Definition
Flux pinning is a phenomenon that occurs when flux vortices in a type‑II superconductor are prevented from moving within the bulk of the superconductor, so that the magnetic field lines are “pinned” to those locations. The superconductor must be a type‑II superconductor because type‑I superconductors cannot be penetrated by magnetic fields. Some type‑I superconductors can experience the effects of flux pinning if they are thin enough. If the material’s thickness is comparable to the London penetration depth, the magnetic field can pass through the material. The act of magnetic penetration is what makes flux pinning possible.
At higher magnetic fields (above the lower critical field \(H_{c1}\) but below the upper critical field \(H_{c2}\)) the superconductor allows magnetic flux to enter in quantized packets surrounded by a superconducting current vortex (see quantum vortex). These sites of penetration are known as flux tubes. The number of flux tubes per unit area is proportional to the magnetic field with a constant of proportionality equal to the magnetic flux quantum.
On a simple 76 mm diameter, 1‑micrometer thick disk, next to a magnetic field of 28 kA/m, there are approximately 100 billion flux tubes that hold 70 000 times the superconductor’s weight. At lower temperatures the flux tubes are pinned in place and cannot move. This pinning is what holds the superconductor in place thereby allowing it to levitate. This phenomenon is closely related to the Meissner effect, though with one crucial difference — the Meissner effect shields the superconductor from all magnetic fields causing repulsion, unlike the pinned state of the superconductor disk which pins flux, and the superconductor in place.
Superconductivity Basics
Superconductivity is a state of matter in which a material exhibits zero electrical resistance and expels magnetic fields below a characteristic critical temperature. In the superconducting state, electrons form Cooper pairs that move without scattering. The expulsion of magnetic flux is known as the Meissner effect. However, the Meissner effect applies strictly to type‑I superconductors, which are completely expelled of magnetic fields below a single critical field.
Type‑II superconductors behave differently. When exposed to magnetic fields between two critical values, they allow magnetic flux to penetrate the material in discrete, quantized units while still maintaining zero resistance. These penetrations occur as flux vortices or flux tubes, each carrying a single quantum of magnetic flux.
Type‑I vs. Type‑II Superconductors
| Feature | Type‑I | Type‑II |
|---|---|---|
| Magnetic field response | Complete expulsion (Meissner effect) | Partial penetration (flux vortices) |
| Critical field | One critical field \(H_c\) | Two critical fields \(H_{c1}\) and \(H_{c2}\) |
| Flux penetration | None below \(H_c\) | Allowed between \(H_{c1}\) and \(H_{c2}\) |
| Flux pinning | Generally not present | Present; key to levitation |
The source states that type‑I superconductors cannot be penetrated by magnetic fields, but that very thin type‑I samples can allow penetration if their thickness is comparable to the London penetration depth. This thin‑film effect is a special case that permits flux pinning in otherwise type‑I materials.
Magnetic Field Penetration and Critical Fields
When a type‑II superconductor is placed in an external magnetic field, two thresholds determine its response:
- Lower critical field \(H_{c1}\) – Below this field, the superconductor remains in the Meissner state, expelling all magnetic flux.
- Upper critical field \(H_{c2}\) – Above this field, superconductivity is destroyed and the material reverts to a normal conductor.
Between \(H_{c1}\) and \(H_{c2}\), magnetic flux enters the superconductor in quantized packets. Each packet is surrounded by a circulating supercurrent, forming a quantum vortex. The vortices arrange themselves in a lattice, and their density is proportional to the external magnetic field.
Flux Vortices and Quantum Vortices
A flux vortex is a region where the superconducting order parameter vanishes, allowing magnetic field lines to thread through. The surrounding supercurrent circulates around the core, creating a magnetic field that cancels the external field inside the bulk. The vortex carries exactly one magnetic flux quantum, \(\Phi_0 = h/2e\), where \(h\) is Planck’s constant and \(e\) is the elementary charge.
Because the core of a vortex is normal (non‑superconducting), it can be thought of as a tiny “hole” in the superconducting matrix. The size of this core is on the order of the coherence length of the superconductor.
Flux Pinning Mechanism
Flux pinning occurs when defects, impurities, or structural irregularities in the superconductor’s lattice create energetically favorable sites for vortex cores. When a vortex aligns with such a defect, it becomes energetically trapped. The vortex can no longer move freely under the influence of external forces, such as the Lorentz force from a current or a changing magnetic field.
Because the vortices are immobilized, the magnetic flux remains fixed in space. This immobilization is the essence of flux pinning. In the pinned state, the superconductor can remain suspended in a magnetic field, effectively “holding” its position relative to the field lines.
