Modern Physics · Advanced Topics

Superconductivity & BCS Theory

Superconductors carry current with zero resistance and expel magnetic fields (Meissner effect). BCS theory explains conventional superconductivity through phonon-mediated Cooper pairing, producing a finite-temperature phase transition into a macroscopic coherent state.

Prerequisites

Solid-state physics (Ch. SS) \cdot Quantum mechanics (Ch. QM) \cdot Statistical mechanics (Ch. SM

\cdot Second quantization

Learning Goals

$Derive the London penetration depth \lambda_L from the London equations and explain the Meissner$ effect.
$State the BCS gap equation and compute \Delta(0) and T_{c} for a given coupling constant N(0)V$ .
Apply the DC and AC Josephson effects to calculate junction current and frequency-to-voltage ratio.
$Classify Type I and Type II superconductors using the Ginzburg-Landau parameter \kappa = \lambda/\xi$ .
$Describe the Abrikosov vortex lattice and derive the upper critical field H_{c2} in GL theor$ y.

SC.1 Phenomenology

Perfect conductivity: resistivity ρ → 0 below T_c. Record high-T_c: HgBa₂Ca₂Cu₃O₈ at 138 K (at ambient pressure), HₓS at 203 K (at 150 GPa), LaH₁₀ at 250 K (under pressure). Room-temperature superconductivity remains elusive.

Meissner effect: below T_c, B = 0 inside a superconductor regardless of whether it was cooled in a field or not. Not merely perfect diamagnetism — it is an active expulsion of flux. The London penetration depth λ_L characterizes the exponential decay of B from the surface: B(x) = B₀ e^(−x/λ_L).

Definition SC.1 — London Equations

F. and H. London (1935) proposed: \partialJ_s/\partialt = (n_{s} e^{2}/m) E and J_{s} = -(n_{s} e^{2}/m) A (London g

auge \nabla\cdotA = 0). The second equation gives the Meissner effect: \nabla^{2}B = B/\lambda_L^{2} where \lambda_L^{2} = m/

(

\mu_{0} n_{s} e^{2}). Typical values: \lambda_L \approx 50 nm (Al), 500 nm (YBCO

Type I vs Type II: Type I (Hg, Al, Pb) have a single critical field H_c. Type II (Nb, YBCO, MgB₂) have H(c1) and H(c2) — between them, magnetic flux enters in quantized vortices (Abrikosov lattice). Flux quantum: Φ₀ = h/(2e) ≈ 2.07×10⁻¹⁵ Wb. The factor of 2e is direct evidence for Cooper pairs.

SC.2 Cooper Pairs and the BCS Ground State

L. Cooper (1956): two electrons near the Fermi surface, interacting via an attractive potential V (mediated by phonons), form a bound state (Cooper pair) for any V > 0 — no matter how weak. This is special to 2D Fermi surface; no bound state in 3D free space for weak attraction.

E_{binding} = -2\hbar\omega_D e^(-2/N(0)V) \qquad (Cooper pair binding energy, N(0) = density of states at E_{F})

(SC.1)

The BCS ground state (Bardeen-Cooper-Schrieffer, 1957):

|BCS⟩ = \prod_k (u_{k} + v_{k} c^\dagger_{k↑} c^\dagger_{-k↓}) |0⟩ \qquad (BCS wavefunction)

(SC.2)

where |u_k|² + |v_k|² = 1, and v_k² = ½(1 − ξ_k/E_k) with E_k = √(ξ_k² + Δ²). The order parameter (gap) Δ = V Σ_k ⟨c_(−k↓) c_(k↑)⟩ satisfies the BCS gap equation.

Theorem SC.1 — BCS Gap Equation

The superconducting gap \Delta satisfies: 1/(N(0)V) = \int(0 to \hbar\omega_D) d\xi/\sqrt(\xi^{2} + \Delta^{2}) \to \Delta(0) \approx 2\hbar\omega_D

e^(-1/(N(0)V)) (weak coupling). Temperature dependence: \Delta(T) \to 0 as T \to T_{c} where k_{B} T_{c}

1.13 \hbar\omega_D e^(-1/(N(0)V)) = \Delta(0)/(1.76). The ratio 2\Delta(0)/(k_{B} T_{c}) = 3.52 is a universal BC

prediction.

Example SC.1 — Aluminum as a BCS Superconductor

$Al has T_{c} = 1.2 K, Debye temperature Θ_D = 428 K (\hbar\omega_D = k_{B} \times 428 K). Estimate \Delta(0), N(0$ )V, and the Fermi velocity from the coherence length.

Gap from T_c:

\Delta(0) = 1.76 k_{B} T_{c} = 1.76 \times (8.617\times10^{-5} eV/K)(1.2 K) = 1.76 \times 1.03\times10^{-4} eV = 1.82\times10^{-4} eV

= 0.18 meV. This matches tunneling spectroscopy measurements on Al junctions.

Coupling constant N(0)V:

From T_{c} = 1.13 Θ_D e^(-1/(N(0)V)): N(0)V = 1/ln(1.13 Θ_D/T_{c}) = 1/ln(1.13 \times 428/1.2) = 1/

ln(402.8) = 1/5.999 \approx 0.167. This is weak coupling (N(0)V ≪ 1) — BCS applies well

Coherence length:

\xi_{0} = \hbarv_F/(\pi\Delta) — the characteristic size of a Cooper pair. For Al: v_{F} = 2.03\times10^{6} m/s. \xi_{0}

= (

1.055\times10^{-34} \times 2.03\times10^{6})/(\pi \times 1.82\times10^{-4} \times 1.6\times10^{-19}) = 2.14\times10^{-28}/9.15\times10^{-23} \approx 2.3\times10^{-6} m =

0 nm. This is much larger than the lattice spacing — Cooper pairs are highly overlapping, not tightly bound molecules.

