6.7 High-temperature superconductivity

Fundamentals of high-temperature superconductivity

High-temperature superconductors (HTS) exhibit zero electrical resistance and perfect diamagnetism at temperatures well above those of conventional superconductors, in many cases above the boiling point of liquid nitrogen (77 K). Their existence challenges the framework of BCS theory and has driven the development of new theoretical and experimental approaches in condensed matter physics.

Critical temperature in HTS

The critical temperature ($T_c$) is the temperature below which a material enters the superconducting state. For known HTS materials, $T_c$ typically ranges from about 30 K to 135 K. The highest confirmed $T_c$ at ambient pressure belongs to mercury-based cuprates ($\text{HgBa}_2\text{Ca}_2\text{Cu}_3\text{O}_{8+\delta}$) at roughly 135 K.

Several factors influence $T_c$:

• Crystal structure and the number of superconducting layers per unit cell
• Doping level, which controls the charge carrier concentration
• Applied pressure, which can further raise $T_c$ (the same Hg-cuprate reaches ~164 K under high pressure)

Experimentally, $T_c$ is determined by measuring the onset of zero resistivity and the diamagnetic response via magnetic susceptibility.

Copper oxide-based superconductors

Cuprate superconductors were the first HTS family, discovered by Bednorz and Müller in 1986. Their defining structural feature is the $\text{CuO}_2$ plane, which serves as the primary superconducting unit.

Notable compounds include:

• YBCO ($\text{YBa}_2\text{Cu}_3\text{O}_{7-\delta}$), with $T_c \approx 93$ K
• BSCCO ($\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_{8+\delta}$), with $T_c \approx 85\text{–}110$ K depending on the number of $\text{CuO}_2$ layers

These materials display strong electron correlations, and their phase diagrams are rich. As you vary doping from the undoped parent compound, you pass through an antiferromagnetic insulating phase, a pseudogap regime, and finally the superconducting dome, with optimal $T_c$ at intermediate doping.

Iron-based superconductors

Discovered in 2008 by Hosono's group, iron-based superconductors form a second major HTS family. Instead of $\text{CuO}_2$ planes, these materials contain iron-pnictide (e.g., FeAs) or iron-chalcogenide (e.g., FeSe) layers as the active superconducting units.

• LaFeAsO$_{1-x}$F$_x$ was the first discovered, with $T_c$ up to ~26 K (higher in related compounds with heavier rare earths, reaching ~55 K)
• FeSe has a modest bulk $T_c \sim 8$ K, but monolayer FeSe on $\text{SrTiO}_3$ shows signatures of $T_c$ above 60 K

Key distinguishing features include multiband superconductivity (multiple Fermi surface sheets participate), potentially very high upper critical fields, and a close interplay between magnetism, nematicity, and superconductivity.

Microscopic mechanisms

Cooper pair formation

Superconductivity requires electrons to form bound pairs (Cooper pairs). In conventional BCS superconductors, lattice vibrations (phonons) mediate an attractive interaction between electrons. In HTS, the pairing "glue" is almost certainly not phonons alone.

Strong electronic correlations dominate the physics of HTS parent compounds. The leading candidates for the pairing mechanism are:

• Spin fluctuations: antiferromagnetic correlations mediate an effective attraction
• Charge fluctuations: density oscillations in the charge sector
• Orbital fluctuations: coupling between different orbital degrees of freedom (especially relevant in iron-based systems)

The stronger effective coupling in these electronic channels is thought to be responsible for the elevated $T_c$ values.

Unconventional pairing symmetry

Conventional superconductors have an isotropic s-wave gap: the energy gap is roughly the same in all momentum directions. HTS materials break this pattern.

• Cuprates exhibit $d_{x^2-y^2}$-wave symmetry. The gap has nodes (zeros) along the diagonal directions of the Brillouin zone, meaning there are low-energy quasiparticle excitations even deep in the superconducting state.
• Iron-based superconductors are often described by $s_\pm$ symmetry, where the gap changes sign between the hole and electron Fermi surface pockets but remains nodeless on each pocket.

These symmetries have been confirmed through phase-sensitive experiments, including corner-junction SQUID interferometry for cuprates and quasiparticle interference measurements for iron-based compounds.

Pseudogap phase

The pseudogap is one of the most debated phenomena in cuprate physics. In underdoped cuprates above $T_c$, a partial suppression of the electronic density of states appears below a characteristic temperature $T^*$.

This is not a full superconducting gap. It opens first near the antinodal regions of the Fermi surface (around $(π, 0)$) while the nodal regions remain gapless, producing disconnected "Fermi arcs."

