Calorimetric evidence for a Fulde-Ferrell-Larkin-Ovchinnikov superconducting state in the layered organic superconductor kappa-(BEDT-TTF)2Cu(NCS)2

Evidence for a finite-momentum Cooper pair in tricolor d-wave superconducting superlattices

Fermionic superfluidity with a nontrivial Cooper-pairing, beyond the conventional Bardeen-Cooper-Schrieffer state, is a captivating field of study in quantum many-body systems. In particular, the search for superconducting states with finite-momentum pairs has long been a challenge, but establishing its existence has long suffered from the lack of an appropriate probe to reveal its momentum. Recently, it has been proposed that the nonreciprocal electron transport is the most powerful probe for the finite-momentum pairs, because it directly couples to the supercurrents. Here we reveal such a pairing state by the non-reciprocal transport on tricolor superlattices with strong spin-orbit coupling combined with broken inversion-symmetry consisting of atomically thin d-wave superconductor CeCoIn5. We find that while the second-harmonic resistance exhibits a distinct dip anomaly at the low-temperature (T)/high-magnetic field (H) corner in the HT-plane for H applied to the antinodal direction of the d-wave gap, such an anomaly is absent for H along the nodal direction. By carefully isolating extrinsic effects due to vortex dynamics, we reveal the presence of a non-reciprocal response originating from intrinsic superconducting properties characterized by finite-momentum pairs. We attribute the high-field state to the helical superconducting state, wherein the phase of the order parameter is spontaneously spatially modulated.

Finite-momentum Cooper pairing in proximitized altermagnets

Finite-momentum Cooper pairing is an unconventional form of superconductivity that is widely believed to require finite magnetization. Altermagnetism is an emerging magnetic phase with highly anisotropic spin-splitting of specific symmetries, but zero net magnetization. Here, we study Cooper pairing in metallic altermagnets connected to conventional s-wave superconductors. Remarkably, we find that the Cooper pairs induced in the altermagnets acquire a finite center-of-mass momentum, despite the zero net magnetization in the system. This anomalous Cooper-pair momentum strongly depends on the propagation direction and exhibits unusual symmetric patterns. Furthermore, it yields several unique features: (i) highly orientation-dependent oscillations in the order parameter, (ii) controllable 0-π transitions in the Josephson supercurrent, (iii) large-oblique-angle Cooper-pair transfer trajectories in junctions parallel with the direction where spin splitting vanishes, and (iv) distinct Fraunhofer patterns in junctions oriented along different directions. Finally, we discuss the implementation of our predictions in candidate materials such as RuO2 and KRu4O8.

Magnetotransport and Fermi surface segmentation in Pauli limited superconductors

We report the first theoretical investigation of the spectroscopic, electrical and optical transport signatures ofd-wave Pauli limited superconductors, based on a non perturbative numerical approach. We demonstrate that the high magnetic field low temperature regime of these materials host a finite momentum paired superconducting phase. Multi-branched dispersion spectra with finite energy superconducting gaps, anisotropic segmentation of the Fermi surface and spatial modulations of the superconducting order characterizes this finite momentum paired phase and should be readily accessible through angle resolved photo emission spectroscopy, quasiparticle interference and differential conductance measurements. Based on the electrical and optical transport properties we capture the non Fermi liquid behavior of these systems at high temperatures, dominated by local superconducting correlations and characterized by resilient quasiparticles which survive the breakdown of the Fermi liquid description. We map out the generic thermal phase diagram of thed-wave Pauli limited superconductors and provide for the first time the accurate estimates of the thermal scales corresponding to the: (a) loss of (quasi) long range superconducting phase coherence (Tc), (b) loss of local pair correlations (Tpg), (c) breakdown of the Fermi liquid theory (Tmax) and cross-over from the non Fermi liquid to the bad metallic phase (TBR). Our thermal phase diagram mapped out on the basis of the spectroscopic and transport properties are found to be in qualitative agreement with the experimental observations on CeCoIn5andκ-BEDT, in terms of the thermodynamic phases and the phase transitions. The results presented in this paper are expected to initiate important transport and spectroscopic experiments on the Pauli limitedd-wave superconductors, providing sharp signatures of the finite momentum Cooper paired state in these materials.

Smectic pair-density-wave order in EuRbFe4As4

The pair density wave (PDW) is a superconducting state in which Cooper pairs carry centre-of-mass momentum in equilibrium, leading to the breaking of translational symmetry^1-4. Experimental evidence for such a state exists in high magnetic field^5-8 and in some materials that feature density-wave orders that explicitly break translational symmetry^9-13. However, evidence for a zero-field PDW state that exists independent of other spatially ordered states has so far been elusive. Here we show that such a state exists in the iron pnictide superconductor EuRbFe₄As₄, a material that features co-existing superconductivity (superconducting transition temperature (T_c) ≈ 37 kelvin) and magnetism (magnetic transition temperature (T_m) ≈ 15 kelvin)^14,15. Using spectroscopic imaging scanning tunnelling microscopy (SI-STM) measurements, we show that the superconducting gap at low temperature has long-range, unidirectional spatial modulations with an incommensurate period of about eight unit cells. Upon increasing the temperature above T_m, the modulated superconductor disappears, but a uniform superconducting gap survives to T_c. When an external magnetic field is applied, gap modulations disappear inside the vortex halo. The SI-STM and bulk measurements show the absence of other density-wave orders, indicating that the PDW state is a primary, zero-field superconducting state in this compound. Both four-fold rotational symmetry and translation symmetry are recovered above T_m, indicating that the PDW is a smectic order.

