1.

Explain the following terms, (i) Fermi energy, (ii) Density of states, (iii)
Superconductivity, (iv) Critical magnetic field (Hc), and (v)
Applications of superconductivity. [10 marks]
Ans:
Sure, here's an explanation of each term:

(i) Fermi Energy:

The Fermi energy, named after the physicist Enrico Fermi, is a concept in quantum
mechanics that represents the highest energy state of a system at absolute zero
temperature. In a system with many fermions (particles with half-integer spin, such
as electrons), like those in a solid, the Fermi energy represents the energy level at
which the probability of finding a particle is 50% at absolute zero. It is a crucial
parameter in understanding the behavior of electrons in materials, especially in
determining electrical conductivity and thermal properties.

(ii) Density of States:

The density of states (DOS) refers to the number of available energy states per unit
volume per unit energy range in a material. It is a fundamental concept in
condensed matter physics and quantum mechanics. DOS provides information
about how electronic or quantum mechanical states are distributed in a material's
energy spectrum. Understanding the DOS is crucial in studying various properties
of materials, such as electronic transport, optical properties, and thermal
conductivity.

(iii) Superconductivity:
Superconductivity is a phenomenon observed in certain materials at low
temperatures where they exhibit zero electrical resistance and the expulsion of
magnetic fields. This means that electric current can flow through these materials
without any energy loss due to resistance, leading to efficient energy transmission
and other applications. Superconductivity is characterized by the formation of
Cooper pairs, which are pairs of electrons that behave as one entity due to their
interaction with the lattice vibrations (phonons) in the material.
(iv) Critical Magnetic Field (Hc):
The critical magnetic field (Hc) is the maximum magnetic field strength that a
superconductor can withstand before it transitions from the superconducting state
to a normal (non-superconducting) state. Above this critical magnetic field,
superconductivity is destroyed, and the material behaves like a regular conductor
with resistance. The critical magnetic field is an important parameter for
determining the practical applications of superconductors, as it dictates the strength
of magnetic fields they can be exposed to while maintaining their superconducting
properties.

(v) Applications of Superconductivity:

Superconductivity has numerous practical applications across various fields,
including:

1. Magnetic Resonance Imaging (MRI) machines in medical diagnostics.

2. Magnetic levitation (Maglev) trains for high-speed transportation.
3. High-field magnets for particle accelerators and research in physics.
4. Superconducting quantum interference devices (SQUIDs) for sensitive
measurements of magnetic fields.
5. Power transmission cables with minimal energy loss.
6. Superconducting fault current limiters for improving grid stability.
7. Quantum computing, where superconducting qubits are used as building blocks
for quantum processors.

These applications leverage the unique properties of superconductors to enhance

performance and efficiency in various technological and scientific endeavors.
2.Discuss the assumptions of classical and quantum free electron theory. [10
marks]
Ans:
Classical and quantum free electron theories are fundamental models used to
describe the behavior of electrons in solids. Here are the assumptions of each
theory:

Classical Free Electron Theory:

1. Free Movement: Electrons are assumed to behave like classical particles moving
freely within a solid lattice. This means they are not influenced by the presence of
other electrons or the lattice ions and can move without experiencing any
collisions.

2. No Interactions: The theory assumes that there are no interactions between

electrons, meaning they do not repel or attract each other. Additionally, there are
no interactions between electrons and the lattice ions.

3. Thermal Equilibrium: The system is assumed to be in thermal equilibrium,

meaning the electrons have a Maxwell-Boltzmann distribution of energies at a
given temperature.

4. Continuum Approximation: The energy levels available to electrons in the solid

are treated as a continuous spectrum rather than discrete energy levels. This
approximation simplifies the calculations but may not accurately represent the
energy levels in a real solid.

Quantum Free Electron Theory:

1. Quantum Mechanics: Unlike the classical theory, the quantum free electron
theory incorporates the principles of quantum mechanics. Electrons are treated as
wavefunctions that satisfy the Schrödinger equation in the periodic potential of the
crystal lattice.