Two broad classes of explanation compete:

• Preformed Cooper pairs: pairs form at $T^*$ but lack long-range phase coherence until $T_c$
• Competing order: a distinct order parameter (charge density wave, spin density wave, or loop currents) opens the gap independently of superconductivity

Experimental probes including ARPES, NMR, and optical conductivity have mapped the pseudogap extensively, but its origin remains unresolved.

Crystal structure and properties

Layered perovskite structure

Most HTS materials share a layered architecture built from alternating charge reservoir layers and superconducting layers. In cuprates, the superconducting layers are the $\text{CuO}_2$ planes; in iron-based compounds, they are the FeAs or FeSe layers.

The charge reservoir layers (e.g., BaO, SrO, or LaO) donate or accept charge carriers, effectively doping the active planes. This separation of structural roles is what makes chemical tuning so powerful: you can substitute atoms in the reservoir layers to control the carrier concentration in the superconducting planes without directly disrupting them.

The strength of interlayer coupling determines how three-dimensional the superconducting state is. Weakly coupled systems (like BSCCO) behave almost two-dimensionally, while more strongly coupled systems (like YBCO) show moderate three-dimensional character.

Anisotropic behavior

The layered structure produces strong anisotropy in nearly every measurable property:

• Resistivity: in-plane resistivity ($\rho_{ab}$) can be orders of magnitude smaller than out-of-plane resistivity ($\rho_c$)
• Coherence length: the in-plane coherence length $\xi_{ab}$ is typically 1–3 nm, while $\xi_c$ can be shorter than the interlayer spacing
• Upper critical field: $H_{c2}$ is much larger for fields applied parallel to the planes than perpendicular to them

This anisotropy has direct consequences for vortex physics. In highly anisotropic cuprates like BSCCO, vortices can decompose into weakly coupled "pancake vortices" confined to individual $\text{CuO}_2$ layers, leading to complex vortex phase diagrams including vortex liquid and vortex glass states.

Doping effects

The undoped parent compounds of both cuprates and iron-based superconductors are not superconducting. Cuprate parents are antiferromagnetic Mott insulators; iron-pnictide parents are antiferromagnetic metals (or semimetals).

Superconductivity is induced by doping, which can be achieved through:

• Chemical substitution (e.g., replacing La with Sr in $\text{La}_{2-x}\text{Sr}_x\text{CuO}_4$)
• Oxygen stoichiometry control (e.g., varying $\delta$ in YBCO)

The result is a characteristic dome-shaped $T_c$ vs. doping curve. Below optimal doping (underdoped), $T_c$ rises with increasing carrier concentration. Above optimal doping (overdoped), $T_c$ decreases. The peak of the dome defines the optimal doping level.

Experimental techniques

Angle-resolved photoemission spectroscopy

ARPES directly measures the momentum-resolved single-particle spectral function $A(\mathbf{k}, \omega)$. In practical terms, it tells you the energy and momentum of electrons near the Fermi surface.

For HTS research, ARPES provides:

• The Fermi surface topology, revealing how it evolves with doping
• The superconducting gap magnitude and symmetry as a function of momentum direction (this is how $d$-wave nodes in cuprates were mapped)
• The pseudogap and its temperature/doping evolution, including the Fermi arc phenomenon

The technique requires high-quality single crystals with clean, cleavable surfaces and ultra-high vacuum conditions. Layered materials like BSCCO cleave easily between weakly bonded layers, making them ideal ARPES samples.

Neutron scattering

Neutron scattering is uniquely suited to HTS because neutrons couple to both the lattice and the magnetic degrees of freedom.

• Elastic neutron scattering determines crystal structure and long-range magnetic order (e.g., confirming antiferromagnetism in parent compounds)
• Inelastic neutron scattering probes spin dynamics and collective excitations, particularly the spin resonance mode that appears below $T_c$ in both cuprates and iron-based superconductors

The spin resonance is a sharp peak in the magnetic excitation spectrum at a specific wavevector and energy. Its appearance in the superconducting state is strong evidence that spin fluctuations are intimately connected to the pairing mechanism. The main practical limitation is that neutron scattering requires large single crystals or well-aligned powder samples.

Scanning tunneling microscopy

STM and scanning tunneling spectroscopy (STS) provide real-space maps of the local density of states with atomic resolution.