GL ratio:

\lambda_L/\xi_{0} \approx 50nm/2300nm \approx 0.022 ≪ 1/\sqrt2 \approx 0.707. Al is a Type I superconductor (Ginzburg-Landa

u parameter \kappa = \lambda/\xi < 1/\sqrt2). For \kappa > 1/\sqrt2 (Type II): Abrikosov vortex lattice forms above

H(c1).

SC.3 Josephson Effect

Two superconductors separated by a thin insulator (tunnel junction). The macroscopic wavefunction is Ψ = √(n_s) e^(iθ). The phase difference δ = θ₁ − θ₂ drives:

I = I_{c} sin(\delta) \qquad (DC Josephson effect) \qquad d\delta/dt = 2eV/\hbar \qquad (AC Josephson effect)

(SC.3)

DC Josephson: supercurrent flows with no voltage (V = 0). Critical current I_c ∝ Δ.AC Josephson: with DC voltage V, the phase oscillates at 2eV/ℏ — microwave radiation at f = 2eV/h ≈ 484 GHz/mV. The Josephson relation defines the voltage-frequency ratio 2e/h — used in metrology (voltage standard).

SQUID (Superconducting Quantum Interference Device): two junctions in parallel. Total current I = 2I_c cos(πΦ/Φ₀) sin(δ_avg) — periodic in the enclosed flux Φ. Sensitivity: δΦ ~ 10⁻⁶ Φ₀ — the most sensitive magnetometer ever built. Applications: brain imaging (MEG), gravitational wave detection, dark matter searches.

SC.4 Ginzburg-Landau Theory

Near T_c, expand the free energy in the order parameter Ψ (complex):

F = F_{n} + \alpha|\Psi|^{2} + \beta/2 |\Psi|^{4} + 1/(2m*)|(-i\hbar\nabla - 2eA)\Psi|^{2} + B^{2}/(2\mu_{0})

(SC.4)

where α = α₀(T − T_c) changes sign at T_c. Minimizing δF/δΨ* = 0 gives the Ginzburg-Landau (GL) equation. Two characteristic lengths emerge: penetration depth λ(T) = λ_L/√(1−T/T_c) and coherence length ξ(T) = ξ₀/√(1−T/T_c). GL parameter κ = λ/ξ: κ < 1/√2 → Type I; κ > 1/√2 → Type II.

Vortex structure (Abrikosov, 1957): in Type II, each vortex carries one flux quantum Φ₀ = h/(2e). Order parameter |Ψ| → 0 at vortex core (radius ξ). Magnetic field decays over λ from the core. The upper critical field H(c2) = Φ₀/(2πξ²) — at H(c2) vortex cores overlap and superconductivity is destroyed. For YBCO: H(c2) ~ 100 T, enabling high-field magnet applications (MRI, LHC).

Definition SC.2 — Common Traps

Zero resistance is not the whole story: the Meissner effect distinguishes superconductors from perfect conductors.
Cooper pairs are correlated states: they are not tiny molecules moving through the lattice.
Type I and type II behave differently in fields: vortices appear only in type II superconductors.
Critical current and field matter: exceeding either destroys superconductivity.

Exercises — SC.1–SC.4 Superconductivity

Derive the London penetration depth \lambda_L from the London equations. Estimate \lambda_L for niobiu

m (density n_{s} \approx 5\times10^{28} m^{-3}

Straightforward

Derive the BCS gap equation at T = 0 and show \Delta(0) = 2\hbar\omega_D e^(-1/(N(0)V)). Find the ratio

2\Delta(0)/(k_{B} T_{c}) = 3.52. Apply to lead (Pb, T_{c} = 7.2 K, Θ_D = 96 K

meV

Intermediate

Explain the AC Josephson effect. Show that the frequency-to-voltage ratio is 2e/h = 483.6

GHz/mV. How is this used to define the volt in the modern SI?

Intermediate

Derive the upper critical field H_{c2} in GL theory and the Abrikosov vortex lattice spacing

Estimate H_{c2} for YBCO (\xi \approx 1.5 nm, \lambda \approx 150 nm) and explain why vortex pinning matters for

pplications.

Challenging

Key Takeaways

$London equations: J_{s} = -(n_{s} e^{2}/m)A \to Meissner effect, field decays over \lambda_L = \sqrt(m/(\mu_{0}n_{s}$ $e^{2}$
$Cooper pairs: phonon attraction binds electrons near E_{F}; any V>0 gives a bound state (spe$ cial to Fermi surface).
$BCS gap: \Delta(0) = 2\hbar\omega_D e^(-1/(N(0)V)), ratio 2\Delta(0)/(k_{B} T_{c}) = 3.52 universal$ .
$DC Josephson: I = I_{c} sin(\delta), supercurrent with no voltage. AC Josephson: f = 2eV/h (volta$ ge standard).
$Type II: flux enters as Abrikosov vortices above H_{c1}. Each carries \Phi_{0} = h/(2e). H_{c2} = \Phi_{0}$ /( $2\pi\xi^{2}$
$GL theory: two lengths \lambda(T) and \xi(T), parameter \kappa = \lambda/\xi. \kappa < 1/\sqrt2: Type I; \kappa > 1/\sqrt2: Type$ II.