11.4 Superconductivity

Fundamentals of superconductivity

Superconductivity is a quantum phenomenon where certain materials lose all electrical resistance and expel magnetic fields when cooled below a critical temperature. Understanding it through statistical mechanics lets you connect what individual electrons are doing (forming pairs, interacting with the lattice) to the bulk properties you can measure in a lab, like zero resistance and specific heat jumps.

Basic properties of superconductors

Four defining features characterize the superconducting state:

• Zero electrical resistance allows current to flow without any energy loss. A current induced in a superconducting loop can persist for years with no measurable decay.
• Perfect diamagnetism (the Meissner effect) means the material actively expels magnetic fields from its interior, not just failing to let new fields in, but pushing out fields that were already there.
• Critical thresholds govern the state: superconductivity only survives below a critical temperature $T_c$, a critical magnetic field $H_c$, and a critical current density $J_c$. Exceed any one of these and the material reverts to normal.
• Macroscopic quantum coherence means all the Cooper pairs share a single quantum wave function $\psi(\mathbf{r})$. This is what makes superconductivity a macroscopic quantum phenomenon rather than just low resistance.

Critical temperature and fields

• Critical temperature $T_c$ is the sharp boundary between normal and superconducting states. For elemental superconductors, $T_c$ ranges from millikelvins (tungsten, ~15 mK) up to about 9.3 K (niobium).
• Upper critical field $H_{c2}$ is the maximum field a Type II superconductor can tolerate before superconductivity is destroyed entirely.
• Lower critical field $H_{c1}$ is the field at which magnetic flux first begins to penetrate a Type II superconductor, forming vortices.
• Critical current density $J_c$ sets the maximum lossless current. Beyond $J_c$, the material's own magnetic field exceeds the critical field and superconductivity breaks down.

Type I vs Type II superconductors

The distinction traces back to the Ginzburg-Landau parameter $\kappa = \lambda / \xi$, where $\lambda$ is the penetration depth and $\xi$ is the coherence length. Type I has $\kappa < 1/\sqrt{2}$; Type II has $\kappa > 1/\sqrt{2}$.

Microscopic theory of superconductivity

The microscopic picture explains why electrons, which normally repel each other, can form bound pairs and condense into a single quantum state. Statistical mechanics then describes how these pairs behave collectively to produce the thermodynamic signatures of superconductivity.

Cooper pair formation

Two electrons near the Fermi surface can form a bound state called a Cooper pair through an indirect attractive interaction:

1. One electron moves through the lattice and slightly distorts it, pulling positive ions toward its path.
2. This lattice distortion (a phonon) creates a region of excess positive charge.
3. A second electron is attracted to that region before the lattice relaxes.
4. The net effect is an attractive interaction between the two electrons, mediated by phonon exchange.

The paired electrons have opposite momenta ($\mathbf{k}$ and $-\mathbf{k}$) and opposite spins ($\uparrow$ and $\downarrow$), giving the pair zero net momentum and zero net spin. This makes Cooper pairs effective bosons, so they can all occupy the same quantum state. The binding energy is small, typically on the order of ~1 meV, which is why superconductivity requires low temperatures.

BCS theory overview

Bardeen-Cooper-Schrieffer (BCS) theory (1957) is the foundational microscopic theory. Its key elements:

• The ground state is a coherent superposition of occupied and unoccupied Cooper pair states, described by the BCS wave function with variational parameters $u_k$ and $v_k$.
• An energy gap $\Delta$ opens at the Fermi level, meaning it costs a minimum energy of $2\Delta$ to break a Cooper pair into two quasiparticles.
• The theory correctly predicts the isotope effect: $T_c \propto M^{-1/2}$, where $M$ is the ionic mass. This confirmed that lattice vibrations are central to conventional superconductivity.
• BCS theory also explains the exponential behavior of electronic specific heat at low $T$ and the coherence peak in NMR relaxation rates.

Energy gap in superconductors

The energy gap $\Delta(T)$ is the order parameter's magnitude and carries direct physical meaning: it's half the minimum energy needed to create an excitation from the superconducting condensate.

• At $T = 0$, the gap reaches its maximum value $\Delta(0)$.
• As temperature increases toward $T_c$, the gap shrinks continuously to zero, following a characteristic BCS curve.
• The BCS relation connects the zero-temperature gap to the critical temperature:

$2\Delta(0) \approx 3.53 \, k_B T_c$

• Tunneling spectroscopy directly measures this gap. For a conventional superconductor like aluminum ($T_c \approx 1.2$ K), $\Delta(0) \approx 0.18$ meV.