Orbital Fulde-Ferrell-Larkin-Ovchinnikov state in an Ising superconductor

In superconductors possessing both time and inversion symmetries, the Zeeman effect of an external magnetic field can break the time-reversal symmetry, forming a conventional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state characterized by Cooper pairings with finite momentum^1,2. In superconductors lacking (local) inversion symmetry, the Zeeman effect may still act as the underlying mechanism of FFLO states by interacting with spin-orbit coupling (SOC). Specifically, the interplay between the Zeeman effect and Rashba SOC can lead to the formation of more accessible Rashba FFLO states that cover broader regions in the phase diagram^3-5. However, when the Zeeman effect is suppressed because of spin locking in the presence of Ising-type SOC, the conventional FFLO scenarios are no longer effective. Instead, an unconventional FFLO state is formed by coupling the orbital effect of magnetic fields with SOC, providing an alternative mechanism in superconductors with broken inversion symmetries^6-8. Here we report the discovery of such an orbital FFLO state in the multilayer Ising superconductor 2H-NbSe₂. Transport measurements show that the translational and rotational symmetries are broken in the orbital FFLO state, providing the hallmark signatures of finite-momentum Cooper pairings. We establish the entire orbital FFLO phase diagram, consisting of a normal metal, a uniform Ising superconducting phase and a six-fold orbital FFLO state. This study highlights an alternative route to achieving finite-momentum superconductivity and provides a universal mechanism to preparing orbital FFLO states in similar materials with broken inversion symmetries.

Unconventional pairing in few-fermion systems at finite temperature

Attractively interacting two-component mixtures of fermionic particles confined in a one-dimensional harmonic trap are investigated. Properties of balanced and imbalanced systems are systematically explored with the exact diagonalization approach, focusing on the finite-temperature effects. Using single- and two-particle density distributions, specific non-classical pairing correlations are analyzed in terms of the noise correlations—quantity directly accessible in state-of-the-art experiments with ultra-cold atoms. It is shown that along with increasing temperature, any imbalanced system hosting Fulde–Ferrel–Larkin–Ovchinnikov pairs crossovers to a standard Bardeen-Cooper-Schrieffer one characterized by zero net momentum of resulting pairs. By performing calculations for systems with different imbalances, the approximate boundary between the two phases on a phase diagram is determined.

Emergent anisotropy in the Fulde-Ferrell-Larkin-Ovchinnikov state

Exotic superconductivity is formed by unconventional electron pairing and exhibits various unique properties that cannot be explained by the basic theory. The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state is known as an exotic superconducting state in that the electron pairs have a finite center-of-mass momentum leading to a spatially modulated pattern of superconductivity. The spatial modulation endows the FFLO state with emergent anisotropy. However, the anisotropy has never been experimentally verified despite numerous efforts over the years. Here, we report detection of anisotropic acoustic responses depending on the sound propagation direction appearing above the Pauli limit. This anisotropy reveals that the two-dimensional FFLO state has a center-of-mass momentum parallel to the nesting vector on the Fermi surface. The present findings will facilitate our understanding of not only superconductivity in solids but also exotic pairings of various particles.

The famous Fulde–Ferrell–Larkin– Ovchinnikov (FFLO) state is a spatially-modulated superconducting state with a predicted spatial anisotropy, but this anisotropy has never been experimentally verified. Here, the authors present ultrasound evidence for anisotropy of the sound velocity in the FFLO state of a 2D organic superconductor.

Exotic Superfluid Phases in Spin-Polarized Fermi Gases in Optical Lattices

Leveraging cutting-edge numerical methodologies, we study the ground state of the two-dimensional spin-polarized Fermi gas in an optical lattice. We focus on systems at high density and small spin polarization, corresponding to the parameter regime believed to be most favorable to the formation of the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. Our systematic study of large lattice sizes, hosting nearly 500 atoms, provides strong evidence of the stability of the FFLO state in this regime, as well as a high-accuracy characterization of its properties. Our results for the density correlation function reveal the existence of density order in the system, suggesting the possibility of an intricate coexistence of long-range orders in the ground state. The ground-state properties are seen to differ significantly from the standard mean-field description, providing a compelling avenue for future theoretical and experimental explorations of the interplay between spin imbalance, strong interactions, and superfluidity in an exotic phase of matter.

Superconducting spin smecticity evidencing the Fulde-Ferrell-Larkin-Ovchinnikov state in Sr2RuO4

Translational symmetry breaking is antagonistic to static fluidity but can be realized in superconductors, which host a quantum-mechanical coherent fluid formed by electron pairs. A peculiar example of such a state is the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, induced by a time-reversal symmetry-breaking magnetic field applied to spin-singlet superconductors. This state is intrinsically accompanied by the superconducting spin smecticity, spin density-modulated fluidity with spontaneous translational-symmetry breaking. Detection of such spin smecticity provides unambiguous evidence for the FFLO state, but its observation has been challenging. Here, we report the characteristic "double-horn" nuclear magnetic resonance spectrum in the layered superconductor Sr₂RuO₄ near its upper critical field, indicating the spatial sinusoidal modulation of spin density that is consistent with superconducting spin smecticity. Our work reveals that Sr₂RuO₄ provides a versatile platform for studying FFLO physics.

