9.5 Type-I and Type-II superconductors

Type-I vs Type-II superconductors

Superconductors fall into two classes based on how they respond to magnetic fields. Type-I superconductors completely expel magnetic flux up to a single critical field, then abruptly lose superconductivity. Type-II superconductors have two critical fields and allow partial flux penetration between them, which lets them remain superconducting in much stronger fields. This distinction controls which materials are useful in real applications and which remain mostly laboratory curiosities.

The classification comes down to a single parameter from Ginzburg-Landau theory: the ratio $\kappa = \lambda / \xi$. Everything else follows from that.

Properties of Type-I superconductors

Perfect diamagnetism below $H_c$

Type-I superconductors exhibit the Meissner effect: they completely expel magnetic flux from their interior when cooled below $T_c$ and kept below the critical field $H_c$. Surface screening currents arise spontaneously and generate a field that exactly cancels the applied field inside the bulk.

This gives a magnetic susceptibility of $\chi = -1$, the most diamagnetic response possible. No flux lines thread the interior at all.

Abrupt transition to the normal state

When the applied field exceeds $H_c$, a Type-I superconductor switches to the normal (resistive) state in a single sharp, first-order phase transition. There is no intermediate regime. The magnetization curve shows a discontinuous jump at $H_c$.

This all-or-nothing behavior makes Type-I materials impractical for any application that involves strong magnetic fields.

Low $T_c$ and $H_c$ values

Type-I superconductors tend to have critical temperatures below about 10 K and critical fields on the order of tens of millitesla (a few hundred gauss). For example:

• Mercury (Hg): $T_c \approx 4.15$ K, $\mu_0 H_c \approx 41$ mT
• Lead (Pb): $T_c \approx 7.2$ K, $\mu_0 H_c \approx 80$ mT
• Aluminum (Al): $T_c \approx 1.2$ K, $\mu_0 H_c \approx 10$ mT
• Tin (Sn): $T_c \approx 3.7$ K, $\mu_0 H_c \approx 31$ mT

These are mostly pure elemental metals. Their low $T_c$ and $H_c$ values confine them largely to fundamental research rather than engineering applications.

Properties of Type-II superconductors

Lower and upper critical fields

Type-II superconductors are defined by two critical fields:

• $H_{c1}$ (lower critical field): Below this, the material is in the full Meissner state with complete flux expulsion, just like a Type-I superconductor.
• $H_{c2}$ (upper critical field): Above this, superconductivity is destroyed entirely and the material becomes normal.

Between $H_{c1}$ and $H_{c2}$ lies the mixed state (also called the vortex state), which is the defining feature of Type-II behavior.

Mixed state between $H_{c1}$ and $H_{c2}$

In the mixed state, magnetic flux penetrates the superconductor, but not uniformly. Instead, it enters as discrete flux vortices (also called fluxons). Each vortex has a small normal-state core of radius $\sim \xi$, surrounded by circulating supercurrents that decay over a length scale $\sim \lambda$.

The bulk of the material between vortices remains superconducting. This is why Type-II superconductors can carry supercurrent even in fields of many tesla.

Magnetic vortices and the Abrikosov lattice

Each vortex carries exactly one flux quantum:

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

The vortices repel each other (their circulating currents interact) and, in a clean system, settle into a regular triangular (hexagonal) lattice predicted by Abrikosov. This Abrikosov vortex lattice has been directly imaged using decoration techniques and scanning tunneling microscopy.

If an external current exerts a Lorentz force on the vortices and they start to move, energy is dissipated and resistance appears. Preventing this motion is the central engineering challenge for Type-II materials.

Higher $T_c$ and $H_{c2}$ values

Type-II superconductors span a wide range of critical parameters:

• NbTi: $T_c \approx 9$ K, $\mu_0 H_{c2} \approx 15$ T
• $\text{Nb}_3\text{Sn}$: $T_c \approx 18$ K, $\mu_0 H_{c2} \approx 30$ T
• $\text{MgB}_2$: $T_c \approx 39$ K
• YBCO ($\text{YBa}_2\text{Cu}_3\text{O}_7$): $T_c \approx 93$ K, $\mu_0 H_{c2} > 100$ T
• BSCCO: $T_c \approx 110$ K

The cuprate high-temperature superconductors are all Type-II, and their $T_c$ values exceed 77 K (liquid nitrogen temperature), which dramatically lowers cooling costs.

Differences in magnetic behavior

Meissner effect in Type-I superconductors

Below $H_c$, a Type-I superconductor is a perfect diamagnet with $\chi = -1$. The magnetization curve is a straight line: $M = -H$. At $H = H_c$, the magnetization drops to zero discontinuously and the material goes normal. There is a latent heat associated with this first-order transition.