Materials that deviate significantly from the 3.53 ratio are called strong-coupling superconductors (like lead and mercury), and they require extensions of BCS theory (Eliashberg theory) for accurate description.

Macroscopic quantum phenomena

Because all Cooper pairs share a single macroscopic wave function $\psi(\mathbf{r}) = |\psi| e^{i\phi}$, quantum effects that are normally invisible at macroscopic scales become directly observable.

Meissner effect

The Meissner effect is the complete expulsion of magnetic flux from a superconductor's interior when it's cooled below $T_c$. This is distinct from what a hypothetical "perfect conductor" would do: a perfect conductor would trap whatever flux was present when resistance dropped to zero, but a superconductor actively expels it.

• Surface supercurrents arise spontaneously to generate a field that exactly cancels the applied field inside the bulk.
• The magnetic susceptibility is $\chi = -1$, the most diamagnetic response possible.
• Fields don't vanish instantly at the surface; they decay exponentially over the London penetration depth $\lambda$, typically 50–500 nm depending on the material.

Flux quantization

If you form a superconducting ring, the total magnetic flux threading it is quantized:

$\Phi = n \Phi_0, \quad \Phi_0 = \frac{h}{2e} \approx 2.07 \times 10^{-15} \, \text{Wb}$

This quantization comes from requiring the macroscopic wave function to be single-valued around the ring. The factor of $2e$ (not $e$) in the flux quantum is direct evidence that the charge carriers are Cooper pairs, not single electrons.

Josephson effect

When two superconductors are separated by a thin barrier (insulator, normal metal, or weak link), Cooper pairs can tunnel across. This is the Josephson effect, and it comes in two forms:

• DC Josephson effect: A supercurrent $I = I_c \sin(\Delta\phi)$ flows across the junction with no applied voltage, where $\Delta\phi$ is the phase difference between the two superconductors and $I_c$ is the critical current of the junction.
• AC Josephson effect: When a DC voltage $V$ is applied, the phase difference evolves in time, producing an oscillating current at frequency $f = 2eV/h$. This relation is so precise that it's used to define the volt in metrology.

Josephson junctions are the building blocks of SQUIDs (Superconducting Quantum Interference Devices) and superconducting qubits.

Thermodynamics of superconductors

The thermodynamic treatment of superconductivity connects the microscopic pairing mechanism to measurable quantities like free energy, entropy, and specific heat.

Free energy considerations

• Below $T_c$, the superconducting state has lower Gibbs free energy than the normal state. The difference is called the condensation energy.
• For a Type I superconductor, the condensation energy density equals $\mu_0 H_c^2 / 2$, which you can extract from magnetization measurements.
• Ginzburg-Landau theory expands the free energy in powers of the order parameter $\psi$ near $T_c$:

$F = F_n + \alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*}|(-i\hbar\nabla - 2e\mathbf{A})\psi|^2 + \frac{\mu_0 H^2}{2}$

Here $\alpha$ changes sign at $T_c$ (negative below, positive above), driving the transition. This phenomenological approach correctly predicts vortex structures, surface effects, and the distinction between Type I and Type II behavior.

Entropy and specific heat

The superconducting state is more ordered than the normal state, so it has lower entropy. At $T_c$, the entropies of the two states are equal (for a second-order transition), but their slopes differ, producing a discontinuity in specific heat.

• The specific heat jump at $T_c$ is predicted by BCS theory: $\Delta C / \gamma T_c \approx 1.43$, where $\gamma T$ is the normal-state electronic specific heat.
• Below $T_c$, the electronic specific heat drops exponentially as $C_{es} \propto e^{-\Delta / k_B T}$, reflecting the energy gap. Thermal excitations across the gap are suppressed at low temperatures.
• This exponential behavior is strong experimental evidence for the gap predicted by BCS theory, and deviations from it can signal unconventional pairing symmetry (like nodes in the gap).

Phase transitions in superconductors

• In zero applied field, the transition at $T_c$ is second order: no latent heat, but a discontinuity in specific heat.
• For a Type I superconductor in an applied field, the transition becomes first order, with a latent heat and a discontinuous change in magnetization.
• Type II superconductors have two transitions: from normal to the mixed (vortex) state at $H_{c2}$, and from the mixed state to the full Meissner state at $H_{c1}$.
• Near $T_c$, mean-field theory (Ginzburg-Landau) works well for most 3D superconductors because the coherence length is large. In low-dimensional or short-coherence-length materials, critical fluctuations become significant and require renormalization group treatment.