Supercurrent diode effect and finite-momentum superconductors

Our work shows a fascinating application of finite-momentum superconductivity, the supercurrent diode effect, which is being reported in a growing number of experiments. We show that, under external magnetic field, Cooper pairs can acquire finite momentum so that critical currents in the direction parallel and antiparallel to the Cooper pair momentum become unequal.

When both inversion and time-reversal symmetries are broken, the critical current of a superconductor can be nonreciprocal. In this work, we show that, in certain classes of two-dimensional superconductors with antisymmetric spin–orbit coupling, Cooper pairs acquire a finite momentum upon the application of an in-plane magnetic field, and, as a result, critical currents in the direction parallel and antiparallel to the Cooper pair momentum become unequal. This supercurrent diode effect is also manifested in the polarity dependence of in-plane critical fields induced by a supercurrent. These nonreciprocal effects may be found in polar SrTiO₃ film, few-layer MoTe₂ in the T_d phase, and twisted bilayer graphene in which the valley degree of freedom plays a role analogous to spin.

The FFLO State in the Dimer Mott Organic Superconductor κ-(BEDT-TTF)2Cu[N(CN)2]Br

The superconducting phase diagram for a quasi-two-dimensional organic superconductor, κ-(BEDT-TTF)2Cu[N(CN)2]Br, was studied using pulsed magnetic field penetration depth measurements under rotating magnetic fields. At low temperatures, Hc2 was abruptly suppressed even by small tilts of the applied fields owing to the orbital pair-breaking effect. In magnetic fields parallel to the conducting plane, the temperature dependence of the upper critical field Hc2 exhibited an upturn and exceeded the Pauli limit field HP in the lower temperature region. Further analyses with the second derivative of the penetration depth showed an anomaly at 31–32 T, which roughly corresponded to HP. The origin of the anomaly should not be related to the orbital effect, but the paramagnetic effect, which is almost isotropic in organic salts, because it barely depends on the field angle. Based on these results, the observed anomaly is most likely due to the transition between the Bardeen-Cooper-Schrieffer (BCS) and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states. Additionally, we discuss the phase diagram and physical parameters of the transition by comparing them with other FFLO candidates.

Signatures of bosonic Landau levels in a finite-momentum superconductor

Charged particles subjected to magnetic fields form Landau levels (LLs). Originally studied in the context of electrons in metals¹, fermionic LLs continue to attract interest as hosts of exotic electronic phenomena^2,3. Bosonic LLs are also expected to realize novel quantum phenomena^4,5, but, apart from recent advances in synthetic systems^6,7, they remain relatively unexplored. Cooper pairs in superconductors-composite bosons formed by electrons-represent a potential condensed-matter platform for bosonic LLs. Under certain conditions, an applied magnetic field is expected to stabilize an unusual superconductor with finite-momentum Cooper pairs^8,9 and exert control over bosonic LLs^10-13. Here we report thermodynamic signatures, observed by torque magnetometry, of bosonic LL transitions in the layered superconductor Ba₆Nb₁₁S₂₈. By applying an in-plane magnetic field, we observe an abrupt, partial suppression of diamagnetism below the upper critical magnetic field, which is suggestive of an emergent phase within the superconducting state. With increasing out-of-plane magnetic field, we observe a series of sharp modulations in the upper critical magnetic field that are indicative of distinct vortex states and with a structure that agrees with predictions for Cooper pair LL transitions in a finite-momentum superconductor^10-14. By applying Onsager's quantization rule¹⁵, we extract the momentum. Furthermore, study of the fermionic LLs shows evidence for a non-zero Berry phase. This suggests opportunities to study bosonic LLs, topological superconductivity, and their interplay via transport¹⁶, scattering¹⁷, scanning probe¹⁸ and exfoliation techniques¹⁹.

Evidence for the Fulde-Ferrell-Larkin-Ovchinnikov state in bulk NbS2

We present measurements of the magnetic torque, specific heat and thermal expansion of the bulk transition metal dichalcogenide (TMD) superconductor NbS₂ in high magnetic fields, with its layer structure aligned strictly parallel to the field using a piezo rotary positioner. The upper critical field of superconducting TMDs in the 2D form is known to be dramatically enhanced by a special form of Ising spin orbit coupling. This Ising superconductivity is very robust to the Pauli paramagnetic effect and can therefore exist beyond the Pauli limit for superconductivity. We find that superconductivity beyond the Pauli limit still exists in bulk single crystals of NbS₂ for a precisely parallel field alignment. However, the comparison of our upper critical field transition line with numerical simulations rather points to the development of a Fulde-Ferrell-Larkin-Ovchinnikov state above the Pauli limit as a cause. This is also consistent with the observation of a magnetic field driven phase transition in the thermodynamic quantities within the superconducting state near the Pauli limit.

Superconductivity is often destroyed under magnetic field larger than a critical value called Pauli limit. Here, the authors report superconductivity beyond the Pauli limit in bulk single crystals of NbS₂, suggesting the development of a Fulde-Ferrell-Larkin-Ovchinnikov state.