2. Pauli Exclusion Principle: The theory takes into account the Pauli exclusion
principle, which states that no two electrons can occupy the same quantum state
simultaneously. This principle affects the distribution of electrons in energy bands
and leads to the formation of Fermi surfaces.

3. Periodic Potential: While electrons are considered free to move within the
crystal, they are subject to the periodic potential created by the lattice ions. This
potential affects the energy levels available to electrons and leads to the formation
of energy bands.

4. Crystal Symmetry: The theory incorporates the symmetry of the crystal lattice,
which influences the allowed energy states and the shape of the energy bands.

5. Fermi-Dirac Statistics: Electrons are treated as fermions, subject to Fermi-Dirac

statistics, which describe the distribution of fermions at absolute zero temperature.
This leads to the formation of Fermi surfaces in momentum space.

In summary, classical free electron theory assumes classical particle behavior and
neglects electron-electron and electron-lattice interactions, while quantum free
electron theory incorporates quantum mechanical principles such as wave-particle
duality, the Pauli exclusion principle, and the periodic potential of the crystal
lattice.
3.Explain how classical free electron theory failed to explain temperature
dependence of resistivity of metals. [10 marks]
Ans:
Classical free electron theory, while providing a useful framework for
understanding certain properties of metals, fails to adequately explain the
temperature dependence of resistivity. Here's why:

1. **Failure to Account for Electron-Electron Interactions:** Classical free

electron theory assumes that electrons move independently of each other and do
not interact with each other. However, at higher temperatures, the motion of
electrons becomes more erratic, leading to increased chances of electron-electron
collisions. These collisions result in an increase in resistivity. Classical theory
neglects this electron-electron interaction entirely, thus failing to explain the
observed temperature dependence of resistivity.

2. **Neglect of Electron-Phonon Interactions:** Classical free electron theory also

overlooks the interaction between electrons and lattice vibrations, known as
phonons. At higher temperatures, the lattice vibrations increase, leading to more
frequent collisions between electrons and phonons. These collisions scatter
electrons, impeding their flow and increasing resistivity. Since classical theory
doesn't consider electron-phonon interactions, it fails to account for the
temperature dependence of resistivity.

3. **Violation of Matthiessen's Rule:** Matthiessen's rule states that the total

resistivity of a metal is the sum of its resistivities due to various scattering
mechanisms, such as impurity scattering, electron-electron scattering, and electron-
phonon scattering. In classical free electron theory, resistivity is assumed to be
solely due to impurity scattering, neglecting other mechanisms like electron-
electron and electron-phonon scattering. As a result, the theory fails to explain why
resistivity increases with temperature, violating Matthiessen's rule.
4. **Specific Heat Discrepancy:** Classical free electron theory predicts a specific
heat capacity that is proportional to temperature, known as the Dulong-Petit law.
However, experimental observations show that the specific heat capacity of metals
decreases at low temperatures, a phenomenon known as the Debye-T^3 law. This
discrepancy further highlights the failure of classical theory to account for
temperature-dependent phenomena in metals.

In summary, classical free electron theory fails to explain the temperature

dependence of resistivity in metals because it neglects electron-electron and
electron-phonon interactions, violates Matthiessen's rule, and inaccurately predicts
the specific heat capacity of metals. Quantum mechanical theories, such as the
Drude model and the more sophisticated quantum free electron theory, have been
developed to address these shortcomings and provide a more accurate description
of electron behavior in metals.
With neat diagrams discuss the temperature dependency of resistivity for
conductor and superconductor. [10 marks]
Ans:
Certainly! Let's discuss the temperature dependency of resistivity for both
conductors and superconductors with neat diagrams.

1. **Conductor:**

In conductors, the resistivity generally increases with temperature due to increased

scattering of electrons by lattice vibrations (phonons) and impurities. This behavior
can be illustrated with the following diagram:


- **Description:** The graph shows the resistivity (ρ) of a conductor plotted

against temperature (T). As temperature increases, the resistivity of the conductor
also increases. This is due to the increased thermal energy causing more frequent
collisions between electrons and lattice vibrations (phonons), leading to higher
resistance to electron flow and thus increased resistivity.