Quantization and Flux Tubes
The number of flux tubes per unit area in a type‑II superconductor is given by:
\[ n = \frac{B}{\Phi_0} \]
where \(B\) is the magnetic induction and \(\Phi_0\) is the magnetic flux quantum. This simple proportionality reflects the fact that each vortex carries exactly one quantum of flux.
In a practical example, a 76 mm diameter, 1‑micrometer thick disk in a magnetic field of 28 kA/m contains about 100 billion flux tubes. These tubes collectively hold 70 000 times the weight of the superconductor. The sheer number of pinned vortices contributes to the remarkable stability of the levitating disk.
Example of Flux Tube Density
The source provides a concrete illustration:
- Disk: 76 mm diameter, 1 µm thick
- Field: 28 kA/m
- Flux tubes: ~100 billion
- Weight supported: 70 000 × the superconductor’s weight
This example demonstrates the extraordinary mechanical support that flux pinning can provide. The pinned vortices act like a rigid scaffold, preventing the superconductor from slipping or rotating under the influence of the magnetic field.
Temperature Dependence
Flux pinning is most effective at lower temperatures. As the temperature approaches the critical temperature from below, thermal fluctuations become significant, allowing vortices to depin and move. At sufficiently low temperatures, the vortices remain locked in place, ensuring a stable pinning landscape. This temperature dependence explains why many demonstrations of magnetic levitation with superconductors are performed at liquid helium or liquid nitrogen temperatures.
Levitation Phenomenon
The ability of a pinned superconductor to levitate is a direct consequence of flux pinning. In a typical demonstration, a disk of type‑II superconductor is placed above a magnet. The magnetic field penetrates the disk in quantized flux tubes. Because these tubes are pinned, the superconductor resists motion and remains suspended. The magnetic field lines are anchored to the superconductor, creating a stable equilibrium position.
Unlike the Meissner effect, which merely repels a superconductor from a magnetic field, flux pinning allows the superconductor to be anchored to the field lines. This distinction is crucial for applications such as magnetic bearings and frictionless transport systems.
Relation to the Meissner Effect
Both the Meissner effect and flux pinning involve the interaction of superconductors with magnetic fields, but they differ fundamentally:
- Meissner effect: The superconductor expels magnetic flux entirely, leading to a repulsive force that pushes the material away from the field.
- Flux pinning: Magnetic flux penetrates the superconductor in quantized vortices that become pinned to defects. The superconductor is held in place by the fixed flux lines rather than repulsion.
Because flux pinning allows the superconductor to remain in a fixed position relative to the magnetic field, it enables levitation and stable magnetic support.
Significance
Flux pinning is the key to many advanced technologies involving superconductors:
- Magnetic levitation: Stable, frictionless bearings for maglev trains and other transport systems.
- Energy storage: Superconducting magnetic energy storage (SMES) devices rely on pinned flux to maintain magnetic fields without losses.
- High‑field magnets: Maintaining stable magnetic fields in fusion reactors and particle accelerators depends on effective flux pinning.
While the source does not mention specific applications beyond levitation, the underlying physics of flux pinning is essential for the development of any technology that requires stable, loss‑free magnetic environments.
Conclusion
Flux pinning is a unique and powerful feature of type‑II superconductors. By preventing the motion of flux vortices, it allows superconductors to lock onto magnetic field lines and remain suspended or firmly anchored. The phenomenon arises only in type‑II materials (or very thin type‑I films) when the magnetic field lies between the lower and upper critical fields. The quantized nature of the vortices, their interaction with material defects, and the resulting mechanical stability make flux pinning a cornerstone of modern superconducting technology.
FAQ
What is the difference between flux pinning and the Meissner effect? The Meissner effect causes a superconductor to repel all magnetic fields, leading to repulsion from a magnet. Flux pinning, on the other hand, allows magnetic flux to penetrate the superconductor in quantized vortices that become pinned to defects; this pins the superconductor in place relative to the magnetic field.
Why does flux pinning only occur in type‑II superconductors? Type‑II superconductors permit magnetic flux to enter between their lower and upper critical fields, forming quantized vortices. Type‑I superconductors expel all flux below a single critical field, so vortices cannot form unless the material is made very thin, comparable to the London penetration depth.
How many flux tubes can a thin superconducting disk contain? In a 76 mm diameter, 1‑micrometer thick disk exposed to a 28 kA/m magnetic field, there can be about 100 billion flux tubes, collectively holding 70 000 times the weight of the superconductor.
At what temperatures does flux pinning become effective? Flux pinning is most effective at lower temperatures, well below the superconductor’s critical temperature, where thermal fluctuations are minimized and vortices remain trapped in defects.
Can flux pinning be used for levitation? Yes. When vortices are pinned, the superconductor can remain suspended above a magnet, as the pinned flux lines create a stable equilibrium that resists motion.