The Zeeman, spin-orbit, and quantum spin Hall interactions in anisotropic and low-dimensional conductors

To construct a microscopic theory of electrons and holes in anisotropic conductors that self-consistently treats their band effective mass anisotropy with their interactions with applied electric and magnetic fields, the Dirac equation is extended for an electron or hole in an orthorhombically-anisotropic conduction band. Its covariance is established both by a modified version of the Klemm-Clem transformations to a space in which it is isotropic, and also in its fully anisotropic form by making the most general proper and improper Lorentz transformations, proving its validity in both the relativistic and non-relativistic limits. The appropriate Foldy-Wouthuysen transformations are extended to expand about the non-relativistic Hamiltonian limit to fourth order in the inverse of the particle's Einstein rest energy. The results have important consequences for magnetic measurements of many classes of clean anisotropic semiconductors, metals, and superconductors. In all of these cases, the Zeeman interaction is found to depend strongly upon the effective mass anisotropy. When an electron or hole is traveling in an atomically thin one-dimensional conduction band, its Zeeman, spin-orbit, and quantum spin Hall interactions are vanishingly small. Accurate expressions for the Zeeman, spin-orbit and quantum spin Hall interactions for two-dimensional conductors are provided.

Strong In-Plane Magnetic Field-Induced Reemergent Superconductivity in the van der Waals Heterointerface of NbSe2 and CrCl3

A magnetic field is generally considered to be incompatible with superconductivity as it tends to spin-polarize electrons and breaks apart the opposite-spin singlet superconducting Cooper pairs. Here, an experimental phenomenon is observed that an intriguing reemergent superconductivity evolves from a conventional superconductivity undergoing a hump-like intermediate phase with a finite electric resistance in the van der Waals heterointerface of layered NbSe₂ and CrCl₃ flakes. This phenomenon merely occurred when the applied magnetic field is parallel to the sample plane and perpendicular to the electric current direction as compared to the reference sample of a NbSe₂ thin flake. The strong anisotropy of the reemergent superconducting phase is pointed to the nature of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state driven by the strong interfacial spin-orbit coupling between NbSe₂ and CrCl₃ layers. The theoretical picture of FFLO state nodes induced by Josephson vortices collectively pinning is presented for well understanding the experimental observation of the reemergent superconductivity. This finding sheds light on an opportunity to search for the exotic FFLO state in the van der Waals heterostructures with strong interfacial spin-orbit coupling.

Pauli limited d-wave superconductors: quantum breached pair phase and thermal transitions

We report a quantum phase transition in Pauli limited d-wave superconductors and give the mean field estimates of the associated quantum critical point. For a population imbalanced d-wave superconductor a stable ground state phase viz quantum breached pair phase has been identified which comprises of spatial coexistence of gapless superconductivity and nonzero magnetization. Based on the thermodynamic and quasiparticle indicators we for the first time analyze this phase, discuss the thermal behavior of Pauli limited d-wave superconductor, give accurate estimates of the thermal scales associated with such systems and map out the pseudogap regime. Our work shows that while the Pauli limited superconductors are known to exhibit exotic modulated superconducting phase at large imbalance of fermion populations; in the regime of weak imbalance an intriguing phase of competing orders is realized. We have established that rather than the superconducting pairing field, it is the average magnetization of the system that quantifies this quantum phase transition. Given that the existing Pauli limited superconductors possess unconventional pairing state symmetry of the superconducting order, our work promises to open up new avenues in the experimental research of these materials. We have also demonstrated an alternate scenario wherein the quantum breached pair phase is a natural outcome for a d-wave superconductor with unequal effective masses of the fermion species.

Evidence for an Fulde-Ferrell-Larkin-Ovchinnikov State with Segmented Vortices in the BCS-BEC-Crossover Superconductor FeSe

We present resistivity and thermal-conductivity measurements of superconducting FeSe in intense magnetic fields up to 35 T applied parallel to the ab plane. At low temperatures, the upper critical field μ_{0}H_{c2}^{ab} shows an anomalous upturn, while thermal conductivity exhibits a discontinuous jump at μ_{0}H^{*}≈24  T well below μ_{0}H_{c2}^{ab}, indicating a first-order phase transition in the superconducting state. This demonstrates the emergence of a distinct field-induced superconducting phase. Moreover, the broad resistive transition at high temperatures abruptly becomes sharp upon entering the high-field phase, indicating a dramatic change of the magnetic-flux properties. We attribute the high-field phase to the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state, where the formation of planar nodes gives rise to a segmentation of the flux-line lattice. We point out that strongly orbital-dependent pairing as well as spin-orbit interactions, the multiband nature, and the extremely small Fermi energy are important for the formation of the FFLO state in FeSe.

Pairing Mechanism in Hund's Metal Superconductors and the Universality of the Superconducting Gap to Critical Temperature Ratio

We analyze a simple model containing the physical ingredients of a Hund's metal, the local spin fluctuations with power-law correlators, (Ω_{0}/|Ω|)^{γ}, with γ greater than one, interacting with electronic quasiparticles. While the critical temperature and the gap change significantly with varying parameters, the 2Δ_{max}/k_{B}T_{c} remains close to twice the BCS value in agreement with experimental observations in the iron-based superconductors (FeSC).

