Jaccarino–Peter Effect in Superconductivity

• The Jaccarino–Peter effect is a compensation mechanism in superconductors where internal exchange fields from magnetic ions offset external magnetic suppression.
• Experimental evidence and modified WHH models reveal non-monotonic Bc2 behavior, including concave curvature and reentrant superconductivity in EuFe2As2 and nickelates.
• This effect enables the design of superconductors with tunable internal fields, promising advancements in high-field applications and magnetically compensated superconductivity.

The Jaccarino–Peter effect is a compensation mechanism in superconducting materials, where an internal exchange field generated by magnetic ions counteracts the detrimental influence of an externally applied magnetic field on the superconducting state. This effect enables the stabilization and enhancement of superconductivity at magnetic field strengths that would normally suppress it, and has been identified in both low-transition-temperature (low-$T_c$) compounds (Chevrel phases, organic and uranium-based systems) and, more recently, in high-temperature superconductors such as infinite-layer nickelates and iron pnictides. The phenomenon manifests through distinct, non-monotonic behavior of the upper critical field $B_{c2}(T)$ as a function of temperature, often resulting in concave curvature or even reentrant superconductivity at extreme fields.

1. Physical Basis of the Jaccarino–Peter Compensation Effect

The core mechanism involves the interaction between localized magnetic moments (e.g., Eu$^{2+}$, Sm$^{3+}$) and superconducting electrons, mediated by antiferromagnetic exchange coupling. When a superconductor contains magnetic ions with sizable moments, these ions generate an internal effective field $H_J$ through exchange interactions. If $H_J$ opposes and partially cancels the applied external magnetic field, the net field experienced by Cooper pairs is reduced, thereby mitigating both orbital pair breaking (vortex formation) and paramagnetic pair breaking (Zeeman splitting).

Mathematically, this compensation can be represented by:

$H_{c2} = H_{c2}^* - a[H_{c2} - |H_J|]^2$

where $H_{c2}^*$ is the orbital critical field, $a$ encapsulates paramagnetic breaking strength, and $H_J$ is determined by the magnetization $M$ of the magnetic sublattice.

2. Manifestation in Magnetic and Superconducting Phase Diagrams

Experimental results in materials such as EuFe$_2$As$_2$ (0908.2280) under pressure and infinite-layer nickelates (SECS)NiO$_2$ reveal superconducting phase diagrams with features indicative of the Jaccarino–Peter effect:

Material	$T_c$ (K)	$B_{c2}$ (T)	Notable Phase Features
EuFe$_2$As$_2$	$\sim$31	Concave $B_{c2}$	AFM phase enclosed by SC; concave $B_{c2}$ vs. $T$
Infinite-layer NiO$_2$	up to $\sim$40	$>65$	Reentrant SC at high fields; high $T_c$

In EuFe$_2$As$_2$, ac susceptibility measurements show 100% diamagnetic shielding signaling the superconducting transition at $T_c\approx31$ K and an anomaly at $T_N\approx21$ K corresponding to antiferromagnetic ordering of Eu$^{2+}$ moments. The antiferromagnetic phase is fully enclosed within the superconducting regime. The $B_{c2}(T)$ curve displays strong concave curvature, characteristic of field compensation: above $T_N$, growing magnetization $M$ leads to incomplete compensation and enhanced pair breaking; below $T_N$, saturated magnetization renders $H_J$ roughly constant, allowing enhanced superconductivity and rapid increase of $B_{c2}$ as temperature decreases.

3. Experimental Evidence and Methodologies

Identification of the Jaccarino–Peter effect requires both magnetotransport and magnetic susceptibility measurements:

• AC magnetic susceptibility (EuFe$_2$As$_2$): detects superconducting shielding and AFM transition, reveals influence of internal exchange field on phase boundaries.
• Electrical resistance under pulsed and DC fields (NiO$_2$): characterizes reentrant superconductivity—suppression at low fields/recovery at high fields, resistivity drops order-of-magnitude at high fields.
• Field sweeps and angular studies: enable mapping of $B_{\text{appl}}$–$T$ phase diagrams, showing distinct regions of superconductivity and transitions between AFM and paramagnetic phases.
• Home-made rotators: probe angular dependence of compensation mechanism.

Comparison with theoretical models (e.g., WHH, Decroux–Fischer) using fitting parameters such as orbital critical field $H_{\text{orb}}$, paramagnetic limiting (Maki parameter), and exchange field $H_J$ demonstrates consistency with observed enhancement and reentrant behavior of $B_{c2}$.

4. Theoretical Modelling: Extensions of WHH and Compensation Formalism

The canonical Werthamer–Helfand–Hohenberg (WHH) approach is modified in these systems to include compensation effects:

$$2t \left[ 1 + \frac{2i(A_{so} - A_m)}{4y} \right] \psi\left(\frac{1}{2} + \frac{h + H_J + i(A_{so} - A_m)}{2t}\right) - \psi\left(\frac{1}{2}\right) = 0$$

with

• $t = T/T_c$,
• $h = 0.281\, H_{\text{orb}}(0)$,
• $h_J = 0.281\, H_J(T)$,
• $\psi$ the digamma function,
• $A_{so}$ and $A_m$ as spin–orbit and magnetic scattering parameters.

The exchange field $H_J$ is modeled via a Brillouin function, typically for $J=7/2$ Eu$^{2+}$ ions. This extension enables simultaneous description of low-field and high-field superconducting regions, with $H_J$ acting to offset the effect of applied fields on Cooper pairing. These theoretical results track with experimental phase diagrams, substantiating the presence and role of Jaccarino–Peter compensation.

5. Comparison Across Material Families

The Jaccarino–Peter effect has historical precedent in Chevrel phase compounds, organic conductors, heavy-fermion uranium systems, and Eu-based chalcogenides, where superconductivity is induced or stabilized at low temperatures ($T_c < 4$ K). Recent discoveries in infinite-layer nickelates (SECS)NiO$_2$ mark the first instance of this effect in a superconductor with significantly higher $T_c$ (up to 40 K). In such systems, strategic doping (Eu, Sm) and oxygen reduction tune the internal exchange field to optimum values, yielding $H_J \sim -70.5$ T and facilitating robust compensation even in ultra-high fields ($B_{c2} > 65$ T). These findings underscore substantial advances compared to cuprates and pnictides, where long-range magnetic order typically prevents effective compensation.

6. Implications and Prospects for Magnetically Compensated Superconductivity

The realization of enhanced and reentrant superconductivity through magnetic compensation opens new possibilities for high-field applications, such as superconducting magnets and quantum sensors. The ability to engineer materials with tunable internal fields—by selective incorporation of magnetic ions or precise control over valence states—provides a route to push the operational limits of superconductors. The success of the Jaccarino–Peter effect in nickelates indicates that the interplay of localized magnetism and itinerant superconductivity can be exploited, especially in layered systems with favorable electronic structures and minimal interfering long-range magnetic order.

A plausible implication is further exploration of other high-$T_c$ superconducting families where magnetic compensation mechanisms might be achievable, potentially driving the development of new superconductors capable of withstanding extreme magnetic environments.