- **Explanation:** At low temperatures, the lattice vibrations are minimal, and

electrons experience less scattering, resulting in lower resistivity. However, as
temperature increases, lattice vibrations become more pronounced, causing more
frequent collisions between electrons and phonons. This increased scattering
impedes the flow of electrons, leading to higher resistivity.

2. **Superconductor:**

In contrast to conductors, superconductors exhibit a dramatic decrease in resistivity

to zero as temperature approaches a critical temperature (Tc). Below this critical
temperature, superconductors conduct electricity with zero resistance, a
phenomenon known as superconductivity. This behavior can be depicted with the
following diagram:


- **Description:** The graph shows the resistivity (ρ) of a superconductor plotted

against temperature (T). As temperature decreases below the critical temperature
(Tc), the resistivity of the superconductor drops abruptly to zero, indicating the
onset of superconductivity.

- **Explanation:** At temperatures above the critical temperature (Tc), a

superconductor behaves like a normal conductor, with finite resistivity. However,
as the temperature decreases below Tc, the material undergoes a phase transition,
and its resistivity drops sharply to zero. In the superconducting state, electrons
form Cooper pairs and exhibit macroscopic quantum coherence, allowing for the
perfect transmission of electrical current without any resistance.
In summary, conductors generally exhibit an increase in resistivity with
temperature due to increased scattering of electrons by lattice vibrations and
impurities, while superconductors experience a dramatic drop in resistivity to zero
below a critical temperature, demonstrating the phenomenon of superconductivity.

Distinguish between Type-I and Type-II superconductors. [10 marks]

Ans:
Type-I and Type-II superconductors are two distinct classes of superconducting
materials, each exhibiting different behaviors in the presence of magnetic fields.
Here's a comparison between them:

1. **Critical Magnetic Field (Hc):**

- Type-I Superconductors: They have a single critical magnetic field (Hc) below
which they exhibit perfect diamagnetism and zero resistance. Above Hc, they
abruptly transition to the normal state, losing their superconducting properties.
- Type-II Superconductors: They have two critical magnetic fields: an upper
critical field (Hc2) and a lower critical field (Hc1). Between Hc1 and Hc2, they
exhibit mixed states where some flux lines penetrate the material, causing partial
loss of superconductivity. Beyond Hc2, they fully revert to the normal state.

2. **Magnetic Flux Penetration:**

- Type-I Superconductors: They expel all magnetic flux from their interior below
Hc and exhibit perfect diamagnetism. However, above Hc, they allow magnetic
flux to penetrate their bulk, leading to a complete loss of superconductivity.
- Type-II Superconductors: They allow partial penetration of magnetic flux lines
within certain ranges of applied magnetic field strength. In the mixed state, flux
lines penetrate the material as quantized vortices, but superconductivity persists in
regions between the vortices.
3. **Critical Temperature (Tc):**
- Both Type-I and Type-II superconductors have a critical temperature (Tc)
below which they exhibit superconductivity. This critical temperature varies for
different materials and is a defining characteristic of superconductors.

4. **Materials:**
- Type-I Superconductors: Examples include elemental superconductors like lead
(Pb), mercury (Hg), and niobium (Nb).
- Type-II Superconductors: Examples include many high-temperature
superconductors (HTS) such as yttrium barium copper oxide (YBCO) and
magnesium diboride (MgB2), as well as certain alloys and compounds.

5. **Applications:**
- Type-I Superconductors: Due to their sensitivity to magnetic fields and limited
applicability in high magnetic fields, they find use in applications like magnetic
shielding, sensitive magnetometers, and levitating superconductors.
- Type-II Superconductors: They are more versatile and find applications in high-
field magnets for MRI machines, particle accelerators, magnetic levitation
(Maglev) trains, and superconducting power transmission lines.

