Spin-orbit coupling (SOC) relates to the interaction between an electron’s motion and its spin and is ubiquitous in solid-state systems. Although the effect of SOC in normal-state phenomena has been extensively studied, its role in superconducting hybrid structures and devices elicits many unexplored questions. In conjunction with broken symmetries and material inhomogeneities within superconducting hybrid structures, SOC may have contributions beyond its effects in homogeneous materials. Notably, even with well-established magnetic or nonmagnetic materials and conventional 𝑠-wave spin-singlet superconductors, SOC leads to emergent phenomena including equal-spin-triplet pairing and topological superconductivity (hosting Majorana states), a modified current-phase relationship in Josephson junctions, and nonreciprocal transport, including superconducting diode effects. SOC is also responsible for transforming quasiparticles in superconducting structures, which enhances the spin Hall effect and changes the spin dynamics. Taken together, SOC in superconducting hybrid structures and the potential for electric tuning of the SOC strength create interesting possibilities to advance superconducting spintronic devices for energy-efficient computing and enable topological fault-tolerant quantum computing. By providing a description of experimental techniques and theoretical methods to study SOC, this Colloquium describes the current understanding of resulting phenomena in superconducting structures and offers a framework to select and design a growing class of materials systems where SOC plays an important role.

We propose that geometric curvature and torsion may be used to probe the quality of an uncompensated antiferromagnetic interface, using the proximity effect. We study a helix of antiferromagnetic wire coupled to a conventional superconductor, and show that a density of states measurement can give information about the quality of an uncompensated interface, crucial for many recently predicted antiferromagnetic proximity effects. Furthermore, we show that geometric curvature alone can result in long-ranged superconducting triplet correlations in the antiferromagnet, and we discuss the impact curvature and torsion can have on the future development of superconducting spintronic devices.

We show that tailoring the geometric curvature profile of magnets can be used for bespoke design of an effective non-relativistic spin–orbit coupling, which may be used to control proximity effects if the magnet is coupled to a superconductor. We consider proximity-coupled one-dimensional magnetic wires with variable curvatures, specifically three distinct shapes classified as J-, C-, and S-type. We demonstrate a chirality-dependent spin polarization of the superconducting correlations, and show the role of curvature in determining the ground state of mixed-chirality junctions. We speculate on how this may be implemented in novel device design, and include analysis of its usage in a spin-triplet SQUID.

Spin -split superconductors offer new functionality compared to conventional superconductors such as diode effects and efficient thermoelectricity. The superconducting state can nevertheless only withstand a small amount of spin splitting. Here, we self -consistently determine the spin transport properties and the phase diagram of a spin -split superconductor in the presence of an injected spin accumulation. Energy and spin relaxation are accounted for in the relaxation time approximation via a single effective inelastic scattering parameter. We find that the spin -splitting field in the superconductor enables a spin diode effect. Moreover, we consider the superconducting phase diagram of a system in contact with a spin accumulation and in the presence of spin relaxation, and find that the inclusion of energy and spin relaxation alters the phase diagram qualitatively. In particular, these mechanisms turn out to induce a superconducting state in large parts of the phase diagram where a normal state would otherwise be the ground state. We identify an Fulde-Ferrel-Larkin-Ovchinnikkov-like state even in the presence of impurity scattering, which can be controllably turned on and off via the electrically induced spin accumulation. We explain the underlying physics from how the superconducting order parameter depends on the nonequilibrium modes in the system as well as the behavior of these modes in the presence of energy and spin relaxation when a spin -splitting field is present.

Majorana zero modes (MZMs) are of central importance for modern condensed matter physics and quantum information due to their non-Abelian nature, which thereby offers the possibility of realizing topological quantum bits. We here show that a grain boundary (GB) defect can host a topological superconductor (SC), with a pair of cohabitating MZMs at its end when immersed in a parent two-dimensional gapped topological SC with the Fermi surface enclosing a nonzero momentum. The essence of our proposal lies in the magnetic-field driven hybridization of the localized MZMs at the elementary blocks of the GB defect, the single lattice dislocations, due to the MZM spin being locked to the Burgers vector. Indeed, as we show through numerical and analytical calculations, the GB topological SC with two localized MZMs emerges in a finite range of both the angle and magnitude of the external magnetic field. Our work demonstrates the possibility of defect-based platforms for quantum information technology and opens up a route for their systematic search in future.