Applications of superconductivity

Superconducting magnets

Superconducting magnets produce strong, stable magnetic fields with essentially no resistive power dissipation. They use Type II superconductors because of their high $H_{c2}$ values.

• MRI machines use NbTi coils cooled to 4.2 K (liquid helium) to generate fields of 1.5–3 T for medical imaging.
• Particle accelerators like the LHC at CERN use $\text{Nb}_3\text{Sn}$ and NbTi magnets to bend and focus proton beams, reaching fields above 8 T.
• Fusion reactors (tokamaks like ITER) rely on superconducting magnets to confine plasma.
• The main engineering challenge is maintaining cryogenic temperatures and protecting against quench events, where a local loss of superconductivity causes rapid heating.

SQUID devices

SQUIDs exploit flux quantization and the Josephson effect to measure magnetic fields with extraordinary sensitivity, down to $\sim 10^{-15}$ T.

• A DC SQUID consists of two Josephson junctions in a superconducting loop. The critical current of the loop oscillates as a function of the applied flux with period $\Phi_0$.
• Applications include magnetoencephalography (mapping brain activity), geophysical surveying, and testing fundamental physics.
• SQUIDs approach the quantum limit of measurement sensitivity set by the uncertainty principle.

High-temperature superconductors

High-$T_c$ superconductors have critical temperatures above 77 K, meaning they can be cooled with liquid nitrogen instead of liquid helium. This dramatically reduces cost.

• The cuprate family (YBCO with $T_c \approx 93$ K, BSCCO with $T_c \approx 110$ K) was discovered in the late 1980s and remains the most studied class.
• Iron-based superconductors, discovered in 2008, form another high-$T_c$ family with $T_c$ values up to ~55 K.
• These materials are not well described by standard BCS theory; their pairing mechanisms likely involve spin fluctuations rather than phonons.
• Practical challenges include brittleness (they're ceramics), anisotropy, and difficulty fabricating long wires with high $J_c$.

Experimental techniques

Resistivity measurements

• The four-probe method is standard: two outer probes supply current while two inner probes measure voltage, eliminating contact resistance from the measurement.
• Sweeping temperature while measuring resistivity reveals $T_c$ as the point where resistance drops to zero.
• Sweeping the applied magnetic field maps out $H_{c2}(T)$.
• For high-frequency characterization, microwave surface resistance measurements detect residual losses even in the superconducting state, which relate to quasiparticle excitations.

Magnetic susceptibility tests

• AC susceptibility measurements detect the onset of diamagnetism at $T_c$. The real part shows the diamagnetic shielding; the imaginary part reveals losses (e.g., from vortex motion).
• SQUID magnetometry offers the highest sensitivity for small or thin-film samples.
• Vibrating sample magnetometers (VSM) measure bulk magnetization as a function of field, producing hysteresis loops that reveal $H_{c1}$, $H_{c2}$, and pinning behavior.
• The London penetration depth $\lambda(T)$ can be extracted from low-field magnetization data.

Tunneling spectroscopy

Tunneling experiments provide the most direct window into the superconducting gap:

1. A planar tunnel junction (superconductor-insulator-normal metal) measures the differential conductance $dI/dV$, which maps the quasiparticle density of states. Peaks at $V = \pm \Delta/e$ directly reveal the gap.
2. Scanning tunneling microscopy (STM) does this with spatial resolution, mapping gap variations across a sample surface.
3. Point-contact spectroscopy measures Andreev reflection at a normal-superconductor interface, where an incoming electron is retroreflected as a hole, converting a Cooper pair into the superconductor. The conductance enhancement at low bias is sensitive to the gap symmetry.

These techniques provided key confirmation of BCS theory and continue to be essential for studying unconventional superconductors with anisotropic or nodal gaps.

Statistical mechanics approach

Partition function for superconductors

Building the statistical mechanics of a superconductor starts from the BCS Hamiltonian:

1. Write the mean-field BCS Hamiltonian with the pairing interaction.
2. Apply the Bogoliubov transformation, which mixes electron creation and annihilation operators to define new quasiparticle operators $\gamma_k$ and $\gamma_k^\dagger$.
3. The transformed Hamiltonian is diagonal: $H = \sum_k E_k \gamma_k^\dagger \gamma_k + \text{const}$, where $E_k = \sqrt{\xi_k^2 + \Delta^2}$ is the quasiparticle energy and $\xi_k$ is the normal-state energy relative to the Fermi level.
4. The partition function is then a product of independent fermionic contributions: $Z = \prod_k \left(1 + e^{-\beta E_k}\right)^2$, where $\beta = 1/k_BT$.
5. From $Z$, you derive the free energy, and minimizing it self-consistently determines $\Delta(T)$, the gap equation.