Evidence for the Presence of the Fulde-Ferrell-Larkin-Ovchinnikov State in CeCu_{2}Si_{2} Revealed Using ^{63}Cu NMR

Nuclear magnetic resonance measurements were performed on CeCu_{2}Si_{2} in the presence of a magnetic field close to the upper critical field μ_{0}H_{c2} in order to investigate its superconducting (SC) properties near pair-breaking fields. In lower fields, the Knight shift and nuclear spin-lattice relaxation rate divided by temperature 1/T_{1}T abruptly decreased below the SC transition temperature T_{c}(H), a phenomenon understood within the framework of conventional spin-singlet superconductivity. In contrast, 1/T_{1}T was enhanced just below T_{c}(H) and exhibited a broad maximum when magnetic fields close to μ_{0}H_{c2}(0) were applied parallel or perpendicular to the c axis; although the Knight shift decreased just below T_{c}(H). This enhancement of 1/T_{1}T, which was recently observed in the organic superconductor κ-(BEDT-TTF)_{2}Cu(NCS)_{2}, suggests the presence of high-density Andreev bound states in the inhomogeneous SC region, a hallmark of the Fulde-Ferrell-Larkin-Ovchinnikov phase.

Finite momentum Cooper pairing in three-dimensional topological insulator Josephson junctions

Unconventional superconductivity arising from the interplay between strong spin–orbit coupling and magnetism is an intensive area of research. One form of unconventional superconductivity arises when Cooper pairs subjected to a magnetic exchange coupling acquire a finite momentum. Here, we report on a signature of finite momentum Cooper pairing in the three-dimensional topological insulator Bi₂Se₃. We apply in-plane and out-of-plane magnetic fields to proximity-coupled Bi₂Se₃ and find that the in-plane field creates a spatially oscillating superconducting order parameter in the junction as evidenced by the emergence of an anomalous Fraunhofer pattern. We describe how the anomalous Fraunhofer patterns evolve for different device parameters, and we use this to understand the microscopic origin of the oscillating order parameter. The agreement between the experimental data and simulations shows that the finite momentum pairing originates from the coexistence of the Zeeman effect and Aharonov–Bohm flux.

Unconventional superconductivity may emerge from the interplay between strong spin–orbit coupling and magnetism. Here, Chen et al. report an anomalous Fraunhofer pattern in three-dimensional topological insulator Bi₂Se₃ and attribute it as a signature of finite momentum Cooper pairing.

Temperature Controlled Fulde-Ferrell-Larkin-Ovchinnikov Instability in Superconductor-Ferromagnet Hybrids

We show that a wide class of layered superconductor-ferromagnet (S/F) hybrids demonstrates the emergence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase well below the superconducting transition temperature. By decreasing the temperature, one can switch the system from uniform to the FFLO state which is accompanied by the damping of the diamagnetic Meissner response down to zero and also by the sign change in the curvature of the current-velocity dependence. Our estimates show that an additional layer of the normal metal (N) covering the ferromagnet substantially softens the conditions required for the predicted FFLO instability, and for existing S/F/N systems, the temperature of the transition into the FFLO phase can reach several kelvins.

Inhomogeneous Superconductivity in Organic and Related Superconductors

Evidence of inhomogeneous superconductivity, in this case superconductivity with a spatially modulated superconducting order parameter, has now been found in many materials and by many measurement methods. Although the evidence is strong, it is circumstantial in the organic superconductors, scant in the pnictides, and complex in the heavy Fermions. However, it is clear some form of exotic superconductivity exists at high fields and low temperatures in many electronically anisotropic superconductors. The evidence is reviewed in this article, and examples of similar measurements are compared across different families of superconductors. An effort is made to find a consistent way to measure the superconducting energy gap across all materials, and use this value to predict the Clogston–Chandrasakhar paramagnetic limit Hp. Methods for predicting the existence of inhomogeneous superconductivity are shown to work for the organic superconductors, and then used to suggest new materials to study.

An unusual continuous paramagnetic-limited superconducting phase transition in 2D NbSe 2

Time reversal and spatial inversion are two key symmetries for conventional Bardeen-Cooper-Schrieffer (BCS) superconductivity ¹ . Breaking inversion symmetry can lead to mixed-parity Cooper pairing and unconventional superconducting properties^1-5. Two-dimensional (2D) NbSe₂ has emerged as a new non-centrosymmetric superconductor with the unique out-of-plane or Ising spin-orbit coupling (SOC)^6-9. Here we report the observation of an unusual continuous paramagnetic-limited superconductor-normal metal transition in 2D NbSe₂. Using tunelling spectroscopy under high in-plane magnetic fields, we observe a continuous closing of the superconducting gap at the upper critical field at low temperatures, in stark contrast to the abrupt first-order transition observed in BCS thin-film superconductors^10-12. The paramagnetic-limited continuous transition arises from a large spin susceptibility of the superconducting phase due to the Ising SOC. The result is further supported by self-consistent mean-field calculations based on the ab initio band structure of 2D NbSe₂. Our findings establish 2D NbSe₂ as a promising platform to explore novel spin-dependent superconducting phenomena and device concepts ¹ , such as equal-spin Andreev reflection ¹³ and topological superconductivity^14-16.

The influence of the dimensionality of the system on the realization of unconventional Fulde-Ferrell-Larkin-Ovchinnikov pairing in ultra-cold Fermi gases

The recent development of experimental techniques in ultracold atomic Fermi gases is extremely helpful in the progress of the realization of the unconventional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase in quasi-one dimensional systems (Liao et al 2010 Nature 467 567). Due to a Fermi surface nesting, which is enhanced in 1D, the low-dimensional systems are particularly good candidates to find the FFLO phase stable. We investigate the influence of a dimensional crossover (from one dimension (1D) to two dimensions (2D) or three dimensions (3D)) on the stability of the FFLO state in the spin-imbalanced attractive Hubbard model.

