Nodal structure of quasi-two-dimensional superconductors probed by a magnetic field

Generalized Anderson's theorem for superconductors derived from topological insulators

A well-known result in unconventional superconductivity is the fragility of nodal superconductors against nonmagnetic impurities. Despite this common wisdom, Bi₂Se₃-based topological superconductors have recently displayed unusual robustness against disorder. Here, we provide a theoretical framework that naturally explains what protects Cooper pairs from strong scattering in complex superconductors. Our analysis is based on the concept of superconducting fitness and generalizes the famous Anderson's theorem into superconductors having multiple internal degrees of freedom with simple assumptions such as the Born approximation. For concreteness, we report on the extreme example of the Cu _x (PbSe)₅(BiSe₃)₆ superconductor. Thermal conductivity measurements down to 50 mK not only give unambiguous evidence for the existence of nodes but also reveal that the energy scale corresponding to the scattering rate is orders of magnitude larger than the superconducting energy gap. This provides the most spectacular case of the generalized Anderson's theorem protecting a nodal superconductor.

Resonances in the Field-Angle-Resolved Thermal Conductivity of CeCoIn_{5}

The thermal conductivity measurement in a rotating magnetic field is a powerful probe of the structure of the superconducting energy gap. We present high-precision measurements of the low-temperature thermal conductivity in the unconventional heavy-fermion superconductor CeCoIn_{5}, with the heat current J along the nodal [110] direction of its d_{x^{2}-y^{2}} order parameter and the magnetic field up to 7 T rotating in the ab plane. In contrast to the smooth oscillations found previously for J∥[100], we observe a sharp resonancelike peak in the thermal conductivity when the magnetic field is also in the [110] direction, parallel to the heat current. We explain this peak qualitatively via a model of the heat transport in a d-wave superconductor. In addition, we observe two smaller but also very sharp peaks in the thermal conductivity for the field directions at angles Θ≈±33° with respect to J. The origin of the observed resonances at Θ≈±33° at present defies theoretical explanation. The challenge of uncovering their source will dictate exploring theoretically more complex models, which might include, e.g., fine details of the Fermi surface, Andreev bound vortex core states, a secondary superconducting order parameter, and the existence of gaps in spin and charge excitations.

Angle-resolved heat capacity of heavy fermion superconductors

Owing to a strong Coulomb repulsion, heavy electron superconductors mostly have anisotropic gap functions which have nodes for certain directions in the momentum space. Since the nodal structure is closely related to the pairing mechanism, its experimental determination is of primary importance. This article discusses the experimental methods of the gap determination by bulk heat capacity measurements in a rotating magnetic field. The basic idea is based on the fact that the quasiparticle density of states in the vortex state of nodal superconductors is field and direction dependent. We present our recent experimental results of the field-orientation dependence of the heat capacity in heavy fermion superconductors CeTIn5 (T  =  Co, Ir), UPt3, CeCu2Si2, and UBe13 and discuss their gap structures.

Nodal to nodeless superconducting energy-gap structure change concomitant with fermi-surface reconstruction in the heavy-fermion compound CeCoIn(5)

The London penetration depth λ(T) was measured in single crystals of Ce_{1-x}R_{x}CoIn_{5}, R=La, Nd, and Yb down to T_{min}≈50  mK (T_{c}/T_{min}∼50) using a tunnel-diode resonator. In the cleanest samples Δλ(T) is best described by the power law Δλ(T)∝T^{n}, with n∼1, consistent with the existence of line nodes in the superconducting gap. Substitutions of Ce with La, Nd, and Yb lead to similar monotonic suppressions of T_{c}; however, the effects on Δλ(T) differ. While La and Nd substitution leads to an increase in the exponent n and saturation at n∼2, as expected for a dirty nodal superconductor, Yb substitution leads to n>3, suggesting a change from nodal to nodeless superconductivity. This superconducting gap structure change happens in the same doping range where changes of the Fermi-surface topology were reported, implying that the nodal structure and Fermi-surface topology are closely linked.

Role of the Fermi-surface anisotropy in angle-dependent magnetic-field oscillations for identifying the energy-gap anisotropy of A(y)Fe(2)Se(2) superconductors

We present a numerical study of the field-angle resolved oscillations of the thermal conductivity and specific heat under a rotated magnetic field in the A(y)Fe(2-x)Se(2) [A = K, Rb, Cs, (Tl, K)] superconductors, using realistic two-band Fermi surface parametrization. Our key finding is that even for isotropic pairing on an anisotropic Fermi surface, the thermodynamic quantities exhibit substantial oscillatory behavior in the superconducting state, even much below the upper critical field. Furthermore, in multiband systems the competition of anisotropies between two Fermi surfaces can cause a double sign reversal of oscillations as a function of temperature, irrespective of gap anisotropy. Our findings put severe constraints on simple interpretations of field-angle resolved measurements widely used to identify the angular structure of the superconducting gap.

