BCS Superconductivity Theory

The Bardeen–Cooper–Schrieffer (BCS) theory explains conventional (low-$T_c$) superconductivity as a Cooper-pair condensate of electrons near the Fermi surface, stabilized by an effective attractive interaction mediated by phonons. It is mean-field many-body physics: one variational ground state captures the essential gap, thermodynamics, and electrodynamics.

Prerequisites: Fermi liquid / free-electron gas, second quantization, basic statistical mechanics at $T=0$ and finite $T$. Scope: BCS weak-coupling mean field and its measurable consequences. Eliashberg strong-coupling numerics, unconventional pairing symmetries, and high-$T_c$ cuprates are out of scope here.

Phenomenology BCS must explain

The order parameter is a complex gap $\Delta(\mathbf{r})$ (or $\Delta_{\mathbf{k}}$ in momentum space): the amplitude of Cooper-pair coherence. In s-wave BCS, $\Delta$ is uniform in $\mathbf{k}$ on the Fermi surface.

Cooper instability

Two electrons above a filled Fermi sea with opposite momenta and opposite spins ($\mathbf{k}\uparrow$, $-\mathbf{k}\downarrow$) can form a bound state if the interaction is net attractive in that channel — even when the bare Coulomb repulsion is large, phonon exchange can win at energies $\hbar\omega_D \ll E_F$ (Debye scale $\ll$ Fermi energy).

Cooper’s variational argument (1956): with an attractive square-well potential of width $2\hbar\omega_D$ around $E_F$, a bound pair exists for arbitrarily weak attraction in 3D. That instability at $T=0$ is the seed of BCS.

flowchart LR
    A[Two electrons near E_F] --> B{Net attraction in pair channel?}
    B -->|Yes| C[Cooper bound state]
    C --> D[Macroscopic pair condensate]
    D --> E[Gap Δ and superfluid response]
    B -->|No| F[Remain normal Fermi liquid]

BCS Hamiltonian (reduced form)

In momentum space, restricting to the pairing channel:

$$ \begin{aligned} \mathcal{H} ={}& \sum_{\mathbf{k},\sigma} \xi_{\mathbf{k}}\thinspace c_{\mathbf{k}\sigma}^\dagger c_{\mathbf{k}\sigma} \\ &\quad + \sum_{\mathbf{k}} \left( \Delta\thinspace c_{\mathbf{k}\uparrow}^\dagger c_{-\mathbf{k}\downarrow}^\dagger + \Delta^* c_{-\mathbf{k}\downarrow} c_{\mathbf{k}\uparrow} \right) \\ &\quad + \frac{|\Delta|^2}{g} \end{aligned} $$

Here $\xi_{\mathbf{k}} = \epsilon_{\mathbf{k}} - \mu$ is measured from the chemical potential (vanishes on the Fermi surface). $\Delta$ is the pair potential (order parameter), determined self-consistently. $g > 0$ is the effective pairing coupling (phonon-mediated attraction in the reduced model).

The quartic interaction that generates $\Delta$ is often written schematically as

$$ \begin{aligned} \mathcal{H}_{\text{int}} &= -\frac{g}{V} \sum_{\mathbf{k}} c_{\mathbf{k}\uparrow}^\dagger c_{-\mathbf{k}\downarrow}^\dagger \\ &\quad c_{-\mathbf{k}\downarrow} c_{\mathbf{k}\uparrow} \end{aligned} $$

with $g$ nonzero only for $|\xi_{\mathbf{k}}| < \hbar\omega_D$ (Debye cutoff). Mean-field decoupling replaces the four-operator term by $\Delta c^\dagger c^\dagger + \text{h.c.}$ plus $|\Delta|^2/g$.

Bogoliubov quasiparticles

Diagonalize $\mathcal{H}$ via the Bogoliubov transformation:

$$ \gamma_{\mathbf{k}\uparrow} = u_{\mathbf{k}} c_{\mathbf{k}\uparrow} - v_{\mathbf{k}} c_{-\mathbf{k}\downarrow}^\dagger, \quad \gamma_{-\mathbf{k}\downarrow}^\dagger = u_{\mathbf{k}} c_{-\mathbf{k}\downarrow}^\dagger + v_{\mathbf{k}} c_{\mathbf{k}\uparrow} $$

with $u_{\mathbf{k}}^2 + v_{\mathbf{k}}^2 = 1$ and $u_{\mathbf{k}} v_{\mathbf{k}} = \Delta / 2E_{\mathbf{k}}$. The Hamiltonian becomes

$$ \mathcal{H} = \sum_{\mathbf{k}} E_{\mathbf{k}} \left( \gamma_{\mathbf{k}\uparrow}^\dagger \gamma_{\mathbf{k}\uparrow} + \gamma_{-\mathbf{k}\downarrow}^\dagger \gamma_{-\mathbf{k}\downarrow} \right) + E_0 $$

Bogoliubov dispersion:

$$ E_{\mathbf{k}} = \sqrt{\xi_{\mathbf{k}}^2 + |\Delta|^2} $$

• $E_{\mathbf{k}} \geq |\Delta|$: the superconducting gap is the minimum excitation energy.
• Creating a real electron at $\mathbf{k}$ costs at least $\Delta$ if $\xi_{\mathbf{k}} = 0$ — scattering that would degrade coherence is suppressed at low $T$, hence zero DC resistance.

