Superconductivity

7.1 Basic Phenomenology

Superconductivity is the complete loss of electrical resistance below a critical temperature $T_c$. Discovered by Onnes in 1911 (mercury, $T_c = 4.2$ K).

Key experimental facts:

1. Zero resistance: $\rho = 0$ for $T \lt T_c$.
2. Meissner effect: Complete expulsion of magnetic flux from the interior: $\mathbf{B} = 0$ inside a superconductor (for $T \lt T_c$ and $B \lt B_c$).
3. Critical magnetic field: Superconductivity is destroyed above $B_c(T) = B_c(0)[1 - (T/T_c)^2]$.
4. Critical current density: Superconductivity is destroyed above a critical current density $J_c$.

7.2 London Equations

The London equations describe the electromagnetic response of a superconductor:

$\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m_e}\mathbf{E}$

$\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m_e}\mathbf{B}$

Where $n_s$ is the density of superconducting electrons.

Combining with Maxwell”s equations:

$\nabla^2 \mathbf{B} = \frac{1}{\lambda_L^2}\mathbf{B}$

Where $\lambda_L = \sqrt{m_e/(\mu_0 n_s e^2)}$ is the London penetration depth.

The solution $\mathbf{B}(x) = B_0 e^{-x/\lambda_L}$ shows that magnetic fields decay exponentially Inside the superconductor, explaining the Meissner effect.

7.3 BCS Theory

BCS theory (Bardeen, Cooper, Schrieffer, 1957) explains superconductivity through the formation Of Cooper pairs.

Cooper pairing. Two electrons with opposite momenta and spins form a bound state via the Electron-phonon interaction (the lattice mediates an effective attractive interaction). The Cooper pair Has charge $2e$ and spin 0 (boson).

The BCS gap equation:

$\Delta = V_{\mathrm{pair} \sum_{\mathbf{k}} \frac{\Delta}{2E_{\mathbf{k}}} \tanh\left(\frac{E_{\mathbf{k}}}{2k_B T}\right)}$

Where $E_{\mathbf{k}} = \sqrt{\xi_{\mathbf{k}}^2 + \Delta^2}$ is the quasiparticle energy, $\xi_{\mathbf{k}}$ Is the normal-state energy relative to $E_F$And $\Delta$ is the superconducting energy gap.

At $T = 0$: $\Delta(0) = 2\hbar\omega_D\, e^{-1/(N(E_F)V_{\mathrm{pair})}}$ (BCS formula).

The critical temperature:

$k_B T_c = 1.13\,\hbar\omega_D\, e^{-1/(N(E_F)V_{\mathrm{pair})}}$

The ratio $2\Delta(0)/(k_B T_c) \approx 3.53$ is a universal BCS prediction.

7.4 Type I and Type II Superconductors

Type I: One critical field $B_c$. Below $B_c$: complete Meissner effect. Above $B_c$: normal State. Examples: Pb, Hg, Al.

Type II: Two critical fields $B_{c1} \lt B_{c2}$. For $B_{c1} \lt B \lt B_{c2}$: mixed state (vortices with normal cores in a superconducting matrix). For $B \gt B_{c2}$: normal state. Examples: Nb, YBCO (high-$T_c$).

7.5 High-Temperature Superconductors

Discovered in 1986 (Bednorz and Müller). Cuprate superconductors such as YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) have $T_c$ up to $\sim 135$ K. These are Type II, layered, and not fully explained by BCS Theory (the pairing mechanism is still debated).

Key properties of high-$T_c$ superconductors:

• d-wave pairing symmetry: Unlike conventional BCS superconductors (s-wave), cuprates have a gap function with $d_{x^2-y^2}$ symmetry: $\Delta(\mathbf{k}) = \Delta_0(\cos k_x - \cos k_y)/2$ which vanishes along the nodal directions $k_x = \pm k_y$.
• Short coherence length: $\xi \sim 1$—$2$ nm (compared with $\sim 100$ nm for conventional superconductors), making them sensitive to defects but allowing high critical current densities.
• Strong anisotropy: Superconducting properties differ dramatically between the $ab$-planes and the $c$-axis direction.
• Pseudogap phase: Above $T_c$ but below a characteristic temperature $T^*$A partial gap opens in the electronic spectrum, suggesting precursive pairing correlations.
• Phase diagram: Doping controls the transition from antiferromagnetic insulator (underdoped) through the superconducting dome to a normal metal (overdoped).

Other families of high-$T_c$ superconductors include iron-based pnictides ($T_c$ up to 56 K), Magnesium diboride MgB$_2$ ($T_c = 39$ K), and the recently discovered nickelates and hydrides ($T_c$ up to $\sim 250$ K under extreme pressure).