Superconducting gap structure of the 115s revisited

Density functional theory calculations of the electronic structure of Ce- and Pu-based heavy fermion superconductors in the so-called 115 family are performed. The gap equation is used to consider which superconducting order parameters are most favorable assuming a pairing interaction that is peaked at (π, π, qz)—the wavevector for the antiferromagnetic ordering found in close proximity. In addition to the commonly accepted dx2−y2 order parameter, there is evidence that an extended s-wave order parameter with nodes is also plausible. We discuss whether these results are consistent with current observations and possible measurements that could help distinguish between these scenarios.

Antiferromagnetic order in Pauli-limited unconventional superconductors

We develop a theory of the coexistence of superconductivity (SC) and antiferromagnetism (AFM) in CeCoIn(5). We show that in Pauli-limited nodal superconductors the nesting of the quasiparticle pockets induced by Zeeman pair breaking leads to incommensurate AFM with the magnetic moment normal to the field. We compute the phase diagram and find a first order transition to the normal state at low temperatures, the absence of normal state AFM, and the coexistence of SC and AFM at high fields, in agreement with experiments. We also predict the existence of a new double-Q magnetic phase.

Nodes versus minima in the energy gap of iron pnictide superconductors from field-induced anisotropy

We develop the formalism for computing the oscillations of the specific heat and thermal transport under rotated magnetic field in multiband superconductors with anisotropic gap and apply it to iron-based materials. We show that these oscillations change sign at low temperatures and fields, which strongly influences the experimental conclusions about the gap structure. We find that recent measurements of the specific heat oscillations indicate that the iron-based superconductors possess an anisotropic gap with deep minima or nodes close to the line connecting electron and hole pockets. We predict the behavior of the thermal conductivity that will help distinguish between these cases.

Quantum criticality and nodal superconductivity in the FeAs-based superconductor KFe2As2

The in-plane resistivity rho and thermal conductivity kappa of the FeAs-based superconductor KFe2As2 single crystal were measured down to 50 mK. We observe non-Fermi-liquid behavior rho(T) approximately T{1.5} at H{c{2}}=5 T, and the development of a Fermi liquid state with rho(T) approximately T{2} when further increasing the field. This suggests a field-induced quantum critical point, occurring at the superconducting upper critical field H{c{2}}. In zero field, there is a large residual linear term kappa{0}/T, and the field dependence of kappa_{0}/T mimics that in d-wave cuprate superconductors. This indicates that the superconducting gaps in KFe2As2 have nodes, likely d-wave symmetry. Such a nodal superconductivity is attributed to the antiferromagnetic spin fluctuations near the quantum critical point.

Andreev reflection and order parameter symmetry in heavy-fermion superconductors: the case of CeCoIn(5)

We review the current status of Andreev reflection spectroscopy on the heavy fermions, mostly focusing on the case of CeCoIn(5), a heavy-fermion superconductor with a critical temperature of 2.3 K. This is a well-established technique to investigate superconducting order parameters via measurements of the differential conductance from nanoscale metallic junctions. Andreev reflection is clearly observed in CeCoIn(5) as in other heavy-fermion superconductors. Considering the large mismatch in Fermi velocities, this observation seemingly appears to disagree with the Blonder-Tinkham-Klapwijk (BTK) theory. The measured Andreev signal is highly reduced to the order of maximum ∼13% compared to the theoretically predicted value (100%). The background conductance exhibits a systematic evolution in its asymmetry over a wide temperature range from above the heavy-fermion coherence temperature down to well below the superconducting transition temperature. Analysis of the conductance spectra using the extended BTK model provides a qualitative measure for the superconducting order parameter symmetry, which is determined to be the d(x(2)-y(2)) wave in CeCoIn(5). It is found that existing models do not quantitatively account for the data, which we attribute to the intrinsic properties of the heavy fermions. A substantial body of experimental data and extensive theoretical analysis point to the existence of two-fluid components in CeCoIn(5) and other heavy-fermion compounds. A phenomenological model is proposed employing a Fano interference effect between two conductance channels in order to explain both the conductance asymmetry and the reduced Andreev signal. This model appears plausible not only because it provides good fits to the data but also because it is highly likely that the electrical conduction occurs via two channels, one into the heavy-electron liquid and the other into the conduction electron continuum. Further experimental and theoretical investigations will shed new light on the mechanism of how the coherent heavy-electron liquid emerges out of the Kondo lattice, a prototypical strongly correlated electron system. Unresolved issues and future directions are also discussed.