Partial flux penetration in Type-II superconductors

A Type-II superconductor also shows $\chi = -1$ for $H < H_{c1}$. At $H_{c1}$, vortices begin to enter and the magnetization starts to decrease in magnitude continuously. As $H$ increases toward $H_{c2}$, more vortices pack in, the normal-core volume fraction grows, and the magnetization smoothly approaches zero. The transition at $H_{c2}$ is second-order (no latent heat).

Flux pinning

In real Type-II materials, crystal defects, grain boundaries, precipitates, and deliberately introduced impurities act as pinning centers that trap vortices in place. Pinned vortices cannot move in response to a transport current, so no energy is dissipated.

Strong pinning is what gives Type-II superconductors a high critical current density $J_c$. Without pinning, even a tiny current would set vortices in motion and the material would show resistance. The entire field of applied superconductivity depends on engineering effective pinning.

Ginzburg-Landau theory

Order parameter and coherence length

Ginzburg-Landau (GL) theory describes superconductivity near $T_c$ using a complex order parameter $\psi(\mathbf{r})$, where $|\psi|^2$ is proportional to the local superfluid density. Two characteristic lengths emerge from the theory:

• Coherence length $\xi$: the distance over which $\psi$ can vary spatially. It sets the size of a vortex core.
• Penetration depth $\lambda$: the distance over which an external magnetic field decays inside the superconductor.

Both $\xi$ and $\lambda$ diverge as $T \to T_c$, but their ratio stays constant.

The $\kappa$ parameter and Type-I vs Type-II classification

The Ginzburg-Landau parameter is:

$\kappa = \frac{\lambda}{\xi}$

The classification rule:

• $\kappa < \frac{1}{\sqrt{2}} \approx 0.71$: Type-I. The coherence length exceeds the penetration depth. The surface energy between a normal and superconducting region is positive, so the system avoids creating interfaces. Flux is either fully expelled or fully admitted.
• $\kappa > \frac{1}{\sqrt{2}}$: Type-II. The penetration depth exceeds the coherence length. The surface energy is negative, so the system gains energy by subdividing flux into as many vortices as possible. This is why the mixed state forms.

The sign of the normal-superconducting interface energy is the physical reason behind the two types. Type-I materials "want" large domains; Type-II materials "want" finely divided flux.

Typical values

Most pure elemental superconductors have $\kappa \ll 1$ (Type-I). Alloys and compounds tend to have shorter mean free paths, which reduces $\xi$ and increases $\kappa$ well above $1/\sqrt{2}$, pushing them into Type-II territory. The cuprate superconductors have $\kappa \sim 100$.

Applications of Type-II superconductors

High-field superconducting magnets

Type-II superconductors are the backbone of high-field magnet technology. NbTi wire is the standard for MRI machines (fields of 1.5–3 T) and particle accelerator dipoles (the LHC uses NbTi at 8.3 T). $\text{Nb}_3\text{Sn}$ is used when fields above ~10 T are needed, such as in fusion reactor magnets and next-generation accelerators.

Superconducting wires and cables

Superconducting power cables made from YBCO or BSCCO tapes can carry current densities orders of magnitude higher than copper with zero resistive loss. Pilot projects for urban power transmission already exist. The wires are typically manufactured as composite tapes with the superconductor deposited on a flexible metal substrate.

Josephson junctions and SQUIDs

A Josephson junction consists of two superconductors separated by a thin barrier (insulator, normal metal, or weak link). Cooper pairs tunnel across the barrier, producing a supercurrent that depends on the phase difference between the two sides.

SQUIDs (Superconducting Quantum Interference Devices) use one or two Josephson junctions in a superconducting loop. They can detect magnetic flux changes as small as a fraction of $\Phi_0$, making them the most sensitive magnetometers known. Applications include biomedical imaging (magnetoencephalography), geological surveying, and qubit readout in quantum computing.

Limitations and challenges

Flux creep and flux flow

Even with pinning, vortices can escape their pinning sites through thermal activation. This process, called flux creep, causes a slow decay of persistent currents over time. At higher currents, when the Lorentz force on vortices exceeds the pinning force, flux flow sets in: vortices move continuously, dissipating energy and producing measurable resistance.

Both effects limit the maximum useful current density and are worse at higher temperatures, which is a particular challenge for the high-$T_c$ cuprates.

AC losses

When a Type-II superconductor carries alternating current or sits in a time-varying field, vortices oscillate back and forth, dissipating energy on each cycle. These AC losses generate heat that must be removed by the cryogenic system. Minimizing AC losses requires careful conductor design: fine filaments, twisted multifilamentary wires, and resistive barriers between filaments.

Fabrication and material issues

Many high-performance Type-II superconductors are brittle intermetallic compounds ($\text{Nb}_3\text{Sn}$) or complex layered ceramics (YBCO, BSCCO). Fabricating them into long, flexible wires with uniform properties is difficult and expensive. Grain boundaries in the cuprates can act as weak links that block supercurrent, so careful texture control (grain alignment) is essential. Ongoing research focuses on raising $T_c$, improving $J_c$ through better pinning, and reducing manufacturing costs.