Thermodynamic Evidence for the Fulde-Ferrell-Larkin-Ovchinnikov State in the KFe_{2}As_{2} Superconductor

We investigate the magnetic phase diagram near the upper critical field of KFe_{2}As_{2} by magnetic torque and specific heat experiments using a high-resolution piezorotary positioner to precisely control the parallel alignment of the magnetic field with respect to the FeAs layers. We observe a clear double transition when the field is strictly aligned in the plane and a characteristic upturn of the upper critical field line, which goes far beyond the Pauli limit at 4.8 T. This provides firm evidence that a Fulde-Ferrell-Larkin-Ovchinnikov state exists in this iron-based KFe_{2}As_{2} superconductor.

Investigation of the superconducting gap structure in κ-(BEDT-TTF)2Cu(NCS)2 and κ-(BEDT-TTF)2Cu[N(CN)2]Br by means of thermal-conductivity measurements

We report temperature-dependent thermal-conductivity, κ, measurements on the layered quasi-two-dimensional organic superconductors κ-(BEDT-TTF)₂Cu(NCS)₂ and κ-(BEDT-TTF)₂Cu[N(CN)₂]Br down to 160 mK. The results for κ-(BEDT-TTF)₂Cu(NCS)₂ may be consistent with a nodal superconducting (SC) gap structure as indicated by a non-negligible remnant linear contribution when [Formula: see text] is extrapolated to [Formula: see text]. For κ-(BEDT-TTF)₂Cu[N(CN)₂]Br, contrary to expectations, higher κ values are observed in the superconducting regime as compared to the normal, high-field state evidencing a dominant phonon contribution to κ in the superconducting state. The strong increase of κ in the normal state below T _c for both samples indicates strong electron-phonon scattering. Our results highlight the need for thermal-conductivity measurements performed down to significantly lower temperatures to determine the symmetry of the SC gap.

Calorimetric Measurements of Magnetic-Field-Induced Inhomogeneous Superconductivity Above the Paramagnetic Limit

We report the first magnetocaloric and calorimetric observations of a magnetic-field-induced phase transition within a superconducting state to the long-sought exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state, first predicted over 50 years ago. Through the combination of bulk thermodynamic calorimetric and magnetocaloric measurements in the organic superconductor κ-(BEDT-TTF)_{2}Cu(NCS)_{2} as a function of temperature, magnetic field strength, and magnetic field orientation, we establish for the first time that this field-induced first-order phase transition at the paramagnetic limit H_{p} is a transition to a higher-entropy superconducting phase, uniquely characteristic of the FFLO state. We also establish that this high-field superconducting state displays the bulk paramagnetic ordering of spin domains required of the FFLO state. These results rule out the alternate possibility of spin-density wave ordering in the high-field superconducting phase. The phase diagram determined from our measurements-including the observation of a phase transition into the FFLO phase at H_{p}-is in good agreement with recent NMR results and our own earlier tunnel-diode magnetic penetration depth experiments but is in disagreement with the only previous calorimetric report.

Exotic magnetic states in Pauli-limited superconductors

Magnetism and superconductivity compete or interact in complex and intricate ways. Here we review the special case where novel magnetic phenomena appear due to superconductivity, but do not exist without it. Such states have recently been identified in unconventional superconductors. They are different from the mere coexistence of magnetic order and superconductivity in conventional superconductors, or from competing magnetic and superconducting phases in many materials. We describe the recent progress in the study of such exotic magnetic phases, and articulate the many open questions in this field.

Frustrated magnets in high magnetic fields-selected examples

An indispensable parameter to study strongly correlated electron systems is the magnetic field. Application of high magnetic fields allows the investigation, modification and control of different states of matter. Specifically for magnetic materials experimental tools applied in such fields are essential for understanding their fundamental properties. Here, we focus on selected high-field studies of frustrated magnetic materials that have been shown to host a broad range of fascinating new and exotic phases. We will give brief insights into the influence of geometrical frustration on the critical behavior of triangular-lattice antiferromagnets, the accurate determination of exchange constants in the high-field saturated state by use of electron spin resonance measurements, and the coupling of magnetic degrees of freedom to the lattice evidenced by ultrasound experiments. The latter technique as well allowed new, partially metastable phases in strong magnetic fields to be revealed.

Microscopic Study of the Fulde-Ferrell-Larkin-Ovchinnikov State in an All-Organic Superconductor

Quasi-two-dimensional superconductors with a sufficiently weak interlayer coupling allow magnetic flux to penetrate in the form of Josephson vortices for in-plane applied magnetic fields. A consequence is the dominance of the Zeeman interaction over orbital effects. In the clean limit, the normal state is favored over superconductivity for fields greater than the paramagnetic limiting field, unless an intermediate, inhomogeneous state is stabilized. Presented here are nuclear magnetic resonance (NMR) studies of the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state for β''-(ET)2SF5CH2CF2SO3. The uniform superconductivity-FFLO transition is identified at an applied field value of 9.3(0.1) T at low temperature (T=130  mK), and evidence for a possible second transition between inhomogeneous states at ∼11  T is presented. The spin polarization distribution inferred from the NMR absorption spectrum compares favorably to a single-Q modulation of the superconducting order parameter.

