9.1 Meissner effect

The Meissner effect, discovered in 1933, is a fundamental property of superconductors. It describes how these materials expel magnetic fields from their interior when cooled below their critical temperature, exhibiting perfect diamagnetism.

This phenomenon arises from the formation of Cooper pairs and is distinct from zero electrical resistance. Understanding the Meissner effect is crucial for applications like magnetic levitation and superconducting bearings, as well as for differentiating between type-I and type-II superconductors.

Discovery of Meissner effect

• Walther Meissner and Robert Ochsenfeld discovered the Meissner effect in 1933 while studying the magnetic properties of superconductors
• Observed that a superconductor expels magnetic fields from its interior when cooled below its critical temperature ($T_c$)
• This perfect diamagnetism is a fundamental property of superconductors distinct from mere zero electrical resistance

Experimental setup for demonstrating Meissner effect

• A superconducting sample is placed in an external magnetic field and cooled below its critical temperature
• As the sample transitions into the superconducting state, it expels the magnetic field from its interior
• This can be observed by measuring the magnetic flux density around the sample using a magnetometer or by the levitation of a magnet above the superconductor

Magnetic field expulsion in superconductors

• In the superconducting state, the material generates surface currents that exactly cancel the applied magnetic field inside the superconductor
• This results in the expulsion of the magnetic field from the interior of the superconductor, with the field lines bending around the sample
• The expulsion of magnetic fields is a consequence of the minimization of the free energy of the superconducting state

Critical magnetic field for destroying superconductivity

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• The Meissner effect persists up to a critical magnetic field ($H_c$) above which superconductivity is destroyed
• The critical field depends on the material and typically decreases with increasing temperature
• Type-I superconductors exhibit a complete Meissner effect up to $H_c$, while type-II superconductors allow partial field penetration above a lower critical field ($H_{c1}$)

Temperature dependence of critical magnetic field

• The critical magnetic field ($H_c$) decreases with increasing temperature, reaching zero at the critical temperature ($T_c$)
• This temperature dependence can be approximated by the empirical formula: $H_c(T) = H_c(0)[1 - (T/T_c)^2]$
• Understanding the temperature dependence of $H_c$ is crucial for designing superconducting devices that operate at specific temperatures and magnetic fields

Microscopic origin of Meissner effect

• The Meissner effect arises from the formation of Cooper pairs, electron pairs bound together by an attractive interaction mediated by lattice vibrations (phonons)
• Cooper pairs have a lower energy than individual electrons and can flow without resistance, leading to superconductivity
• The expulsion of magnetic fields is a consequence of the collective behavior of Cooper pairs

Role of Cooper pairs in magnetic field expulsion

• Cooper pairs have a net electric charge and can be accelerated by electric fields, generating supercurrents
• These supercurrents flow on the surface of the superconductor, creating a magnetization that exactly cancels the applied magnetic field inside the material
• The coherent motion of Cooper pairs is essential for maintaining the Meissner effect

London equations for describing Meissner effect

• The London equations, developed by brothers Fritz and Heinz London in 1935, provide a phenomenological description of the Meissner effect
• The first London equation relates the supercurrent density to the vector potential, while the second equation describes the magnetic field expulsion
• The London penetration depth ($\lambda_L$) characterizes the distance over which the magnetic field decays inside the superconductor

Thermodynamics of Meissner effect

• The Meissner effect can be understood thermodynamically as a consequence of the minimization of the Gibbs free energy in the superconducting state
• The expulsion of magnetic fields reduces the magnetic energy contribution to the Gibbs free energy, making the superconducting state more stable
• The transition from the normal to the superconducting state is accompanied by a discontinuous change in the magnetization and a latent heat

Gibbs free energy considerations

• The Gibbs free energy of a superconductor in a magnetic field includes contributions from the condensation energy and the magnetic energy
• Below the critical field, the superconducting state has a lower Gibbs free energy than the normal state, favoring the expulsion of magnetic fields
• Above the critical field, the magnetic energy contribution dominates, and the normal state becomes more stable

Entropy change during superconducting transition

• The transition from the normal to the superconducting state is accompanied by a decrease in entropy
• This entropy change is associated with the formation of Cooper pairs and the reduction of available electronic states
• The latent heat of the superconducting transition is related to the entropy change through the Clausius-Clapeyron relation

