Tru Physics

Superconductivity

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Tru Physics
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Introduction

Superconductivity is a quantum mechanical phenomenon where certain materials can conduct electric current with zero electrical resistance. This state occurs below a certain temperature, known as the critical temperature (T_c).

BCS Theory

The underlying theory of superconductivity is the BCS theory (Bardeen–Cooper–Schrieffer theory). This theory explains superconductivity as a state in which electrons with opposite momenta and spins are able to pair up, forming Cooper pairs. This happens due to an attractive interaction mediated by lattice vibrations (phonons).

The energy gap equation of the BCS theory is given by:

\Delta = - V \sum_{k} \dfrac{\Delta}{2E_k} \tanh\left(\dfrac{E_k}{2k_B T}\right)

where:

  • V is the interaction potential,
  • E_k = \sqrt{(\varepsilon_k - \mu)^2 + |\Delta|^2} is the energy of the quasiparticles,
  • \varepsilon_k is the single-particle energy,
  • \mu is the chemical potential,
  • \Delta is the superconducting gap, and
  • k_B is the Boltzmann constant.

Meissner Effect

The Meissner effect is a defining characteristic of superconductivity. When a material makes the transition from a normal to a superconducting state, it actively excludes magnetic fields from its interior, a phenomenon known as perfect diamagnetism. This expulsion of magnetic fields is described by the London equations:

\nabla^2 \vec{B} = \dfrac{1}{\lambda^2} \vec{B}

\dfrac{\partial \vec{B}}{\partial t} = \nabla \times (\lambda^2 \vec{J})

where \lambda is the London penetration depth, \vec{B} is the magnetic field, and \vec{J} is the current density.

Applications

Superconductivity has wide-ranging applications in science and technology. Due to their zero resistance, superconductors are used in producing strong magnetic fields for magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR). They are also used in power cables, transformers, particle accelerators, and in the construction of supercomputers. Furthermore, superconductors are pivotal in quantum computing, particularly in the making of qubits, the fundamental units of quantum information.

The study of superconductivity continues to be a major field of research in condensed matter physics, with ongoing efforts to discover and understand materials that exhibit high-temperature superconductivity.

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