IntRoduction to solid state physics

Week 8: Superconductivity (Continued)

Lecture Topics:

1. Superconducting Energy Gap

   • Energy gap in superconductors:

     • The BCS theory predicts the formation of an energy gap (Δ) in the electronic density of states.

     • This gap represents the minimum energy required to break Cooper pairs and excite electrons out of the superconducting state.

   • Temperature dependence of the energy gap:

     • The energy gap Δ(T) decreases as temperature approaches the critical temperature TC​.

     • At T=0, the energy gap reaches its maximum value, and at T=TC​, the gap closes, and superconductivity is lost.

     • The energy gap can be approximated as:

• where kB​ is the Boltzmann constant, and TC is the critical temperature.

• Experimental verification of the energy gap:

  1. Techniques like tunneling spectroscopy, where the current-voltage characteristics of a superconducting junction are analyzed, confirm the presence of the energy gap.

Flux Quantization and Vortex States

• Flux quantization:

  • In superconductors, the magnetic flux passing through a superconducting loop is quantized, meaning it can only take discrete values.

  • The quantum of magnetic flux (Φ0) is given by:

where h is Planck's constant and e is the electron charge.

• 1. This phenomenon is observed in superconducting rings and is crucial for understanding the behavior of Type II superconductors.

• Vortex states in Type II superconductors:

  1. In Type II superconductors, between the lower (HC1​) and upper (HC2​) critical fields, magnetic flux enters the superconductor in discrete quanta, forming vortices.

  2. Each vortex carries one quantum of magnetic flux Φ0​ and has a normal core where superconductivity is destroyed.

  3. The superconducting current flows around the vortex, and the arrangement of vortices forms a vortex lattice.

  4. Vortex pinning: Defects in the material can pin vortices in place, preventing them from moving. This enhances the critical current and is crucial for practical applications like superconducting magnets.

Josephson Effect

• Introduction to the Josephson effect:

  • The Josephson effect occurs when two superconductors are separated by a thin insulating barrier, forming a Josephson junction.

  • Supercurrent: A current can flow across the junction without any applied voltage as long as the phase difference between the superconducting wavefunctions on either side is constant.

• DC Josephson effect:

  • A supercurrent flows across the junction with zero voltage drop.

  • The current depends on the phase difference ϕ between the superconductors:

where IC​ is the critical current of the junction.

AC Josephson effect:

• When a constant voltage V is applied across the junction, the phase difference ϕ changes with time, generating an alternating supercurrent.

• The frequency of the AC is given by:

• 1. where e is the electron charge and h is Planck’s constant.

  2. Applications of the Josephson effect:

     1. SQUIDs (Superconducting Quantum Interference Devices): Extremely sensitive magnetometers based on the Josephson effect and used in medical diagnostics, geophysical research, and quantum computing.

     2. Quantum metrology: The Josephson effect defines the voltage standard based on the fundamental constants of nature.

• High-Temperature Superconductors (HTS)

  1. Discovery of high-temperature superconductors:

     1. In 1986, Bednorz and Müller discovered the first high-temperature superconductor, a ceramic material based on lanthanum copper oxide, with a critical temperature of 35 K.

     2. Subsequent materials, such as YBCO (Yttrium Barium Copper Oxide), exhibit TC​ values above the boiling point of liquid nitrogen (77 K).

  2. Properties of high-temperature superconductors:

     1. Unlike conventional superconductors, HTS materials are typically layered perovskite compounds.

     2. They exhibit anisotropic superconductivity, meaning the superconducting properties differ depending on the direction of the applied magnetic field and current.

  3. Challenges in understanding HTS:

     1. The mechanism behind high-temperature superconductivity has yet to be fully understood. However, it is believed to involve strong electron-electron correlations and magnetically mediated pairing rather than the phonon-mediated pairing seen in BCS theory.

  4. Applications of HTS:

     1. HTS materials are used in high-field magnets, magnetic levitation, and energy storage systems.

     2. Their ability to operate at higher temperatures makes them more practical for large-scale applications, reducing the cooling costs associated with conventional superconductors.

• Applications of Superconductors in Technology

  1. Magnetic Levitation (Maglev):

     1. Superconductors can be used to create magnetic levitation systems due to the Meissner effect.

     2. Maglev trains, which use superconducting magnets to levitate and propel the train without friction, are a notable application.

  2. Superconducting Magnetic Energy Storage (SMES):

     1. SMES systems store energy in the magnetic field generated by a superconducting coil. Since no energy loss is due to resistance, they offer highly efficient energy storage.

  3. Superconducting cables:

     1. Superconducting cables can transmit electrical power with zero losses, making them ideal for power grids and energy distribution.

  4. Quantum Computing:

     1. Superconducting qubits, based on Josephson junctions, are used in quantum computers, offering a platform for creating and controlling quantum states with high precision.

Examples:

• Calculate the energy gap in a superconductor at T=0 using the BCS theory.

• Analysis of flux quantization in a superconducting ring and its implications for SQUID magnetometers.

• Explain the Josephson effect in practical applications like quantum computing and voltage standards.

• Discussion of the advantages and challenges of high-temperature superconductors in technology.

Homework/Exercises:

1. Explain the significance of the energy gap in superconductors and how it varies with temperature.

2. Calculate the frequency of the AC Josephson current for a given applied voltage.

3. Compare the properties and applications of conventional superconductors and high-temperature superconductors.

4. Design a magnetic levitation system using a high-temperature superconductor and explain the role of the Meissner effect.

Suggested Reading:

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

• Research articles on high-temperature superconductors and the practical challenges of integrating them into technology.

Key Takeaways:

• The energy gap in superconductors is a key feature of the BCS theory and is crucial for understanding the zero-resistance state.

• Flux quantization and vortex states play significant roles in Type II superconductors, particularly in high-field applications.

• The Josephson effect underpins a range of important technologies, from highly sensitive magnetometers to quantum computing devices.

• High-temperature superconductors represent a major technological advancement, with applications in energy storage, magnetic levitation, and quantum computing.

This week completes the study of superconductivity, focusing on advanced concepts such as the energy gap, Josephson effect, and high-temperature superconductors. The applications of these phenomena in modern technology are also explored.