Superconductivity

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 $J_c$.

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.

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.

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-$T_c$).

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).

• BCS gap ratio: $2\Delta(0)/(k_BT_c) \approx 3.53$ for conventional superconductors. Deviations indicate strong coupling or unconventional pairing.
• London penetration depth: $\lambda_L = \sqrt{m_e/(\mu_0 n_s e^2)}$. Typical values are 20—100 nm. $\lambda_L$ diverges as $T \to T_c$ because $n_s \to 0$.
• Coherence length: $\xi_0 = \hbar v_F/(\pi\Delta_0)$ is the BCS coherence length, setting the scale over which the gap varies spatially.
• Ginzburg-Landau parameter: $\kappa = \lambda_L/\xi_0$. Type I: $\kappa < 1/\sqrt{2}$; Type II: $\kappa > 1/\sqrt{2}$.
• Flux quantisation: The magnetic flux through a superconducting loop is quantised in units of $\Phi_0 = h/(2e) \approx 2.07 \times 10^{-15}$ Wb. The factor $2e$ reflects the charge of a Cooper pair.

• Confusing $B_c$ with $B_{c1}$ and $B_{c2}$: For Type II superconductors, $B_{c1}$ marks the onset of flux penetration (mixed state) while $B_{c2}$ marks the complete destruction of superconductivity. $B_c$ is only meaningful for Type I.
• Assuming zero resistance means infinite conductivity: The London equations show that superconductors have a frequency-dependent response. At finite frequency there is a reactive (lossless) current, not infinite DC conductivity.
• Neglecting the isotope effect: $T_c \propto M^{-\alpha}$ with $\alpha \approx 0.5$ for conventional superconductors. A deviation from $\alpha = 0.5$ suggests a non-phonon pairing mechanism.
• Overlooking metastable states: Type II superconductors can trap vortices (flux pinning). The critical current density $J_c$ depends on the pinning force, not just the upper critical field.

• MRI magnets: Superconducting magnets provide the strong, stable magnetic fields (1.5—7 T) required for magnetic resonance imaging. Niobium-titanium wire cooled to 4.2 K is the standard.
• Particle accelerators: The Large Hadron Collider at CERN uses over 1,200 superconducting dipole magnets (NbTi at 1.9 K) to bend proton beams at 6.5 TeV.
• SQUIDs: Superconducting quantum interference devices exploit flux quantisation and Josephson tunnelling to measure magnetic fields as small as $10^{-15}$ T, used in magnetoencephalography.
• Lossless power transmission: High-temperature superconducting cables (YBCO) are being deployed in urban grids to transmit large currents with zero resistive losses, though cooling costs must be offset.

A Josephson junction consists of two superconductors separated by a thin insulating barrier. Cooper pairs can tunnel through the barrier, producing remarkable effects:

• DC Josephson effect: A supercurrent $I = I_c \sin\phi$ flows across the junction even at zero applied voltage, where $\phi$ is the phase difference of the order parameter across the barrier and $I_c$ is the critical current.
• AC Josephson effect: When a constant voltage $V$ is applied, the phase evolves as $d\phi/dt = 2eV/\hbar$, producing an oscillating current with frequency $f = 2eV/h \approx 483.6$ MHz/$\mu$V. This provides an exact voltage-to-frequency conversion.

The Josephson effects are the basis for SQUIDs, voltage standards, and superconducting qubits used in quantum computing.