Feedback spin resonance in superconducting CeCu2Si2 and CeCoIn5

We show that the recently observed spin resonance modes in heavy-fermion superconductors CeCoIn5 and CeCu2Si2 are magnetic excitons originating from superconducting quasiparticles. The wave vector Q of the resonance state leads to a powerful criterion for the symmetry and node positions of the unconventional gap function. The detailed analysis of the superconducting feedback on magnetic excitations reveals that the symmetry of the superconducting gap corresponds to a singlet d_{x;{2}-y;{2}} state symmetry in both compounds. In particular this resolves the long-standing ambiguity of the gap symmetry in CeCoIn5. We demonstrate that in both superconductors the resonance peak shows a significant dispersion away from Q which can be checked experimentally. Our analysis reveals the similar origin of the resonance peaks in the two heavy-fermion superconductors and in layered cuprates.

Probing the nodal gap in the pressure-induced heavy fermion superconductor CeRhIn5

We report field-orientation specific heat studies of the pressure-induced heavy-fermion superconductor CeRhIn5. These experiments provide the momentum-dependent superconducting gap function for the first time in any pressure-induced superconductor. In the coexisting phase of superconductivity and antiferromagnetism, field rotation within the Ce-In plane reveals fourfold modulation in the density of states, which favors a d-wave order parameter and constrains a theory of the interplay between superconductivity and magnetism.

Kramer-Pesch approximation for analyzing field-angle-resolved measurements made in unconventional superconductors: a calculation of the zero-energy density of states

By measuring the angular-oscillations behavior of the heat capacity with respect to the applied field direction, one can detect the details of the gap structure. We introduce the Kramer-Pesch approximation as a new method to analyze the field-angle-dependent experiments, which improves the previous Doppler-shift technique. We show that the Fermi-surface anisotropy is an indispensable factor for identifying the superconducting gap symmetry.

Thermal conductivity evidence for a dx2-y2 pairing symmetry in the heavy-fermion CeIrIn5 superconductor

The phase diagram of the quasi-2D Ce(Ir,Rh)In5 system contains two distinct superconducting domes. By the thermal transport measurements in rotating magnetic fields H, we pinned down the superconducting gap structure of CeIrIn5 in the second dome, located distant from the first dome in proximity to an antiferromagnetic quantum critical point. Clear fourfold oscillation was observed when H is rotated within the ab plane, while no oscillation was observed within the bc plane. In sharp contrast to previous reports, our results are most consistent with dx2-y2 symmetry, implying that the superconductivity in the second phase is also mediated by antiferromagnetic spin fluctuations.

Andreev reflection in heavy-fermion superconductors and order parameter symmetry in CeCoIn5

Differential conductance spectra are obtained from nanoscale junctions on the heavy-fermion superconductor CeCoIn5 along three major crystallographic orientations. Consistency and reproducibility of characteristic features among the junctions ensure their spectroscopic nature. All junctions show a similar conductance asymmetry and Andreev reflectionlike conductance with a reduced signal ( approximately 10%-13%), both commonly observed in heavy-fermion superconductor junctions. Analysis using the extended Blonder-Tinkham-Klapwijk model indicates that our data provide the first spectroscopic evidence for d_{x;{2}-y;{2}} symmetry. To quantify our conductance spectra, we propose a model by considering the general phenomenology in heavy fermions, the two-fluid behavior, and an energy-dependent density of states. Our model fits to the experimental data remarkably well and should invigorate further investigations.

Hybrid gap structure of the heavy-fermion superconductor CeIrIn5

The thermal conductivity kappa of the heavy-fermion superconductor CeIrIn5 was measured as a function of temperature down to T(c)/8, for current directions parallel (J parallel c) and perpendicular (J parallel a) to the tetragonal c axis. For J parallel a, a sizable residual linear term kappa(0)/T is observed, as previously, which confirms the presence of line nodes in the superconducting gap. For J parallel c, on the other hand, kappa/T-->0 as T-->0. The resulting precipitous decline in the anisotropy ratio kappa(c)/kappa(a) at low temperature rules out a gap structure with line nodes running along the c axis, such as the d-wave state favored for CeCoIn5, and instead points to a hybrid gap of E(g) symmetry.

Field dependent coherence length in the superclean, high-kappa superconductor CeCoIn5

Using small-angle neutron scattering, we have studied the flux-line lattice (FLL) in the superclean, high-kappa superconductor CeCoIn5. The FLL undergoes a first-order symmetry and reorientation transition at approximately 0.55 T at 50 mK. In addition, the FLL form factor in this material is found to be independent of the applied magnetic field, in striking contrast to the exponential decrease usually observed in superconductors. This result is consistent with a strongly field-dependent coherence length, proportional to the vortex separation.