We present a mechanism allowing for superconductivity at high magnetic fields, beyond the Pauli-Chandrasekhar-Clogston limit. We consider spin splitting induced by an in-plane external magnetic field in a superconductor with two relevant bands close to the Fermi level. The magnetic field therefore controls which bands are available for Cooper pair formation. The presence of interband superconducting pairing, i.e., Cooper pairs formed by electrons with different band indices, produces high-field reentrant superconducting domains, whose critical magnetic field violates the Pauli-Chandrasekhar-Clogston limit. We analyze how the interband superconducting domains are influenced by the band parameters, and show that, for a certain range of parameters, the system presents two separate superconducting regions, for low and high magnetic field.

We show that corner Majorana zero modes in a two-dimensional p + id topological superconductor can be controlled by the manipulation of the parent p-wave superconducting order. Assuming that the p-wave superconducting order is in either a chiral or helical phase, we find that when a dx2−y2 wave superconducting order is induced, the system exhibits quite different behavior depending on the nature of the parent p-wave phase. In particular, we find that while in the helical phase, a localized Majorana mode appears at each of the four corners, in the chiral phase, it is localized along only two of the four edges. We furthermore demonstrate that the Majoranas can be directly controlled by the form of the edges, as we explicitly show in the case of circular edges. We argue that the application of strain may provide additional means of fine-tuning the Majorana zero modes in the system; in particular, it can partially gap them out. Our findings may be relevant for probing the topology in two-dimensional mixed-pairing superconductors.

Coupling a conventional s-wave superconductor to a ferromagnet allows us, via the proximity effect, to generate superconducting triplet correlations. This feature can be employed to achieve a superconducting triplet spin-valve effect in superconductor-ferromagnet (SF) hybrid structures, for example by switching the magnetizations of the ferromagnets between parallel and antiparallel configurations in F1SF2 and SF1F2 trilayers, or in SF bilayers with both Rashba and Dresselhaus spin-orbit coupling (SOC). It was recently reported that geometric curvature can control the generation of long-ranged triplets. We use this property to show that the superconducting critical temperature of an SF hybrid nanowire can be tuned by varying the curvature of the ferromagnetic side alone, with no need of another ferromagnet or SOC. We show that the variation of the critical temperature as a function of the curvature can be exploited to obtain a robust, curvature-controlled, superconducting triplet spin-valve effect. Furthermore, we perform an analysis with the inclusion of spin-orbit coupling and explain how it modifies the spin-valve effect both quantitatively and qualitatively.

Nonlinear devices, such as transistors, enable contemporary computing technologies. We theoretically investigate nonlinear effects, bearing a high fundamental scientific and technical relevance, in magnonics with emphasis on superconductor-ferromagnet hybrids. Accounting for a finite magnon chemical potential, we theoretically demonstrate magnonic spin Joule heating, the spin analog of conventional electronic Joule heating. Besides suggesting a key contribution to magnonic heat transport in a broad range of devices, it provides insights into the thermal physics of nonconserved bosonic excitations. Considering a spin-split superconductor self-consistently, we demonstrate its interface with a ferromagnetic insulator to harbor large tunability of spin and thermal conductances. We further demonstrate hysteretic rectification I-V characteristics in this hybrid, where the hysteresis results from the superconducting state bistability.

The quasiclassical theory of superconductivity provides a methodology to study emergent phenomena in hybrid structures comprised of superconductors interfaced with other materials. A key component in this theory is the boundary condition that the Green functions describing the materials must satisfy. Recently, progress has been made toward formulating such a boundary condition for interfaces with spin-orbit coupling, the latter playing an important role for several phenomena in spintronics. Here we derive a boundary condition for spin-orbit coupled interfaces that includes gradient terms, which enables the description of spin-Hall-like effects with superconductors due to such interfaces. As an example, we show that the boundary conditions predict that a supercurrent flowing through a superconductor that is coupled to a normal metal via a spin-orbit interface can induce a nonlocal magnetization in the normal metal.