Nodal Gap Structure Research Articles

Atomic-Scale Andreev Probe of Unconventional Superconductivity.

Recent emergence of low-dimensional unconventional superconductors and their exotic interface properties calls for new approaches to probe the pairing symmetry, a fundamental and frequently elusive property of the superconducting condensate. Here, we introduce the unique capability of tunneling Andreev reflection (TAR) to probe unconventional pairing symmetry, utilizing the sensitivity of this technique to specific Andreev reflections. Specifically, suppression of the lowest-order Andreev reflection due to quantum interference but emergence of the higher-order Andreev processes provides direct evidence of the sign-changing order parameter in the paradigmatic FeSe superconductor. TAR spectroscopy also reveals two superconducting gaps, points to a possibility of a nodal gap structure, and directly confirms that superconductivity is locally suppressed along the nematic twin boundary, with preferential and near-complete suppression of the larger energy gap. Our findings therefore enable new, atomic-scale insight into microscopic, inhomogeneous, and interfacial properties of emerging quantum materials.

Observations of the effect of strong Pauli paramagnetism on the vortex lattice in superconducting CeCu2Si2

We present the results of a study of the vortex lattice in the heavy fermion superconductor ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$, using small-angle neutron scattering (SANS). In this material at temperatures well below ${T}_{\mathrm{c}}\ensuremath{\sim}0.6$ K, the value of the upper critical field ${B}_{\mathrm{c}2}\ensuremath{\sim}2.2$ T is strongly limited by the Pauli paramagnetism of the heavy fermions. In this temperature region, our SANS data show an increase in the magnetization of the flux line cores with field, followed by a rapid fall near ${B}_{\mathrm{c}2}$. This behavior is the effect of Pauli paramagnetism and we present a theory-based model, which can be used to describe this effect in a range of materials. The pairing in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ appears to arise from the effect of magnetic fluctuations, but the evidence for a $d$-wave order parameter is rather weak. We find that the vortex lattice structure in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ is close to regular hexagonal. There are no phase transitions to square or rhombic structures; such transitions are expected for $d$-wave superconductors and observed in ${\mathrm{CeCoIn}}_{5}$; however, the temperature dependence of the SANS intensity indicates that both large and small gap values are present, most likely due to multiband $s$-wave superconductivity, rather than a nodal gap structure.

Local observation of linear- T superfluid density and anomalous vortex dynamics in URu2Si2

The heavy fermion superconductor URu$_2$Si$_2$ is a candidate for chiral, time-reversal symmetry-breaking superconductivity with a nodal gap structure. Here, we microscopically visualized superconductivity and spatially inhomogeneous ferromagnetism in URu$_2$Si$_2$. We observed linear-$T$ superfluid density, consistent with d-wave pairing symmetries including chiral d-wave, but did not observe the spontaneous magnetization expected for chiral d-wave. Local vortex pinning potentials had either four- or two-fold rotational symmetries with various orientations at different locations. Taken together, these data support a nodal gap structure in URu$_2$Si$_2$ and suggest that chirality either is not present or does not lead to detectable spontaneous magnetization.

Multigap nodeless superconductivity in CsCa2Fe4As4F2 probed by heat transport

Recently, a new family of iron-based superconductors called 12442 was discovered and the muon spin relaxation ($\mu$SR) measurements on KCa$_2$Fe$_4$As$_4$F$_2$ and CsCa$_2$Fe$_4$As$_4$F$_2$ polycrystals, two members of the family, indicated that both have a nodal superconducting gap structure with $s+d$ pairing symmetry. Here we report the ultralow-temperature thermal conductivity measurements on CsCa$_2$Fe$_4$As$_4$F$_2$ single crystals ($T_c$ = 29.3 K). A negligible residual linear term $\kappa_0/T$ in zero field and the field dependence of $\kappa_0/T$ suggest multiple nodeless superconducting gaps in CsCa$_2$Fe$_4$As$_4$F$_2$. This gap structure is similar to CaKFe$_4$As$_4$ and moderately doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$, but contrasts to the nodal gap structure indicated by the $\mu$SR measurements on CsCa$_2$Fe$_4$As$_4$F$_2$ polycrystals.

Evidence of nodal gap structure in the basal plane of the FeSe superconductor

Identifying the symmetry of the wave function describing the Cooper pairs is pivotal in understanding the origin of high-temperature superconductivity in iron-based superconductors. Despite nearly a decade of intense investigation, the answer to this question remains elusive. Here we use the muon spin rotation/relaxation (muSR) technique to investigate the underlying symmetry of the pairing state of the FeSe superconductor, the basic building block of all iron-chalcogenide superconductors. Contrary to earlier muSR studies on powders and crystals, we show that while the superconducting gap is most probably anisotropic but nodeless along the crystallographic c-axis, it is nodal in the ab-plane, as indicated by the linear increase of the superfluid density at low temperature. We further show that the superconducting properties of FeSe display a less pronounced anisotropy than expected.

