The Cluster's BCS Theory
A page about BCS theory — the microscopic theory of superconductivity.
The BCS theory
BCS theory (Bardeen, Cooper, Schrieffer, 1957) explains conventional superconductivity as a condensate of Cooper pairs. The key insight: electrons in a metal, despite Coulomb repulsion, experience an attractive interaction mediated by phonons (lattice vibrations). This attraction overcomes the repulsion when the electrons are near the Fermi surface.
The BCS ground state is a coherent superposition of paired states: |BCS> = prod_k (u_k + v_k c^+k_up c^--k_down) |0> where v_k^2 is the probability that the state k is occupied and u_k^2 that it is empty.
In the cluster, the edit BCS theory explains an edit superconductivity as an edit condensate of edit pairs.
The Cooper problem
Cooper (1956) showed that two electrons added to a Fermi sea with any attractive interaction, no matter how weak, form a bound state. The binding energy is: Delta_E = 2 hbar omega_D exp(-2 / N(0) V) where omega_D is the Debye frequency, N(0) is the density of states at the Fermi level, and V is the effective attractive interaction. In the cluster, the edit Cooper problem shows an edit bound state.
The BCS gap
The energy gap at T = 0 is: Delta(0) = hbar omega_D exp(-1 / N(0) V) ~ 1.76 k_B T_c The gap closes at T_c: Delta(T_c) = 0. For a typical superconductor (T_c = 1 K): Delta(0) ~ 0.17 meV.
The gap suppresses scattering: an electron must break a Cooper pair (energy cost 2 Delta) before it can scatter. This is why there is zero resistance below T_c.
In the cluster, the edit BCS gap gives an edit energy cost.
The temperature dependence
Delta(T) / Delta(0) varies as: Delta(T) / Delta(0) ~ sqrt(1 - T / T_c) near T_c Delta(T) / Delta(0) ~ 1 - sqrt(2 pi Delta(0) / (k_B T)) exp(-Delta(0) / (k_B T)) for T << T_c
The specific heat shows an exponential suppression: C ~ exp(-Delta(0) / (k_B T)).
In the cluster, the edit temperature dependence gives an edit energy suppression.
The predictions
BCS theory predicts:
- Isotope effect: T_c ~ M^{-alpha}, alpha ~ 0.5 (omega_D ~ M^{-1/2})
- Energy gap: Delta = 1.76 k_B T_c
- Specific heat jump: Delta C / C_N = 1.43 at T_c
- Coherence length: xi_0 ~ hbar v_F / (pi Delta(0))
- London penetration depth: lambda_L ~ sqrt(m / (mu_0 n_s e^2))
In the cluster, the edit predictions include:
- edit Isotope effect
- edit Energy gap
- edit Specific heat jump
- edit Coherence length
- edit London penetration depth
This theory
This page is about BCS theory. Delta(0) = hbar omega_D exp(-1/N(0)V). Delta(0) = 1.76 k_B T_c. Zero resistance from energy gap. Isotope effect: T_c ~ M^{-0.5}. The theory is real.