Bogoliubov–de Gennes: Quasiparticle Spectrum

BCS Quasiparticles

In a BCS superconductor, the pairing interaction transforms electrons and holes into Bogoliubov quasiparticles: superpositions of electron (probability u²) and hole (probability v²) with u² + v² = 1.

The quasiparticle energy is E(k) = √(ξₖ² + Δ²), where ξₖ = ℏ²k²/2m − μ is the normal-state dispersion and Δ is the superconducting gap. The gap is visible as a minimum at k = kF.

The coherence factors u(k) and v(k) determine the electron/hole character. At k = kF both are 1/√2 — maximum mixing.

E(k) = √(ξ(k)² + Δ²)
u²(k) = ½(1 + ξₖ/Eₖ)
v²(k) = ½(1 − ξₖ/Eₖ)
DOS: N(E) = N₀ · E/√(E²−Δ²)