PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Condensed matter physics → Superconductivity ↓


BCS principle


The BCS theory, named after its creators John Bardeen, Leon Cooper and Robert Schrieffer, is a fundamental theory that explains superconductivity in condensed matter physics. Superconductivity is a quantum mechanical phenomenon where materials exhibit zero electrical resistance and expulsion of magnetic fields below a certain critical temperature. Let us take a deeper look at this remarkable theory that has reshaped our understanding of condensed matter physics.

Introduction to superconductivity

Before we explore the BCS theory, let's briefly review superconductivity. Discovered by Heike Kamerlingh Onnes in 1911, superconductivity refers to a phase of certain materials where they exhibit zero electrical resistance and perfect diamagnetism. This behavior occurs at a specific critical temperature (Tc). In normal conductive materials, resistance arises due to the scattering of electrons by impurities and lattice vibrations. However, in superconductors, electrons move without any resistance.

The path of BCS theory

For decades after its discovery, superconductivity was poorly understood. The field was revolutionized when Bardeen, Cooper, and Schrieffer presented their microscopic theory in 1957. The main breakthrough of their work was to show how electron pairing (known as Cooper pairing) could lead to a macroscopic quantum state that enables superconductivity.

Understanding the BCS principle

BCS theory centres on the concept of Cooper pairs, which are pairs of electrons with opposite spins that are coupled due to attractive interactions mediated by lattice vibrations or phonons. When these pairs form a coherent quantum state, together they form a 'superfluid' of charge that experiences no resistance to electric current.

The Cooper couple

Normally, electrons repel each other because they are negatively charged. However, when a lattice of positive ions (from the material's crystal structure) is involved, an effective attractive force can arise between two electrons. This occurs because as an electron passes through the lattice, it creates a wake that can attract another electron, leading to a net attractive interaction.

Mathematical framework

The BCS Hamiltonian describes the system of interacting electrons:

            H = Σ(k,σ) ε_k c_kσ† c_kσ + Σ(k,k') V(k,k') c_k↑† c_-k↓† c_-k'↓ c_k'↑
            H = Σ(k,σ) ε_k c_kσ† c_kσ + Σ(k,k') V(k,k') c_k↑† c_-k↓† c_-k'↓ c_k'↑

In this equation:

  • ε_k are the energy levels of the electrons.
  • c_kσ† and c_kσ are the creation and annihilation operators for an electron with momentum k and spin σ.
  • V(k,k') represents the effective interaction potential between pairs of electrons.

Solving the BCS Hamiltonian gives the gap equation that describes the energy required to break a Cooper pair:

            Δ = Σ(k') V(k,k') Δ / (2E_k') tanh(E_k' / (2k_B T))
            Δ = Σ(k') V(k,k') Δ / (2E_k') tanh(E_k' / (2k_B T))

Here, Δ is the energy gap, E_k' is the energy of the electrons, k_B is the Boltzmann constant, and T is the temperature.

Temperature and superconducting state

In BCS theory, the critical temperature Tc marks the phase transition to the superconducting state. Below Tc, the electrons form Cooper pairs, resulting in an energy gap Δ that characterizes superconductivity. The energy gap depends on temperature and closes as the system approaches Tc.

Energy gap Δ T_C

As the temperature increases, the Cooper pairs break down, thereby narrowing the energy gap until it vanishes at the critical temperature, and normal conduction occurs.

Key predictions and experiments

The BCS theory made several important predictions that were confirmed experimentally, resulting in its widespread acceptance:

  • Exponential increase in electronic heat capacity at Tc.
  • The isotope effect, where the critical temperature depends on the mass of the lattice ions, supports phonon-mediated coupling.
  • Presence of zero resistance and expulsion of magnetic field (Meissner effect).

Legacy and influence

The BCS theory was a major milestone not only in superconductivity, but also in the quantum theory of many-body systems. It demonstrated the power of macroscopic wavefunction and symmetry-breaking concepts. This theory opened avenues for understanding other quantum phenomena in condensed matter, such as superconductivity and the quantum Hall effect.

Conclusion

The BCS theory remains a cornerstone of theoretical physics despite the emergence of high-temperature superconductors and the need for new theories beyond its traditional framework. This is a testament to the beauty and depth of quantum mechanics, which provides a subtle understanding of one of the most fascinating states of matter.


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BCS principle

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