In HTS materials, STM has revealed:

• Spatial inhomogeneity in the superconducting gap, particularly in BSCCO, where the gap varies on nanometer scales
• Vortex core structure, showing how the local density of states changes inside and around vortices
• Impurity effects, where individual atomic defects create characteristic patterns in the quasiparticle interference that can be used to reconstruct the gap symmetry

STM requires atomically flat, clean surfaces and operates under low-temperature, ultra-high vacuum conditions.

Theoretical models

BCS theory vs. HTS

BCS theory explains conventional superconductivity through phonon-mediated pairing of electrons into Cooper pairs with s-wave symmetry. It works well for materials like Al, Nb, and Pb, but it fails to account for several key features of HTS:

• $T_c$ values far exceed the BCS ceiling predicted for phonon-mediated pairing
• Coherence lengths are extremely short (1–3 nm), comparable to the lattice spacing
• Pairing symmetry is unconventional ($d$-wave or $s_\pm$)
• The normal state above $T_c$ is itself anomalous (strange metal behavior, pseudogap)

These failures point to a fundamentally different pairing mechanism, likely electronic rather than phononic, and require theoretical frameworks that can handle strong correlations.

Resonating valence bond theory

P.W. Anderson proposed the resonating valence bond (RVB) theory shortly after the discovery of cuprate superconductors. The central idea starts from the Mott insulating parent compound, where each Cu site has one localized electron.

In the RVB picture, these electrons form a quantum superposition of spin-singlet pairs (valence bonds) that resonate among many configurations rather than freezing into a static antiferromagnetic pattern. When you dope the Mott insulator by removing electrons (introducing holes), these preexisting singlet pairs become mobile and can carry charge without resistance.

RVB theory naturally predicts $d$-wave pairing symmetry and connects superconductivity to the strong antiferromagnetic correlations of the parent compound. However, making the theory quantitatively predictive has proven difficult, and debate continues about whether the RVB state is the correct ground state.

Spin fluctuation models

Spin fluctuation models propose that the exchange of antiferromagnetic spin fluctuations between electrons provides the pairing interaction, analogous to how phonon exchange works in BCS theory but with a magnetic origin.

The key predictions are:

• Superconductivity should appear near antiferromagnetic phases in the phase diagram (confirmed in both cuprates and iron-based systems)
• The pairing symmetry should be $d$-wave for cuprates (repulsive interaction at $(\pi, \pi)$) and $s_\pm$ for iron-based superconductors (repulsive interaction between hole and electron pockets)
• A spin resonance mode should appear in the superconducting state at the antiferromagnetic wavevector

Neutron scattering experiments have confirmed the spin resonance in multiple HTS families, providing strong support for these models.

Applications of HTS

Superconducting magnets

HTS materials can generate high magnetic fields while operating at temperatures accessible with liquid nitrogen cooling (77 K) rather than liquid helium (4.2 K). This dramatically reduces cooling costs and complexity.

Applications include:

• MRI machines: HTS inserts can boost field strength beyond what conventional NbTi or Nb$_3$Sn magnets achieve alone
• Particle accelerators: compact, high-field dipole and quadrupole magnets
• Fusion reactors: HTS tape (particularly REBCO) is being used in next-generation tokamak designs (e.g., SPARC) for high-field magnets

Challenges remain in fabricating long-length conductors, managing the thermal stability of HTS windings (quench protection is more complex than for low-temperature superconductors), and reducing manufacturing costs.

Power transmission

HTS cables can carry 3–5 times more current than copper cables of the same cross-section, with negligible resistive losses. They operate at liquid nitrogen temperatures, which is far cheaper to maintain than liquid helium.

Several pilot projects have demonstrated HTS power cables in urban grids, where space constraints make high current density valuable. The main barriers to widespread deployment are the cost of HTS tape, the need for reliable cryogenic cooling infrastructure along the cable length, and competition with incremental improvements in conventional grid technology.

Magnetic levitation

HTS bulk materials can trap magnetic flux through flux pinning, creating a stable levitation effect over a permanent magnet without any active feedback control. The trapped flux acts like a magnetic spring, providing both lift and lateral stability.

Applications include:

• Frictionless bearings for flywheels and other rotating machinery
• Maglev transportation: HTS-based systems can levitate vehicles with passive stability
• Practical implementation requires large, uniform HTS bulks or coated conductor tapes and reliable cryogenic systems

Challenges and future directions

Room-temperature superconductivity

Achieving superconductivity at ambient temperature and pressure remains the central goal of the field. Recent work on hydrogen-rich compounds (hydrides) under extreme pressures has produced striking results: $\text{H}_3\text{S}$ shows $T_c \approx 203$ K at ~150 GPa, and $\text{LaH}_{10}$ reaches ~250 K at ~170 GPa.