Meissner effect vs perfect diamagnetism

• The Meissner effect is often referred to as perfect diamagnetism, but there are subtle differences between the two phenomena
• Perfect diamagnetism refers to the complete expulsion of magnetic fields from a material, regardless of the applied field strength or the material's history
• The Meissner effect, on the other hand, is a spontaneous expulsion of magnetic fields that occurs only when a superconductor is cooled below its critical temperature in the presence of a weak applied field
• Additionally, the Meissner effect is associated with zero electrical resistance and the formation of Cooper pairs, which are not necessarily present in perfect diamagnets

Type-I vs type-II superconductors

• Superconductors can be classified into two types based on their magnetic properties: type-I and type-II
• Type-I superconductors exhibit a complete Meissner effect up to a single critical field ($H_c$), above which superconductivity is abruptly destroyed (examples: mercury, lead, tin)
• Type-II superconductors have two critical fields: a lower critical field ($H_{c1}$) and an upper critical field ($H_{c2}$). Between these fields, they allow partial penetration of magnetic flux in the form of quantized vortices (examples: niobium, vanadium, high-temperature superconductors)

Differences in magnetic properties

• Type-I superconductors have a single critical field ($H_c$) and exhibit a complete Meissner effect below this field
• Type-II superconductors have a lower critical field ($H_{c1}$) below which they exhibit a complete Meissner effect, and an upper critical field ($H_{c2}$) above which superconductivity is destroyed
• Between $H_{c1}$ and $H_{c2}$, type-II superconductors are in a mixed state, where magnetic flux partially penetrates the material in the form of quantized vortices

Abrikosov vortex lattice in type-II superconductors

• In the mixed state of type-II superconductors, magnetic flux penetrates the material in the form of quantized vortices, known as Abrikosov vortices
• Each vortex carries a single quantum of magnetic flux ($\Phi_0 = h/2e$) and consists of a normal core surrounded by circulating supercurrents
• The vortices arrange themselves in a regular triangular lattice, known as the Abrikosov vortex lattice, to minimize their mutual repulsion and the overall energy of the system

Applications exploiting Meissner effect

• The Meissner effect has found numerous applications in various fields, leveraging the unique properties of superconductors
• These applications take advantage of the ability of superconductors to expel magnetic fields, conduct electricity without resistance, and trap magnetic flux
• Some notable applications include magnetic levitation, superconducting bearings, and superconducting quantum interference devices (SQUIDs)

Magnetic levitation using superconductors

• The Meissner effect allows superconductors to levitate magnets or be levitated by magnets
• This principle is used in maglev trains, where superconducting magnets on the train interact with electromagnets on the track, providing a frictionless and efficient means of transportation
• Superconducting levitation is also used in research for studying the behavior of materials in low-gravity environments

Superconducting bearings and flywheels

• Superconducting bearings exploit the Meissner effect to create a stable, contactless, and low-friction support for rotating systems
• A magnet is levitated above a superconductor, and the magnetic field is trapped in the superconductor, providing a strong restoring force that keeps the magnet in place
• Superconducting magnetic bearings are used in high-precision instruments, such as flywheel energy storage systems and high-speed centrifuges

Limitations of Meissner effect

• While the Meissner effect is a fundamental property of superconductors, it has certain limitations that need to be considered in practical applications
• These limitations arise from factors such as the finite thickness of superconducting samples, the presence of impurities or defects, and the geometry of the superconductor
• Understanding these limitations is crucial for designing efficient superconducting devices and optimizing their performance

Thickness dependence of magnetic field penetration

• The expulsion of magnetic fields in superconductors is not perfect, as the field can penetrate a short distance into the superconductor, characterized by the London penetration depth ($\lambda_L$)
• For thin superconducting films with a thickness comparable to or smaller than $\lambda_L$, the Meissner effect is weakened, and the magnetic field can partially penetrate the sample
• This thickness dependence limits the effectiveness of the Meissner effect in thin-film superconducting devices and needs to be accounted for in their design

Demagnetization effects in non-ellipsoidal samples

• The Meissner effect is most effective in superconductors with an ellipsoidal shape, where the magnetic field lines can smoothly bend around the sample
• In non-ellipsoidal samples, such as rectangular or irregular shapes, demagnetization effects can arise due to the non-uniform distribution of magnetic fields
• These demagnetization effects can lead to local enhancements of the magnetic field near corners or edges, potentially exceeding the critical field and destroying superconductivity in those regions
• Careful design and shaping of superconducting samples are necessary to minimize demagnetization effects and ensure optimal performance