Determination of the hyperfine coupling tensor in organic conductors κ-(BEDT-TTF)₂X (X=Cu[N(CN)₂]Br, Cu(NCS)₂) on central ¹³C sites

Although the organic superconductors κ-(BEDT-TTF)2X (X=Cu[N(CN)2]Br and Cu(NCS)2) have been studied by NMR spectroscopy, hyperfine coupling tensors are required to quantify NMR spectra. Angle dependences of NMR spectra were measured to determine hyperfine coupling tensors applicable to further NMR assessments of attractive physical phenomena on κ-salts. The tensors of κ-(BEDT-TTF)2X and β(')-(BEDT-TTF)2ICl2 salts were compared to determine the hyperfine coupling mechanism in organic metals, with the results indicating that off-site dimer contribution should be considered. We also report the electron correlation of these salts and further application of the tensors.

Multiple phase transitions in Pauli-limited iron-based superconductors

Specific heat measurements have been successfully used to probe unconventional superconducting phases in one-band heavy-fermion and organic superconductors. We extend the method to study successive phase transitions in multi-band materials such as iron-based superconductors. The signatures are multiple peaks in the specific heat, at low temperatures and high magnetic field, which can lead to the experimental verification of unconventional superconducting states with non-zero total momentum.

Spin-polarized supercurrents for spintronics: a review of current progress

During the past 15 years a new field has emerged, which combines superconductivity and spintronics, with the goal to pave a way for new types of devices for applications combining the virtues of both by offering the possibility of long-range spin-polarized supercurrents. Such supercurrents constitute a fruitful basis for the study of fundamental physics as they combine macroscopic quantum coherence with microscopic exchange interactions, spin selectivity, and spin transport. This report follows recent developments in the controlled creation of long-range equal-spin triplet supercurrents in ferromagnets and its contribution to spintronics. The mutual proximity-induced modification of order in superconductor-ferromagnet hybrid structures introduces in a natural way such evasive phenomena as triplet superconductivity, odd-frequency pairing, Fulde-Ferrell-Larkin-Ovchinnikov pairing, long-range equal-spin supercurrents, [Formula: see text]-Josephson junctions, as well as long-range magnetic proximity effects. All these effects were rather exotic before 2000, when improvements in nanofabrication and materials control allowed for a new quality of hybrid structures. Guided by pioneering theoretical studies, experimental progress evolved rapidly, and since 2010 triplet supercurrents are routinely produced and observed. We have entered a new stage of studying new phases of matter previously out of our reach, and of merging the hitherto disparate fields of superconductivity and spintronics to a new research direction: super-spintronics.

FFLO superfluids in 2D spin-orbit coupled Fermi gases

We show that the combination of spin-orbit coupling and in-plane Zeeman field in a two-dimensional degenerate Fermi gas can lead to a larger parameter region for Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases than that using spin-imbalanced Fermi gases. The resulting FFLO superfluids are also more stable due to the enhanced energy difference between FFLO and conventional Bardeen-Cooper-Schrieffer (BCS) excited states. We clarify the crucial role of the symmetry of Fermi surface on the formation of finite momentum pairing. The phase diagram for FFLO superfluids is obtained in the BCS-BEC crossover region and possible experimental observations of FFLO phases are discussed.

Pauli-limited multiband superconductivity in KFe2As2

The upper critical field H(c2)(T) of the multiband superconductor KFe2As2 has been studied via low-temperature thermal expansion and magnetostriction measurements. We present compelling evidence for Pauli-limiting effects dominating H(c2)(T) for H || a, as revealed by a crossover from second- to first-order phase transitions to the superconducting state in the magnetostriction measurements down to 50 mK. Corresponding features were absent for H || c. To our knowledge, this crossover constitutes the first confirmation of Pauli limiting of the H(c2)(T) of a multiband superconductor. The results are supported by modeling Pauli limits for single-band and multiband cases.

The Fulde-Ferrell-Larkin-Ovchinnikov state in layered d-wave superconductors: in-plane anisotropy and resonance effects in the angular dependence of the upper critical field

We study the anisotropy of the in-plane upper critical magnetic field coupled to the orbital motion and the spins of electrons in a layered d(x2-y2) organic superconductor in the spatially modulated Fulde-Ferrell-Larkin-Ovchinnikov phase. We show that the interplay between the nodal structure of the order parameter and its spatial modulation results in the very peculiar angular dependence of the onset of superconductivity in the high-field regime. The principal axis of the field-direction dependence of the onset of superconductivity is tilted by π/4 in the temperature range 0.056 < or approximately equal T < 0.56. In some cases the resonance between the modulation wavevector and the vector potential of a parallel magnetic field may lead to anomalous cusps in the temperature and in-plane angular dependences of the onset of superconductivity. The obtained results support the interpretation of the recent experiments as evidence of the FFLO state.

First-order superconducting transition of Sr2RuO4

By means of the magnetocaloric effect, we examine the nature of the superconducting-normal (S-N) transition of Sr(2)RuO(4), a most promising candidate for a spin-triplet superconductor. We provide thermodynamic evidence that the S-N transition of this oxide is of first order below approximately 0.8 K and only for magnetic field directions very close to the conducting plane, in clear contrast to the ordinary type-II superconductors exhibiting second-order S-N transitions. The entropy release across the transition at 0.2 K is 10% of the normal-state entropy. Our result urges an introduction of a new mechanism to break superconductivity by magnetic field.