For a s-wave order parameter, $|\Delta_{\mathbf{k}}| = \Delta$ is constant on the Fermi surface; angle-dependent gaps appear in anisotropic or unconventional superconductors.

Gap equation (self-consistency)

At $T = 0$, the self-consistency condition for $\Delta$ is

$$ 1 = g \sum_{\mathbf{k}} \frac{1}{2E_{\mathbf{k}}} \quad \Rightarrow \quad 1 = N(0)\thinspace g \int_0^{\hbar\omega_D} \frac{d\xi}{\sqrt{\xi^2 + \Delta^0}} $$

with $N(0)$ the normal-state density of states at the Fermi level. Evaluating the integral gives the BCS gap equation:

$$ \Delta^0 = 2\hbar\omega_D \exp\left(-\frac{1}{N(0)\thinspace g}\right) $$

At finite $T$, thermal quasiparticle occupancy smears the gap:

$$ 1 = N(0)\thinspace g \int_0^{\hbar\omega_D} d\xi\thinspace \frac{\tanh(E/2k_B T)}{\sqrt{\xi^2 + \Delta^2(T)}} $$

The gap vanishes at $T_c$ where the linearized equation yields

$$ k_B T_c = \frac{2 e^\gamma}{\pi}\thinspace \hbar\omega_D \exp\left(-\frac{1}{N(0)\thinspace g}\right), \quad \gamma \approx 0.5772\quad \text{(Euler–Mascheroni)} $$

Weak-coupling BCS ratios (useful sanity checks):

$$ \frac{\Delta^0}{k_B T_c} \approx 1.764, \qquad \frac{C_s - C_n}{C_n}\bigg|_{T_c} \approx 1.43 $$

The isotope effect $T_c \propto M^{-1/2}$ follows because $\hbar\omega_D \propto M^{-1/2}$ when phonons mediate pairing.

Ginzburg–Landau and electrodynamics

Near $T_c$, a Ginzburg–Landau (GL) expansion in $|\psi|^2$ captures the same order parameter with a coherence length $\xi \sim v_F / \Delta$ and penetration depth $\lambda_L$. BCS microscopically fixes the GL coefficients.

London equation (local limit, $T \ll T_c$):

$$ \nabla \times \mathbf{j}_s = -\frac{n_s e^2}{m}\thinspace \mathbf{B} $$

Persistent supercurrents screen magnetic fields over $\lambda_L$ — the Meissner effect. A superconductor in an applied field is not merely a zero-resistance conductor; it is a perfect diamagnet (up to flux quantization and vortex physics in Type II materials).

Flux quantization: magnetic flux through a superconducting loop is quantized in units of $\Phi_0 = h/2e$, reflecting the $2e$ charge of Cooper pairs.

Josephson effect (device-relevant)

Two superconductors separated by a thin insulator form a Josephson junction. The DC Josephson relation:

$$ I = I_c \sin(\phi) $$

where $\phi$ is the difference of the superconducting phases across the barrier and $I_c$ depends on $\Delta$ and tunneling. The AC Josephson relation $\dot\phi = 2eV/\hbar$ links phase evolution to voltage — the basis of SQUIDs, voltage standards, and superconducting qubits.

For someone coming from nanostructures and transport, the same pairing physics appears when a normal metal or semiconductor is proximitized by a superconductor: induced gaps, Andreev reflection, and subgap conductance are mesoscopic signatures of the BCS order parameter.

Density of states

The quasiparticle DOS (per spin) is

$$ N_s(E) = N(0)\thinspace \frac{|E|}{\sqrt{E^2 - \Delta^2}}, \qquad |E| > \Delta $$

with a square-root van Hove singularity at $|E| = \Delta$. Tunneling spectroscopy (STM on superconductors, or planar junction $dI/dV$) measures this directly — a clean experimental handle on $\Delta$ and, with strong coupling, phonon structure (Eliashberg regime).

Limitations and extensions

BCS is the reference frame for conventional superconductivity: Cooper pairing, broken U(1) symmetry, gapped quasiparticles, and macroscopic phase coherence. Modern circuit QED and superconducting qubits still live in this picture, with junction nonlinearity and charge noise layered on top.

Related notes

• Quantum transport and mesoscopic signatures: pair tunneling, Andreev reflection (not yet a dedicated page here).
• Minkowski Space — unrelated physically, but the same “state a convention, then compute” style applies.

References (standard): Bardeen, Cooper & Schrieffer, Phys. Rev. 108, 1175 (1957); de Gennes, Superconductivity of Metals and Alloys; Tinkham, Introduction to Superconductivity.