In summary, Type-I and Type-II superconductors differ in their response to

magnetic fields, critical magnetic fields, magnetic flux penetration behavior,
materials, and applications. Type-II superconductors, with their ability to sustain
superconductivity in the presence of higher magnetic fields, have broader
technological implications compared to Type-I superconductors.
Explain the following terms, (i) Meissner effect, (ii) Type-II superconductors,
(iii), Critical current density (Jc), (iv) Wiedemann-Franz law, and (v)
Electrical conductivity. [10 marks]
Ans:
Sure, here's a compressed version:
(i) Meissner Effect:
- Phenomenon in superconductors.
- Expels almost all magnetic fields when cooled below critical temperature (Tc).
- Results in perfect diamagnetism.

(ii) Type-II Superconductors:

- Exhibit two critical magnetic fields: Hc1 and Hc2.
- Below Hc1: Expel magnetic flux, perfect diamagnetism.
- Between Hc1 and Hc2: Enter mixed state with flux penetration as vortices.
- More robust against magnetic fields.

(iii) Critical Current Density (Jc):

- Maximum current density before transition to normal state.
- Determines practical applications of superconductors.
- Dictates maximum current-carrying capacity.

(iv) Wiedemann-Franz Law:

- Ratio of electrical conductivity (σ) to thermal conductivity (κ) in metals.
- Proportional to absolute temperature (T).
- Describes behavior of charge carriers and phonons in conductors.

(v) Electrical Conductivity:

- Measure of material's ability to conduct electric current.
- Denoted by σ, expressed in S/m or Ω·m.
- Influenced by material composition, temperature, and impurities.
Describe how BCS theory explains superconductivity. [10 marks]
Ans:
The Bardeen-Cooper-Schrieffer (BCS) theory provides a comprehensive
explanation of superconductivity by describing the formation of Cooper pairs,
which are pairs of electrons with opposite momentum and spin. Here's a detailed
explanation of how the BCS theory explains superconductivity:

1. **Electron-Phonon Interaction:**
- In the BCS theory, superconductivity arises from the interaction between
electrons and lattice vibrations (phonons) in the crystal lattice of the
superconducting material.
- When an electron moves through the lattice, it attracts nearby positive lattice
ions, causing the lattice to deform or create a slight positive charge imbalance in
the lattice.
- This deformation or positive charge imbalance acts as an attractive potential for
other electrons, leading to the formation of Cooper pairs.

2. **Formation of Cooper Pairs:**

- Cooper pairs are formed when two electrons with opposite spins and nearly
opposite momenta are bound together by the attractive potential created by lattice
deformations.
- Due to the lattice vibrations, the energy required to form Cooper pairs is
lowered, allowing electrons to overcome their mutual Coulomb repulsion and form
bound states.
- These paired electrons effectively move through the lattice as a single entity,
with the lattice vibrations providing the binding energy necessary to maintain the
pair.

3. **BCS Wavefunction:**
 - The BCS theory describes the wavefunction of Cooper pairs using a modified
BCS wavefunction, which accounts for the correlated motion of paired electrons.
- The BCS wavefunction describes the probability amplitude of finding two
electrons at certain positions and momenta within the superconducting material.

4. **Macroscopic Quantum Coherence:**

- Cooper pairs exhibit macroscopic quantum coherence, meaning that all paired
electrons behave collectively as a single quantum entity.
- This coherence allows Cooper pairs to move through the material without
experiencing scattering from impurities or lattice defects, leading to zero electrical
resistance.

5. **Energy Gap:**
- Due to the formation of Cooper pairs, a gap opens up in the energy spectrum of
the superconductor, known as the superconducting energy gap (Δ).
- This energy gap represents the minimum energy required to break apart a
Cooper pair and is responsible for the zero resistance and perfect diamagnetism
observed in superconductors below their critical temperature (Tc).

