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  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Condensed matter physics → Superconductivity ↓


BCS theory and Cooper pairs


Superconductivity is a unique phenomenon that occurs in certain materials at low temperatures, where they conduct electricity without any resistance. This extraordinary behaviour was first observed by Heike Kamerlingh Onnes in 1911. It took several decades for theoretical physicists to explain this effect. The breakthrough came in 1957 with the BCS theory proposed by John Bardeen, Leon Cooper and Robert Schrieffer, which provided a comprehensive explanation for superconductivity. Central to this theory is the concept of Cooper pairs.

Understanding superconductivity

Before we dive deep into the BCS theory and Cooper pairings, it is important to understand the fundamentals of superconductivity. In ordinary conductors such as metals, electrical resistance arises from the scattering of conduction electrons due to lattice imperfections, impurities, and vibrations (phonons). However, in a superconductor, this resistance drops to zero below a critical temperature.

Superconductors also exhibit the Meissner effect, in which they expel magnetic fields from within them, a phenomenon that distinguishes superconductivity from perfect conductivity. This effect provides an important clue to the microscopic understanding of superconductors.

Role of Cooper pairs

A fundamental concept of BCS theory is the formation of Cooper pairs. These are pairs of electrons that move in a correlated manner through a lattice. At first glance, the concept of electron pairing seems paradoxical because electrons are negatively charged particles and naturally repel each other. However, in the case of Cooper pairs, this attraction comes indirectly through lattice interactions.

How are Cooper pairs formed?

When an electron passes through a crystal lattice, it creates a tiny distortion in the lattice. This distortion can attract another electron. While the attraction between the two electrons in a Cooper pair is extremely weak compared to other forces, it proves to be quite strong under the right conditions—such as low temperatures.

The energy saved by forming this pair is less than the energy of the two separated electrons. Therefore, at sufficiently low temperatures, it becomes energetically favorable for the electrons to form pairs rather than remain unpaired. This pairing essentially opens an energy gap at the Fermi surface, which prevents the scattering processes that lead to electrical resistance.

Cooper pairing: mathematical framework

The consequences of Cooper pairing can best be understood using quantum mechanics. In a normal metal, electrons obey Fermi-Dirac statistics and occupy all energy states up to the Fermi level. However, in a superconductor, the electrons form bonded states – Cooper pairs – which behave as a single unit and condense into a collective ground state.

Ψ(k) = a_kψ(k) + a_{-k}ψ(-k)

The above equation is a simplified version of the wave function that describes the Cooper pair in terms of the quantum states of two electrons with momenta k and -k. The variational coefficients, a_k and a_{-k}, describe the probability amplitudes.

An important aspect is that Cooper pairs are bosons. This means that they do not obey the Pauli exclusion principle and can thus all occupy the same ground state. This results in the creation of a macroscopic quantum state, which gives rise to superconductivity.

Visualization of Cooper pair interaction

To see how Cooper pairs interact with the lattice, consider a one-dimensional line of positive ions representing the lattice. An electron moving through the lattice distorts nearby ions, creating a tiny "well" that can capture another electron:

Electron 1 Electron 2 Lattice interactions

In the diagram above, electron 1 distorts the lattice as it moves. This distortion is a temporary attraction point for electron 2, effectively pairing them as a Cooper pair.

Band gap and superconductivity

An essential aspect of superconductivity is the energy gap that emerges as a result of Cooper pairing. Unlike normal conductors, superconductors have a range of energies around the Fermi level where no electron states can exist. This is known as the superconducting energy gap and is responsible for the zero-resistance state.

band gap energy levels

As shown, this energy gap is an important hallmark of superconductivity. It prevents the electrons from scattering and thus circumvents the usual mechanism that creates resistance in ordinary conductors.

Effect of critical temperature and BCS theory

The formation of Cooper pairs and the associated superconducting energy gap are possible only at temperatures below a certain critical value, called the critical temperature T_c. BCS theory provides predictions about T_c and describes how it depends on various factors such as material properties and lattice structures.

BCS theory also predicts the effects of magnetic fields and impurities on superconductivity. Superconductors can be classified into two types based on their responses to external magnetic fields: type I and type II.

Type I superconductors exhibit a complete Meissner effect, expel all magnetic fields, and show superconductivity up to a critical field strength below T_c. On the other hand, Type II superconductors allow partial penetration of magnetic fields through vortices when exposed to strong fields.

Influence of the BCS theory on modern physics

BCS theory has had profound influence in many areas of modern physics, materials science and engineering. Its principles extend beyond superconductivity and have influenced the understanding of superfluidity, Bose–Einstein condensates and other phenomena involving collective quantum states.

In addition, the BCS theory laid the groundwork for the development of superconductor-dependent technology, such as MRI machines, maglev trains, and more. Despite its complex mathematical formulas, the underlying concept of Cooper coupling provides a beautifully simple explanation for a complex and intriguing phenomenon.

Explaining superconducting systems provides fascinating insights not only into condensed matter physics but also into the quantum-scale wonders that govern the universe. The cooperative behavior of electrons in superconductors is evidence of the mysterious and magnificent laws of quantum mechanics.


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BCS theory and Cooper pairs

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