Quasiparticle entropy in the high-field superconducting phase of CeCoIn(5)

The heavy-fermion superconductor CeCoIn(5) displays an additional transition within its superconducting (SC) state, whose nature is characterized by high-precision studies of the isothermal field dependence of the entropy, derived from combined specific heat and magnetocaloric effect measurements at temperatures T≥100  mK and fields H≤12  T aligned along different directions. For any of these conditions, we do not observe an additional entropy contribution upon tuning at constant temperature by magnetic field from the homogeneous SC into the presumed Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) SC state. By contrast, for H∥[100] a reduction of entropy was found that quantitatively agrees with the expectation for spin-density-wave order without FFLO superconductivity. Our data exclude the formation of a FFLO state in CeCoIn(5) for out-of-plane field directions, where no spin-density-wave order exists.

Angle-dependent evolution of the Fulde-Ferrell-Larkin-Ovchinnikov state in an organic superconductor

We report magnetic-field and angular-dependent high-resolution specific-heat measurements of the organic superconductor β''-(BEDT-TTF)2SF5CH2CF2SO3. When the magnetic field is aligned precisely within the conducting BEDT-TTF layer, at low temperatures a clear upturn of the upper critical field beyond the Pauli limit of 9.73 T is observed, hinting at the emergence of a Fulde-Ferrell-Larkin-Ovchinnikov state. This upturn disappears when the field is oriented out of plane by more than ∼0.5  deg. For smaller out-of-plane angles, the specific-heat anomaly at T(c) sharpens and a second peaky phase transition appears within the superconducting state.

In-plane magnetic field anisotropy of the Fulde-Ferrell-Larkin-Ovchinnikov state in layered superconductors

There is strong experimental evidence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state formation in layered organic superconductors in a parallel magnetic field. We study theoretically the interplay between the orbital effect and the FFLO modulation in this case and demonstrate that the in-plane critical field anisotropy drastically changes at the transition to the FFLO state. The very peculiar angular dependence of the superconducting onset temperature which is predicted may serve for unambiguous identification of the FFLO modulation. The obtained results permit us to suggest the modulated phase stabilization as the origin of the magnetic-field angle dependence of the onset of superconductivity experimentally observed in (TMTSF)2ClO4 organic conductors.

Vortex lattice studies in CeCoIn5 with H is orthogonal to c

We present small angle neutron scattering studies of the vortex lattice (VL) in CeCoIn5 with magnetic fields applied parallel (H) to the antinodal [100] and nodal [110] directions. For H is parallel to [100], a single VL orientation is observed, while a 90° reorientation transition is found for H is parallel to [110]. For both field orientations and VL configurations we find a distorted hexagonal VL with an anisotropy, Γ=2.0±0.05. The VL form factor shows strong Pauli paramagnetic effects similar to what have previously been reported for H is parallel to [001]. At high fields, above which the upper critical field (H(c2)) becomes a first-order transition, an increased disordering of the VL is observed.

Superconductivity in 4-Angstrom carbon nanotubes--a short review

We give an up-to-date review of the superconducting phenomena in 4-Angstrom carbon nanotubes embedded in aligned linear pores of the AlPO(4)-5 (AFI) zeolite, first discovered in 2001 as a fluctuation Meissner effect. With the introduction of a new approach to sample synthesis around 2007, new data confirming the superconductivity have been obtained. These comprise electrical, specific heat, and magnetic measurements which together yield a consistent yet complex physical picture of the superconducting state, largely owing to the one-dimensional (1D) nature of the 4-Angstrom carbon nanotubes. For the electrical transport characteristics, two types of superconducting resistive behaviors were reproducibly observed in different samples. The first type is the quasi 1D fluctuation superconductivity that exhibits a smooth resistance drop with decreasing temperature, initiating at 15 K. At low temperatures the differential resistance also shows a smooth increase with increasing bias current (voltage). Both are unaffected by an applied magnetic field up to 11 Tesla. These manifestations are shown to be consistent with those of a quasi 1D superconductor with thermally activated phase slips as predicted by the Langer-Ambegaokar-McCumber-Halperin (LAMH) theory. The second type is the quasi 1D to 3D superconducting crossover transition, which was observed to initiate at 15 K with a slow resistance decrease switching to a sharp order of magnitude drop at ∼7.5 K. The latter exhibits anisotropic magnetic field dependence and is attributed to a Berezinskii-Kosterlitz-Thouless (BKT)-like transition that establishes quasi-long-range order in the plane transverse to the c-axis of the aligned nanotubes, thereby mediating a 1D to 3D crossover. The electrical data are complemented by magnetic and thermal specific heat bulk measurements. By using both the SQUID VSM and the magnetic torque technique, the onset of diamagnetism was observed to occur at ∼15 K, with a rapid increase of the diamagnetic moment below ∼7 K. The zero-field-cooled and field-cooled branches deviated from each other below 7 K, indicating the establishment of a 3D Meissner state with macroscopic phase coherence. The superconductivity is further supported by the specific heat measurements, which show an anomaly with onset at 15 K and a peak at 11-12 K. In the 3D superconducting state, the nanotube arrays constitute a type-II anisotropic superconductor with H(c1)≈ 60 to 150 Oe, coherence length ξ≈ 5 to 15 nm, London penetration length λ≈ 1.5 µm, and Ginzburg-Landau κ≈ 100. We give a physical interpretation to the observed phenomena and note the challenges and prospects ahead.