Flux quantization and vortices are key concepts in superconductivity. They explain how magnetic fields interact with superconductors, leading to quantized magnetic flux and the formation of vortices in certain materials. Understanding these phenomena is crucial for developing practical applications of superconductors.

These concepts build on the foundational theories of superconductivity, connecting microscopic quantum effects to macroscopic behavior. They help explain why some superconductors can carry high currents in strong magnetic fields, which is essential for many real-world uses of these materials.

Flux quantization in superconductors

Magnetic flux quantum and its origin

• In superconductors, magnetic flux is quantized in units of the magnetic flux quantum, $\Phi₀ = h/2e ≈ 2.07 × 10⁻¹⁵ Wb$, where $h$ is Planck's constant and $e$ is the electron charge
• Flux quantization arises from the requirement that the superconducting wave function maintains single-valuedness, leading to the quantization of the magnetic flux enclosed by a superconducting loop
• The quantization of magnetic flux in superconductors is a direct consequence of the macroscopic quantum state and the coherence of the superconducting wave function

Experimental observation and significance

• Flux quantization can be observed experimentally through the measurement of magnetic flux trapped in superconducting rings or the observation of flux jumps in magnetization measurements (SQUID)
• The concept of flux quantization is crucial for understanding the behavior of superconductors in the presence of magnetic fields and the formation of superconducting vortices
• Flux quantization has practical applications in superconducting quantum interference devices (SQUIDs) for highly sensitive magnetic field measurements and in superconducting quantum computing (flux qubits)

Formation of superconducting vortices

Structure and properties of vortices

• Superconducting vortices, also known as Abrikosov vortices or fluxons, are local regions where superconductivity is suppressed, and magnetic flux penetrates the superconductor in quantized units of $\Phi₀$
• Each vortex consists of a normal core, where superconductivity is suppressed, surrounded by a circulating supercurrent that generates a quantized magnetic flux
• The size of the vortex core is characterized by the coherence length, $\xi$, while the extent of the circulating supercurrent is determined by the magnetic penetration depth, $\lambda$

Vortex formation and lattice arrangement

• Vortices form in type-II superconductors when the applied magnetic field exceeds the lower critical field, $H_{c1}$, but is below the upper critical field, $H_{c2}$
• Vortices arrange themselves in a regular lattice structure, known as the Abrikosov vortex lattice, to minimize the overall energy of the system
• The vortex lattice can exhibit different symmetries (triangular, square) depending on the material properties and the applied magnetic field
• The motion of vortices under the influence of an applied current leads to dissipation and limits the critical current density in type-II superconductors

Superconductors in magnetic fields

Meissner effect and critical fields

• The behavior of superconductors in the presence of magnetic fields is characterized by the Meissner effect, where the superconductor expels magnetic flux from its interior
• Type-I superconductors exhibit perfect diamagnetism and complete flux expulsion up to a critical field, $H_c$, above which superconductivity is destroyed
• Type-II superconductors have two critical fields: the lower critical field, $H_{c1}$, below which the superconductor is in the Meissner state, and the upper critical field, $H_{c2}$, above which superconductivity is destroyed

Mixed state and vortex dynamics

• Between $H_{c1}$ and $H_{c2}$, type-II superconductors are in a mixed state, where magnetic flux partially penetrates the superconductor in the form of quantized vortices
• The presence of an applied current in a superconductor generates a Lorentz force on the vortices, causing them to move if the force exceeds the pinning force
• The motion of vortices under the influence of an applied current leads to dissipation and limits the critical current density, $J_c$, which is the maximum current density a superconductor can sustain without dissipation

Vortices and critical current density

Vortex pinning and its importance

• The critical current density, $J_c$, is a key parameter that determines the maximum current a superconductor can carry without dissipation
• In type-II superconductors, the motion of vortices under the influence of an applied current is the primary source of dissipation and limits the critical current density
• To enhance the critical current density, vortices must be immobilized or pinned by introducing defects or artificial pinning centers in the superconductor

Factors influencing critical current density

• Vortices experience a Lorentz force, $F_L = J × B$, where $J$ is the applied current density and $B$ is the magnetic field, which causes them to move if the force exceeds the pinning force
• The strength of the pinning force depends on the size, density, and distribution of the pinning centers, as well as the coherence length and penetration depth of the superconductor
• Pinning centers can be in the form of material defects, such as dislocations, grain boundaries, or impurities, or artificially engineered structures, such as nanoparticles or nanorods (artificial pinning centers)
• Optimizing the pinning landscape in superconductors is crucial for achieving high critical current densities and enabling practical applications, such as high-field magnets (MRI) and power transmission lines (superconducting cables)