These results are exciting but come with caveats. The pressures required are enormous (millions of atmospheres), making practical applications impossible with current technology. Claims of near-room-temperature superconductivity in carbonaceous sulfur hydride (~288 K at 267 GPa) have faced scrutiny and reproducibility challenges. The path forward involves both verifying existing claims and searching for materials that superconduct at high temperatures under accessible pressures.

Material synthesis and optimization

Improving the critical parameters of HTS materials ($T_c$, critical current density $J_c$, upper critical field $H_{c2}$) requires advances in synthesis:

• Thin film growth via pulsed laser deposition or molecular beam epitaxy allows precise control of stoichiometry and strain
• Artificial pinning centers (nanoparticle inclusions, columnar defects) enhance $J_c$ by immobilizing vortices
• Heterostructures and interfaces can produce enhanced or novel superconducting states (e.g., the FeSe/SrTiO$_3$ interface)

Scaling these techniques from laboratory samples to industrial production of long-length wires and tapes remains a major engineering challenge.

Computational predictions

Modern computational methods play an increasing role in HTS research:

• Density functional theory (DFT) predicts crystal structures and electronic properties of candidate materials
• Dynamical mean-field theory (DMFT) and other many-body methods address the strong correlations that DFT handles poorly
• Machine learning approaches accelerate screening of large chemical spaces for promising superconducting compounds

The main computational challenge is that the same strong correlations responsible for HTS make these systems extremely difficult to simulate accurately. Integrating computational predictions with experimental validation is essential for efficient material discovery.

Comparison with conventional superconductors

Transition temperature

Coherence length

HTS materials have coherence lengths $\xi \sim 1\text{–}3$ nm, compared to tens or hundreds of nanometers in conventional superconductors. This short coherence length has several consequences:

• HTS materials are strongly type-II, with very high $H_{c2}$ values
• They are more sensitive to defects: a single atomic-scale defect can significantly affect the local superconducting order parameter
• Fabricating high-quality Josephson junctions is harder because the junction barrier must be controlled on the scale of $\xi$

Magnetic field effects

HTS materials have upper critical fields that can exceed 100 T (extrapolated to $T = 0$), far surpassing conventional superconductors. However, the irreversibility field $H_{\text{irr}}$, above which vortices become mobile and dissipation sets in, is often much lower than $H_{c2}$.

The vortex phase diagram in HTS is richer than in conventional superconductors due to strong thermal fluctuations and anisotropy. Distinct phases include:

• Vortex lattice: ordered Abrikosov lattice at low temperatures and fields
• Vortex liquid: thermally disordered vortex state, especially prominent in highly anisotropic cuprates
• Vortex glass: disordered but frozen vortex state pinned by defects

For applications, the irreversibility field (not $H_{c2}$) sets the practical operating limit.

Current research frontiers

Cuprate vs. iron-based superconductors

Comparative studies of these two HTS families aim to identify universal features of high-temperature superconductivity versus material-specific details.

• Iron-based superconductors are less anisotropic than cuprates, making them more promising for wire and magnet applications
• Cuprates achieve higher $T_c$ values but are brittle ceramics that are difficult to form into flexible conductors
• Both families show superconductivity emerging from or near antiferromagnetic parent states, suggesting magnetism plays a central role in both
• The pairing symmetries differ ($d$-wave vs. $s_\pm$), raising the question of whether a single theoretical framework can explain both

Topological superconductivity

Some HTS materials or their heterostructures may host topological superconducting states, which are predicted to support Majorana fermions at surfaces, edges, or vortex cores. Majorana fermions are their own antiparticles and obey non-Abelian exchange statistics, making them candidates for topological quantum computation.

Research focuses on:

• Surface states of cuprate and iron-based superconductors
• Engineered heterostructures combining HTS with topological insulators
• Vortex-bound states in $\text{FeTe}_{1-x}\text{Se}_x$, where evidence for Majorana zero modes has been reported

Quantum critical points

A quantum critical point (QCP) is a continuous phase transition at zero temperature, driven by a non-thermal control parameter like doping, pressure, or magnetic field. In several HTS systems, optimal $T_c$ appears to coincide with a QCP where antiferromagnetic or nematic order is suppressed to zero temperature.

Near a QCP, quantum fluctuations extend over all energy scales and may be responsible for both the strange metal behavior (linear-in-$T$ resistivity) observed in the normal state and the strong pairing interaction that produces high $T_c$. Disentangling quantum critical fluctuations from classical thermal fluctuations remains a significant experimental and theoretical challenge.