where 'n' is the electron density and 'v' is their average velocity.

• Average kinetic energy at A = (3/2)kT
• Average kinetic energy at B = (3/2)k(T-dT)

The net amount of energy transferred from A to B per unit area per unit time (Q) is:

From the basic definition of thermal conductivity, the heat conducted per unit time per unit area is Q = K(dT/λ). Equating this with (8):

Since the mean free path λ = vτ (where τ is collision time, assumed equal to relaxation time), we can substitute this into (9):

This is the classical expression for thermal conductivity of a metal.

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(iii) Deduction of Wiedemann-Franz Law

The Wiedemann-Franz law states that for metals, the ratio of thermal conductivity (K) to electrical conductivity (σ) is directly proportional to the absolute temperature (T).

To deduce this, we divide equation (10) by equation (7):

From the kinetic theory of gases, the kinetic energy of an electron is (1/2)mv² = (3/2)kT. Substituting this into (11):

Comparing this with K/σ = LT, we find the Lorentz number L:

Since k (Boltzmann's constant) and e (electron charge) are constants, L is a constant. Thus, the law is proved: K/σ = LT, showing the ratio is directly proportional to the absolute temperature T.

Q2: What is density of states? Derive an expression for the density of states and carrier concentration in a metal.

(i) Density of States (Z(E))

Definition: Density of states, Z(E), is defined as the number of available electron states per unit volume in an energy interval between E and E+dE.

The Fermi function F(E) only gives the *probability* of an energy state being filled, but Z(E) tells us how many states *exist* at that energy level.

(ii) Derivation of Density of States

We consider a 3D space with quantum numbers n_x, n_y, and n_z as axes. A radius vector 'n' corresponds to a specific energy E, where n² = n_x² + n_y² + n_z². All points on a sphere of radius 'n' have the same energy E.

Since quantum numbers can only be positive integers, we consider only one octant (1/8th) of the sphere.

The number of available states N(E)dE in the energy interval dE (i.e., in the shell between n and n+dn) is found by subtracting these:

Neglecting higher powers of dn (dn², dn³):

The energy of an electron in a cubical metal piece of side 'a' is:

From this, we find n and n dn:

Substitute (2) and (3) into (1), noting N(E)dE = (π/2) * n * (ndn):

According to Pauli's exclusion principle, each state can hold two electrons (opposite spins), so we multiply by 2:

Density of states Z(E)dE is the number of states N(E)dE per unit volume (V = a³):

(iii) Derivation of Carrier Concentration (n_c)

Carrier concentration (n_c) is the number of electrons per unit volume. It is found by integrating the product of the density of states (how many states exist) and the Fermi function (probability they are filled).

For a metal at absolute zero (T=0K), the Fermi function F(E) = 1 for all energies below the Fermi energy (E_F) and F(E) = 0 for all energies above it.

Therefore, we integrate Z(E) from 0 to E_F:

This equation gives the carrier concentration in metals at 0K.

Q3: (i) Write a short note on Tight Binding Approximation for energy band in solids. (ii) Derive the expression for effective mass of an electron.

(i) Tight Binding Approximation

The tight-binding model is an approach to understand energy bands in solids that starts from the opposite extreme of the "nearly free electron model".

• It assumes the electrons in the crystal behave much like they do in an assembly of individual, constituent atoms.
• The process starts by imagining the atoms are independent and far apart. In this state, the electrons are "tightly bound" to their respective atoms and have fixed, discrete energy levels.
• As these atoms are brought closer together to form a solid, their outer (valence) shell electrons begin to overlap.
• Due to this overlap and the Pauli exclusion principle, the discrete energy levels must split into a large number of closely spaced energy levels.
• This set of closely spaced levels is what forms an energy band.
• As the inter-atomic spacing continues to decrease, the inner shell electrons also begin to overlap, and their corresponding energy levels split as well.

(ii) Derivation of Effective Mass (m*)

In the periodic potential of a crystal, an electron's mass is different from its free-space rest mass. This new mass is called the effective mass (m*). It is defined as the mass of the electron when it is accelerated within this periodic potential.

According to de Broglie, a moving electron has an associated wave, and its velocity (v) is the group velocity (v_g) of this wave.

From quantum mechanics, the energy E = ħω (where ħ = h/2π). Thus, ω = E/ħ. Substituting this into (1):

The acceleration 'a' of the electron is the time derivative of its velocity (v_g):

The momentum 'p' of an electron in quantum theory is p = ħk. The force 'F' on the electron is the time derivative of momentum:

From this, we find dk/dt:

Now, substitute (4) into (3):

We compare this to Newton's second law, F = ma. For an electron in a crystal, we write F = m*a. Rearranging (5) to solve for F:

By comparing F = m*a with equation (6), we get the expression for effective mass:

This shows that the effective mass is not constant but depends on the curvature (d²E/dk²) of the E-k energy band.

