IntRoduction to solid state physics

Week 7: Superconductivity

Lecture Topics:

1. Introduction to Superconductivity

   • Discovery of superconductivity: In 1911, Heike Kamerlingh Onnes discovered superconductivity while studying mercury at low temperatures, which exhibited zero electrical resistance.

   • Key characteristics of superconductors:

     • Zero electrical resistance: When cooled below a critical temperature (TC​), superconductors exhibit no electrical resistance.

     • Perfect diamagnetism (Meissner effect): Superconductors expel magnetic fields from their interior, a phenomenon known as the Meissner effect.

2. Meissner Effect

   • Definition: The expulsion of magnetic fields from a superconductor as it transitions into the superconducting state below TC​.

   • Implications of the Meissner effect: Distinguishes superconductors from perfect conductors. In a perfect conductor, a magnetic field would remain trapped, whereas in a superconductor, it is expelled.

   • Demonstration: A superconducting material cooled below its TC repels a magnetic field, causing a magnet placed above it to levitate.

   • Type I and Type II superconductors:

     • Type I superconductors: Exhibit complete Meissner effect and lose superconductivity above a critical magnetic field. Examples include mercury and lead.

     • Type II superconductors: Exhibit partial Meissner effect with two critical magnetic fields (HC1​ and HC2​). Between these fields, they allow magnetic flux to penetrate in quantized vortices while maintaining superconductivity. Examples include niobium-titanium alloys and high-temperature superconductors.

3. Critical Temperature, Critical Field, and Critical Current

   • Critical temperature (TC): The temperature below which a material becomes superconducting. Different materials have different TC values.

   • Critical magnetic field (HC): The maximum magnetic field a superconductor can withstand before losing its superconductivity. Type II superconductors have two critical fields: HC1​ (lower critical field) and HC2​ (upper critical field).

   • Critical current (IC​): The maximum current a superconductor can carry without losing its superconducting state. Beyond this current, the material reverts to its normal resistive state.

4. London Equations

   • London equations: Developed by Fritz and Heinz London in 1935, these two equations describe the electromagnetic properties of superconductors.

   • First London equation: Describes the zero electrical resistance property by relating the electric field and the supercurrent density: 

• where JS is the supercurrent density, ns is the density of superconducting electrons, e is the electron charge, me​ is the electron mass, and E is the electric field.

• Second London equation: Describes the Meissner effect by relating the magnetic field and supercurrent density:

where B is the magnetic field, this equation shows that the magnetic field decays exponentially inside a superconductor.

• BCS Theory

  1. Introduction to BCS theory: The first successful microscopic theory of superconductivity was proposed by John Bardeen, Leon Cooper, and Robert Schrieffer in 1957, known as the BCS theory.

  2. Cooper pairs: At temperatures below TC​, electrons in a superconductor form pairs, called Cooper pairs, through an attractive interaction mediated by lattice vibrations (phonons).

     1. The Cooper pairs condense into a collective quantum state that flows without resistance.

     2. Energy gap: The formation of Cooper pairs leads to an energy gap in the electronic density of states. This gap must be overcome for the material to return to a normal (non-superconducting) state.

  3. BCS wavefunction: Describes the ground state of a superconductor, which involves the coherent superposition of all possible configurations of Cooper pairs.

• Type I vs. Type II Superconductors

  1. Type I superconductors:

     1. Exhibit a sharp transition to the superconducting state below TC​.

     2. Show complete expulsion of the magnetic field (Meissner effect) up to a critical field HC​, beyond which they revert to a normal state.

     3. Example materials: Lead (Pb), Mercury (Hg), and Tin (Sn).

  2. Type II superconductors:

     1. Show two critical magnetic fields: HC1​ (below which the Meissner effect is observed) and HC2​ (above which superconductivity is destroyed).

     2. Between HC1​ and HC2​, magnetic flux penetrates the material in quantized vortices.

     3. Used in high-field applications like magnets for MRI machines.

     4. Example materials: Niobium-titanium (NbTi), YBCO (Yttrium barium copper oxide).

• Applications of Superconductivity

  1. Superconducting magnets: Used in Magnetic Resonance Imaging (MRI), particle accelerators (e.g., CERN), and magnetic levitation trains.

  2. Superconducting quantum interference devices (SQUIDs): Extremely sensitive magnetometers used in medical diagnostics and geophysical research.

  3. Lossless power transmission: Superconductors can carry electricity without energy losses, making them ideal for efficient power transmission.

  4. High-temperature superconductors (HTS): Materials with TC values above the boiling point of liquid nitrogen (77 K). These materials are revolutionizing applications in energy and transportation.

Examples:

• Calculate the critical temperature and magnetic field for a given superconductor.

• Analysis of the Meissner effect using experimental data and explaining its implications for practical applications.

• Explanation of the role of Cooper pairs in the zero-resistance state of superconductors using the BCS theory.

• Comparing the behavior of Type I and Type II superconductors in the presence of an external magnetic field.

Homework/Exercises:

1. Explain the London equations and their role in describing the electromagnetic properties of superconductors.

2. Calculate the critical current for a superconducting wire given the critical magnetic field and dimensions.

3. Compare the Meissner effect in Type I and Type II superconductors and explain how the penetration depth differs between the two.

4. Discuss the role of Cooper pairs in superconductivity and how the BCS theory explains the energy gap in the electronic structure.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 12: Superconductivity.

• Research papers on high-temperature superconductors and their technological applications.

Key Takeaways:

• Superconductors exhibit unique properties such as zero electrical resistance and the expulsion of magnetic fields (Meissner effect).

• The BCS theory provides a microscopic explanation of superconductivity, highlighting the role of Cooper pairs.

• The distinction between Type I and Type II superconductors is critical for their practical applications, with Type II superconductors used in high-field technologies.

• Superconductivity has significant technological applications, from medical devices to lossless power transmission, and continues to be an active area of research.

This week provides an in-depth understanding of superconductivity, including its fundamental principles, theoretical models, and real-world applications.