Near $T_c$, Gaussian fluctuations around the mean-field solution become relevant, contributing corrections to conductivity (paraconductivity) and specific heat.

Order parameter and symmetry breaking

The superconducting order parameter $\psi(\mathbf{r}) = |\psi|e^{i\phi}$ is the expectation value of the Cooper pair field. Below $T_c$, the system spontaneously chooses a phase $\phi$, breaking the $U(1)$ gauge symmetry.

• $|\psi|^2$ is proportional to the superfluid density $n_s$.
• Ginzburg-Landau theory describes the spatial variation of $\psi$ near $T_c$, predicting vortex solutions, surface superconductivity, and the two characteristic length scales $\lambda$ and $\xi$.
• In unconventional superconductors, the order parameter has lower symmetry. For example, $d$-wave pairing in cuprates means $\Delta(\mathbf{k}) \propto \cos k_x - \cos k_y$, with nodes (zero gap) along certain directions. This has profound consequences for low-temperature thermodynamics and transport.

Fluctuations near critical point

Mean-field theory (BCS/Ginzburg-Landau) works well far from $T_c$ or in 3D systems with long coherence lengths. But fluctuations matter in several regimes:

• Gaussian fluctuations above $T_c$ produce measurable paraconductivity (Aslamazov-Larkin contribution) and excess diamagnetism.
• The Ginzburg criterion estimates when fluctuations become non-perturbative. For conventional superconductors, the Ginzburg number is tiny ($10^{-8}$), so mean-field theory works almost up to $T_c$. For high-$T_c$ cuprates, it's much larger ($10^{-2}$), and fluctuation effects are prominent.
• In 2D superconductors, the transition is described by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism: vortex-antivortex pairs unbind at the transition temperature, destroying superfluidity without a conventional order parameter onset.
• Renormalization group methods provide the systematic framework for studying these critical phenomena beyond mean-field approximations.

Advanced topics

Unconventional superconductivity

Not all superconductors follow the BCS phonon-mediated picture. Unconventional superconductors have pairing mechanisms and/or gap symmetries that differ from the standard $s$-wave BCS model.

• Cuprate high-$T_c$ superconductors have $d_{x^2-y^2}$ pairing symmetry, confirmed by phase-sensitive experiments. The gap has nodes, leading to power-law (not exponential) temperature dependence of specific heat and penetration depth.
• Heavy fermion superconductors (like CeCoIn$_5$, UPt$_3$) have effective electron masses 100–1000 times the free electron mass, and their pairing is likely mediated by spin fluctuations.
• Iron-based superconductors show $s_\pm$ pairing, where the gap changes sign between different Fermi surface sheets but remains nodeless on each sheet.
• Many unconventional superconductors show coexistence or competition with magnetism, charge density waves, or nematic order, making their phase diagrams rich and complex.

Topological superconductors

Topological superconductors combine superconducting pairing with non-trivial band topology, leading to protected boundary states.

• The key prediction is Majorana zero modes at boundaries, vortex cores, or domain walls. These are their own antiparticles and obey non-Abelian exchange statistics.
• Candidate systems include $p$-wave superconductors (like Sr$_2$RuO$_4$, though its pairing symmetry remains debated), semiconductor nanowires with strong spin-orbit coupling proximitized by an $s$-wave superconductor, and topological insulator surfaces with induced superconductivity.
• Majorana modes are of intense interest for topological quantum computing, where quantum information is encoded non-locally in pairs of Majoranas, making it inherently resistant to local decoherence.

Superconducting qubits

Superconducting circuits form one of the leading platforms for quantum computing. The key ingredient is the Josephson junction, which provides a nonlinear, dissipationless circuit element.

• Transmon qubits (the most common type today) are charge qubits designed to be insensitive to charge noise, using a large shunting capacitance across the junction.
• Flux qubits encode information in the direction of persistent current in a superconducting loop interrupted by Josephson junctions.
• Coherence times have improved from nanoseconds in early devices to hundreds of microseconds in state-of-the-art transmons, driven by better materials, junction fabrication, and circuit design.
• Companies like IBM, Google, and others use superconducting qubits as the basis of their quantum processors, with Google's Sycamore chip (53 qubits) and IBM's Eagle chip (127 qubits) as notable milestones.