Q4: Write a short note on the following:
(i) Fermi Dirac Distribution
(ii) Fermi energy at T = 0K and T > 0K
(iii) Significance of Fermi energy

(i) Fermi-Dirac Distribution

The Fermi-Dirac distribution function, F(E), describes the probability that a given electron energy state 'E' is occupied by an electron at an absolute temperature 'T'. This statistical model applies to particles with half-integral spin, known as Fermions (e.g., electrons).

The distribution function is given by the formula:

Where:

• E = The energy of the energy level whose occupancy is being considered.
• E_F = The energy of the Fermi level (Fermi energy).
• k = Boltzmann's constant.
• T = Absolute temperature.

The probability F(E) always has a value between 0 (vacant) and 1 (occupied).

(ii) Fermi Energy at T = 0K and T > 0K

Case 1: At T = 0K (Absolute Zero)

• If E < E_F (Energy below Fermi level):
  The term (E - E_F) is negative. As T = 0, (E - E_F)/kT → -∞.
  F(E) = 1 / (1 + e^-∞) = 1 / (1 + 0) = 1.
  This means there is a 100% chance all energy levels below the Fermi energy are filled.
• If E > E_F (Energy above Fermi level):
  The term (E - E_F) is positive. As T = 0, (E - E_F)/kT → +∞.
  F(E) = 1 / (1 + e^+∞) = 1 / (1 + ∞) = 0.
  This means there is a 0% chance any energy level above the Fermi energy is occupied.

At T=0K, the Fermi energy E_F is the highest occupied energy level.

Case 2: At T > 0K (Above Absolute Zero)

• When temperature is raised, electrons gain thermal energy (≈ kT).
• Only the electrons within the range of kT *below* the Fermi level absorb this energy and jump to higher, empty energy states *above* the Fermi level.
• The distribution function "smears out". It smoothly decreases from 1 to 0 around the Fermi energy.
• If E = E_F:
  The exponential term becomes e⁰ = 1.
  F(E) = 1 / (1 + 1) = 0.5.
  This means at any temperature above 0K, the Fermi level E_F is the energy level that has a 50% probability of being occupied.

(iii) Uses of Fermi Distribution Function

The Fermi function is significant for several reasons:

• It gives the probability of occupation for any given energy level at a specific temperature.
• It provides a clear idea about filled and unfilled energy states.
• It is essential for calculating the number of free electrons per unit volume (carrier concentration) in a material at a given temperature.
• It is used to calculate the Fermi energy of a metal.

Q5: (i) Contrast the term degenerate and non-degenerate states. (ii) The thermal conductivity of a metal is 123.93 Wm^-1 K^-1. Find the electrical conductivity and Lorentz number when the metal posses relaxation time 10^-14 sec at 300K.(Density of electron= 6*10²⁸ per m³)

(i) Degenerate vs. Non-Degenerate States

• Degenerate States: A state is called degenerate if there are multiple different combinations of quantum numbers (and thus different Eigenfunctions or wave functions) that result in the same Eigenvalue (Energy level).
  • Example: For a 3D particle in a box, the quantum number combinations (1, 1, 2), (1, 2, 1), and (2, 1, 1) all produce the same energy value: E = 6h² / 8ma². However, their wave functions (Ψ₁₁₂, Ψ₁₂₁, Ψ₂₁₁) are all different. This energy level is therefore degenerate.
• Non-Degenerate States: A state is non-degenerate if it has a unique energy level (Eigenvalue) corresponding to a single, unique set of quantum numbers (and a single Eigenfunction).
  • Example: The ground state (1, 1, 1) has a unique energy E = (1²+1²+1²)h² / 8ma² = 3h² / 8ma². No other combination of positive integers produces this energy, so it is non-degenerate.

(ii) Numerical Problem

Given:

• Thermal Conductivity (K) = 123.93 Wm^-1 K^-1
• Relaxation Time (τ) = 10^-14 s
• Temperature (T) = 300 K
• Electron Density (n) = 6 × 10²⁸ m^-3

Constants:

• Charge of electron (e) = 1.6 × 10^-19 C
• Mass of electron (m) = 9.11 × 10^-31 kg (standard value, not in text)

Step 1: Find Electrical Conductivity (σ)

We use the classical formula for electrical conductivity:

The electrical conductivity (σ) is 1.686 × 10⁷ Ω^-1m^-1.

Step 2: Find Lorentz Number (L)

The Lorentz number (L) relates K, σ, and T through the Wiedemann-Franz law:

The Lorentz number (L) is 2.45 × 10^-8 WΩK^-2. (This calculated value is in excellent agreement with the experimental value of 2.44 × 10^-8 WΩK^-2 mentioned in the source document).