Microscopic BCS Theory

Ginzburg–Landau theory leaves the coefficients $\alpha$ and $\beta$ and the pairing charge $2e$ as inputs. Bardeen, Cooper, and Schrieffer (1957) derived them from the electron–phonon interaction. Two steps carry the argument: Cooper's demonstration that the Fermi sea is unstable against pairing, and the BCS variational state that pairs every electron near the Fermi surface self-consistently. The weak-coupling solution predicts the energy gap, the critical temperature, and their universal ratio.

The phonon-mediated attraction

An electron distorts the lattice it moves through, drawing the positive ions inward and leaving a trailing region of excess positive charge. A second electron is attracted to that region. In second-order perturbation theory the exchange of a virtual phonon produces an effective electron–electron interaction that is attractive when the energy transfer is less than the phonon energy. BCS model this by a constant attractive matrix element within a shell of width $\hslash {\omega }_D$ (the Debye energy) about the Fermi surface,

where ${\xi }_{\vec{k}}={\epsilon }_{\vec{k}}-E_F$ is the single-electron energy measured from the Fermi level and $V>0$. The cutoff $\hslash {\omega }_D$ carries the ion mass through ${\omega }_D\propto M^{-1/2}$, which is the origin of the isotope effect.

The Cooper problem

Add two electrons to a filled Fermi sea at $T=0$, letting them interact through $V_{\vec{k}{\vec{k}}^{\prime }}$ while the sea electrons stay inert and merely block the states below $k_F$. A pair of zero total momentum and opposite spins has the wave function

The two-body Schrödinger equation, with the pair energy $E$, reads

so $g_{\vec{k}}=VC/(2{\xi }_{\vec{k}}+2E_F-E)$. Summing over $\vec{k}$ in the shell and cancelling $C$ gives the consistency condition, converted to an integral over the density of states $N(0)$ at the Fermi surface,

Writing the binding energy $E_b=2E_F-E>0$ and integrating,

The last form holds in the weak-coupling limit $N(0)V\ll 1$. The binding energy is positive for any $V>0$: however weak the attraction, the pair binds. The result is non-analytic in $V$ — no order of perturbation theory reproduces $e^{-2/N(0)V}$ — so the normal Fermi sea is unstable, and a new ground state is needed.¹

The BCS ground state and the gap equation

Because the instability involves every electron near $E_F$, the ground state pairs them all at once. BCS proposed the variational wave function

where $v_{\vec{k}}^2$ is the probability that the pair $(\vec{k}\uparrow ,-\vec{k}\downarrow )$ is occupied. Minimizing the expectation of the reduced Hamiltonian $H={\sum }_{\vec{k}}{\xi }_{\vec{k}}{c}^{†}c-V{\sum }_{\vec{k}{\vec{k}}^{\prime }}{c}_{\vec{k}\uparrow }^{†}{c}_{-\vec{k}\downarrow }^{†}{c}_{-{\vec{k}}^{\prime }\downarrow }{c}_{{\vec{k}}^{\prime }\uparrow }$ introduces the gap parameter

The minimization yields the excitation energy and the coherence factors

The minimum energy to add a single excitation (a Bogoliubov quasiparticle) is $E_{\vec{k}}\ge \Delta$: there is an energy gap $\Delta$ between the paired ground state and the lowest excitation. Substituting $u_{\vec{k}}{v}_{\vec{k}}$ into the definition of $\Delta$ gives the BCS gap equation,

At $T=0$ the integral is ${\sinh }^{-1}(\hslash {\omega }_D/\Delta )$, so

The gap equation is solved graphically by intersecting the decreasing curve $N(0)V{\sinh }^{-1}(\hslash {\omega }_D/\Delta )$ with the horizontal line at $1$; a stronger coupling $N(0)V$ raises the curve and pushes the intersection to a larger gap.

Density of states and the gap in the spectrum

The pairing removes all single-electron states within $\Delta$ of $E_F$ and piles them into sharp coherence peaks at the gap edges. Conserving states, $N_s(E)\mathbf{d}E=N(0)\mathbf{d}\xi$, gives the superconducting density of states

with $E$ measured from $E_F$. The gap $2\Delta$ is directly visible in single-particle tunnelling, and the coherence peaks are its signature.

Critical temperature and the universal ratios

At finite temperature the quasiparticle states are thermally populated with the Fermi function $f(E_{\vec{k}})$, which weakens the pairing. The factor $1-2f(E_{\vec{k}})=\tanh (E_{\vec{k}}/2k_{B}T)$ enters the gap equation,

The gap shrinks as $T$ rises and vanishes at $T_c$. Setting $\Delta =0$ there,

with $\gamma =0.577$ the Euler constant. Both $\Delta (0)$ and $k_B{T}_c$ carry the same exponential $e^{-1/N(0)V}$, so their ratio is a pure number independent of the material,

The energy gap at zero temperature is $E_g(0)=2\Delta (0)=3.53k_B{T}_c$, matching the phenomenological value $E_g\approx 3.5k_B{T}_c$. Because ${\omega }_D\propto M^{-1/2}$, the theory reproduces the isotope effect $T_c\propto M^{-1/2}$ with exponent $\alpha =\frac{1}{2}$.

The gap closes with a square-root law near $T_c$, $\Delta (T)\approx 1.74\Delta (0)\sqrt{1-T/T_c}$, and is nearly flat at low temperature.

Thermodynamics

The gap in the excitation spectrum controls the thermal properties. At low temperature the quasiparticle population is set by the Boltzmann factor across the gap, so the electronic heat capacity and the thermal conductivity fall off exponentially,

in contrast to the normal-state $\gamma T$. This exponential activation is direct evidence for a gap. At $T_c$ the heat capacity jumps, and BCS predicts the ratio of the jump to the normal electronic value as a second universal number,

independent of the material within weak coupling. Measured jumps in tin, aluminium, and other classic superconductors cluster near this value; deviations upward (lead, mercury) mark strong-coupling materials where $N(0)V$ is not small.

BCS theory accounts for the gap, the exponential $T_c$, the isotope effect, and the universal ratios from one attractive interaction. It also justifies Ginzburg–Landau theory near $T_c$, fixing the pairing charge at $2e$ and giving $\alpha$ and $\beta$ in terms of $N(0)$ and $T_c$. What it does not explain are the cuprates, whose critical temperatures and pairing symmetry lie outside the phonon weak-coupling picture, taken up in the final lesson.