In summary, the BCS theory explains superconductivity by describing the

formation of Cooper pairs through electron-phonon interactions, their collective
behavior as a macroscopic quantum entity, and the opening of a superconducting
energy gap in the material's energy spectrum. This theory provides a robust
framework for understanding the phenomenon of superconductivity and has been
successfully applied to various superconducting materials.
Derive an expression for electrical conductivity based on quantum free
electron theory. [10 marks]
Ans:
Certainly, here are the short answers:
1. **Electrical Conductivity (σ):**
- Conductivity (σ) = ne^2τ/m
- Where: n = electron density, e = elementary charge, τ = relaxation time, m =
electron mass.

2. **Electron Mobility (μe):**

- Mobility (μe) = eτ/m
- Where: τ = relaxation time, e = elementary charge, m = electron mass.

3. **Current Density (J):**

- Current Density (J) = nev_d
- Where: n = electron density, e = elementary charge, vd = drift velocity.

4. **Drift Velocity (vd):**

- Drift Velocity (vd) = μeE
- Where: μe = electron mobility, E = electric field.

5. **Ohm's Law:**
- Ohm's Law: J = σE
- Where: J = current density, σ = conductivity, E = electric field.

These formulas summarize the key relationships between electrical conductivity,

electron mobility, current density, drift velocity, and Ohm's law in the context of
quantum free electron theory.
Distinguish between low and high temperature superconductors [10 marks]
Ans:
Certainly, here's a comparison between low-temperature superconductors (LTS)
and high-temperature superconductors (HTS):

1. **Critical Temperature (Tc):**

- LTS: Low-temperature superconductors have critical temperatures typically
below 30 K (-243°C). Examples include elemental superconductors like lead (Pb)
and niobium-titanium (NbTi) alloys.
- HTS: High-temperature superconductors have critical temperatures above 30 K,
with some materials reaching as high as 138 K (-135°C). Examples include cuprate
superconductors like yttrium barium copper oxide (YBCO) and iron-based
superconductors.

2. **Material Composition:**
- LTS: Low-temperature superconductors are often metallic elements or alloys,
with simple crystal structures and relatively low critical temperatures.
- HTS: High-temperature superconductors are typically complex compounds
consisting of transition metals, oxygen, and other elements. They have more
intricate crystal structures and exhibit superconductivity at higher temperatures.

3. **Mechanism of Superconductivity:**
- LTS: The mechanism of superconductivity in low-temperature superconductors
is primarily explained by the BCS theory, which involves electron-phonon
interactions and the formation of Cooper pairs.
- HTS: The mechanism of superconductivity in high-temperature
superconductors is still not fully understood. It is believed to involve
unconventional pairing mechanisms, such as spin fluctuations, charge density
waves, and strong electron correlations.

4. **Critical Magnetic Field (Hc):**

 - LTS: Low-temperature superconductors have relatively low critical magnetic
fields (Hc), typically in the range of a few tesla (T). They are sensitive to magnetic
fields and lose their superconducting properties when subjected to magnetic fields
above Hc.
- HTS: High-temperature superconductors have higher critical magnetic fields,
often exceeding tens of tesla (T) or more. They are more robust against magnetic
fields and can maintain superconductivity at higher magnetic field strengths.

5. **Applications:**
- LTS: Low-temperature superconductors are widely used in applications
requiring low temperatures, such as in superconducting magnets for medical
imaging (MRI), particle accelerators, and research experiments.
- HTS: High-temperature superconductors have potential applications in power
generation and transmission, magnetic levitation (Maglev) trains, and high-field
magnets for scientific research and industrial processes. However, practical
applications are still limited by material fabrication challenges and cost.

6. **Cost and Availability:**

- LTS: Low-temperature superconductors are well-established and relatively
inexpensive to produce. However, they require cryogenic cooling systems to
maintain the low temperatures required for superconductivity.
- HTS: High-temperature superconductors are more challenging and costly to
produce but offer the advantage of operating at higher temperatures, reducing the
need for extreme cryogenic cooling. However, production costs and material
availability remain significant challenges for widespread adoption.