Glossary

2.4 Flux Quantization and Vortices

4 min read•august 14, 2024

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

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  • In superconductors, magnetic flux is quantized in units of the , Φ0=h/2e≈2.07×10−15Wb\Phi₀ = h/2e ≈ 2.07 × 10⁻¹⁵ Wb, where hh is Planck's constant and ee 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 () for highly sensitive magnetic field measurements and in superconducting (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 Φ0\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, ξ\xi, while the extent of the circulating supercurrent is determined by the magnetic penetration depth, λ\lambda

Vortex formation and lattice arrangement

  • Vortices form in type-II superconductors when the applied magnetic field exceeds the lower critical field, Hc1H_{c1}, but is below the upper critical field, Hc2H_{c2}
  • Vortices arrange themselves in a regular lattice structure, known as the lattice, to minimize the overall energy of the system
  • The 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 , where the superconductor expels magnetic flux from its interior
  • Type-I superconductors exhibit perfect diamagnetism and complete flux expulsion up to a critical field, HcH_c, above which superconductivity is destroyed
  • Type-II superconductors have two critical fields: the lower critical field, Hc1H_{c1}, below which the superconductor is in the Meissner state, and the upper critical field, Hc2H_{c2}, above which superconductivity is destroyed

Mixed state and vortex dynamics

  • Between Hc1H_{c1} and Hc2H_{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, JcJ_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, JcJ_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, FL=J×BF_L = J × B, where JJ is the applied current density and BB 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)

Key Terms to Review (16)

Abrikosov vortex: An Abrikosov vortex is a quantized magnetic flux line that forms in type-II superconductors when they are subjected to an external magnetic field. This phenomenon occurs due to the interplay between the superconducting state and magnetic fields, leading to the formation of vortices that allow for partial penetration of the magnetic field while maintaining superconductivity. The presence of these vortices is crucial for understanding the behavior of type-II superconductors, particularly in applications like magnetic levitation and superconducting electronics.
Alexei Abrikosov: Alexei Abrikosov was a prominent Soviet physicist known for his groundbreaking contributions to the understanding of superconductivity, particularly through the development of the concept of magnetic vortices in type-II superconductors. His work provided crucial insights into flux quantization and the behavior of superconducting materials under external magnetic fields, which are essential for advancing technologies in superconducting devices and metamaterials.
Critical magnetic field: The critical magnetic field is the maximum magnetic field strength that a superconductor can withstand before it transitions into the normal (non-superconducting) state. This phenomenon is crucial for understanding how superconductors behave under external magnetic influences, and it is tightly linked to concepts such as critical temperature and current, coherence length, and penetration depth.
Flux pinning: Flux pinning is a phenomenon in superconductors where magnetic flux lines are trapped or 'pinned' within the material, preventing them from moving freely. This effect allows superconductors to maintain their zero-resistance state in the presence of external magnetic fields, enhancing their stability and performance in various applications.
Fluxoid: A fluxoid is a quantized magnetic flux that can exist in a superconducting ring or loop, representing the fundamental unit of magnetic flux in superconductors. It arises due to the quantization of magnetic field lines in type-II superconductors, where the magnetic flux can only take discrete values that are integer multiples of a fundamental unit called the magnetic flux quantum. The existence of fluxoids is crucial for understanding phenomena such as flux pinning and vortex dynamics in superconductors.
Ginzburg-Landau Theory: The Ginzburg-Landau Theory is a theoretical framework used to describe superconductivity in terms of a complex order parameter, which captures the macroscopic quantum behavior of superconductors. This theory connects crucial concepts like the formation of Cooper pairs and the response of superconductors to magnetic fields, providing insights into phenomena such as flux quantization and the Meissner effect.
London Equations: The London Equations are a set of fundamental equations that describe the electromagnetic properties of superconductors, specifically how they respond to magnetic fields. They provide a mathematical framework that explains phenomena such as perfect diamagnetism and the behavior of supercurrents in superconducting materials, linking closely with concepts like coherence length and penetration depth.
Magnetic Flux Quantum: The magnetic flux quantum, often denoted as $$\Phi_0$$, represents the smallest amount of magnetic flux that can pass through a superconducting loop and is quantized in units of $$\Phi_0 = \frac{h}{2e}$$, where $$h$$ is Planck's constant and $$e$$ is the elementary charge. This quantization is fundamental to understanding the behavior of superconductors, particularly how they interact with magnetic fields and form vortices in type-II superconductors.
Magnetometry: Magnetometry is the measurement of magnetic fields, often used to analyze materials and their magnetic properties. This technique is crucial for understanding superconducting phenomena, where the magnetic behavior of materials changes dramatically under certain conditions, like temperature or when exposed to a magnetic field. Through magnetometry, one can observe key phenomena such as flux quantization, the Meissner effect, and the operational principles of devices like SQUIDs.
Meissner Effect: The Meissner Effect is the phenomenon where a superconducting material expels magnetic fields as it transitions into the superconducting state, allowing it to exhibit perfect diamagnetism. This effect is fundamental to understanding how superconductors interact with magnetic fields and is crucial for applications like magnetic levitation.
Quantum Computing: Quantum computing is a revolutionary computing paradigm that uses the principles of quantum mechanics to process information in ways that classical computers cannot. By leveraging quantum bits, or qubits, these systems can perform complex calculations at unprecedented speeds and tackle problems considered intractable for traditional computers, making them highly relevant to advanced fields like superconductivity.
SQUIDs: Superconducting Quantum Interference Devices (SQUIDs) are highly sensitive magnetometers that exploit the quantum mechanical effects of superconductivity. They are capable of measuring extremely weak magnetic fields, making them invaluable tools in various applications including medical imaging and fundamental physics research. Their operation is fundamentally linked to principles of superconductivity, quantum mechanics, and the behavior of magnetic fields in superconductors.
Type-I Superconductor: A type-I superconductor is a material that exhibits superconductivity with a complete expulsion of magnetic fields, known as the Meissner effect, when cooled below a critical temperature. These superconductors are characterized by a single critical magnetic field, above which superconductivity is destroyed, and are typically pure elemental metals or simple alloys. Their behavior in magnetic fields leads to important phenomena like flux quantization and the generation of vortices.
Type-II Superconductor: A type-II superconductor is a material that exhibits superconductivity at higher temperatures and under stronger magnetic fields than type-I superconductors. These materials allow magnetic flux to penetrate them in quantized vortices, creating a mixed state where both superconducting and normal states coexist. This unique characteristic plays a crucial role in applications involving magnetic fields, such as MRI machines and particle accelerators.
Vitaly Ginzburg: Vitaly Ginzburg was a prominent Soviet physicist who made significant contributions to the field of superconductivity, particularly known for his development of the Ginzburg-Landau theory. This theory provides a theoretical framework for understanding superconductivity and describes the behavior of superconductors in terms of a complex order parameter, which is fundamental to understanding phenomena such as flux quantization and vortex dynamics.
Vortex lattice: A vortex lattice is a periodic arrangement of quantized magnetic vortices that form in type-II superconductors when exposed to a magnetic field. These vortices, which are regions where the magnetic flux penetrates the superconductor, organize into a structured pattern due to their interactions and the underlying superconducting order parameter. Understanding the behavior and dynamics of this lattice is crucial for applications in superconducting devices and materials.
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