Pairing mechanism for nodal s -wave superconductivity in BaFe2(As,P)2 : Analysis beyond Migdal-Eliashberg formalism

The pairing mechanism and gap structure in Ba122 pnictides have been hotly discussed for long time as one of the central issues in Fe-based superconductors. Here, we attack this problem by taking account of the vertex corrections (VCs) for the Coulomb interaction $U$ ($U$-VCs), which are totally dropped in conventional Migdal-Eliashberg formalism. The $U$-VC in the charge susceptibility induces strong orbital fluctuations, and the $U$-VC also enlarges the orbital-fluctuation-mediated attractive interaction. By analyzing the effective multiorbital Hubbard model for Ba122 pnictides, we find that the orbital fluctuations develop in all four $d$-orbitals ($t_{2g}$- and $z^2$-orbitals), by which the FSs are composed. For this reason, nearly isotropic gap function appears on all the hole-type FSs, including the outer hole-FS around Z-point composed of $z^2$-orbital. In contrast, nodal gap structure appears on the electron-type FSs for wide parameter range. The obtained nodal $s$-wave state changes to fully-gapped $s$-wave state without sign-reversal ($s_{++}$-wave state) by introducing small amount of impurities, accompanied by small reduction in $T_c$. The present microscopic theory naturally explains the important characteristics of the gap structure of both hole- and electron-FSs in Ba122 pnictides, without introducing any phenomenological pairing interaction.

Superconducting symmetries and magnetic responses of uranium heavy-fermion systems UBe13 and UPd2Al3

Low-temperature thermodynamic investigation for UBe13 and UPd2Al3 were performed in order to gain insight into their unusual ground states of 5f electrons. Our heat-capacity data for the cubic UBe13 strongly suggest that nodal quasiparticles are absent and its superconducting (SC) gap is fully open over the Fermi surface. Moreover, two unusual thermodynamic anomalies are also observed in UBe13 at ∼ 3T and ∼ 9T; the lower-field anomaly is seen only in the SC mixed state by dc magnetization M(H) as well as heat-capacity C(H), while the higher-field anomaly appears for C(H) in the normal phase above the upper critical field. On the other hand, field-orientation dependence of the heat capacity in the hexagonal UPd2Al3 shows a significantly anisotropic behavior of C(H)∝H1/2, reflecting the nodal gap structure of this system. Our result strongly suggests the presence of a horizontal line node on the Fermi surface with heavy effective mass in UPd2Al3.

Nodal Superconducting Gap Structure in the Quasi-One-Dimensional Cs2Cr3As3 Investigated Using μSR Measurements

The superconducting ground state of the newly discovered superconductor Cs$_2$Cr$_3$As$_3$ with a quasi-one-dimensional crystal structure ($T_{\bf c}\sim$ 2.1(1) K) has been investigated using magnetization and muon-spin relaxation or rotation ($\mu$SR), both zero-field (ZF) and transverse-field (TF), measurements. Our ZF $\mu$SR measurements reveal the presence of spin fluctuations below 4 K and the ZF relaxation rate ($\lambda$) shows enhancement below $T_{\bf c}\sim$ 2.1 K, which might indicate that the superconducting state is unconventional. This observation suggests that the electrons are paired via unconventional channels such as spin fluctuations, as proposed on the basis of theoretical models. Our analysis of the TF $\mu$SR results shows that the temperature dependence of the superfluid density is fitted better with a nodal gap structure than an isotropic s-wave model for the superconducting gap. The observation of a nodal gap in Cs$_2$Cr$_3$As$_3$ is consistent with that observed in the isostructural K$_2$Cr$_3$As$_3$ compound through TF $\mu$SR measurements. Furthermore, from our TF $\mu$SR study we have estimated the magnetic penetration depth $\lambda_{\mathrm{L}}$$(0)$ = 954 nm, superconducting carrier density $n_s = 4.98 \times 10^{26}~ $m$^{-3}$, and carrier's effective-mass enhancement $m^*$ = 1.61m$_{e}$.

Nodal gap structure of the heavy-fermion superconductor URu2Si2 revealed by field-angle-dependent specific-heat measurements

URu2Si2 is a promising candidate for a chiral d-wave superconductor whose gap is composed of a horizontal line node at equator and point nodes at the north and south poles. However, previous measurements of the specific heat and the thermal conductivity did not detect quasiparticle excitations from a horizontal line node in heavy-mass bands. Here, we have provided strong evidence for the presence of a horizontal line node from field-angle-dependent specific-heat measurements by using a high-quality single crystal of URu2Si2. We observed the behavior of the specific heat at low fields which does not depend on inplane field angle ϕ but shows a shoulder-like anomaly in the polar-angle θ dependence around θ ∼ 45°. This feature is more clearly seen at lower temperatures, suggesting the detection of low-energy quasiparticle excitations reflecting the gap structure. From theoretical analyses based on microscopic calculations, we have demonstrated that this anomaly can be explained by the existence of a horizontal line node at kz = 0. Thus, the gap structure of URu2Si2 matches well with the Eg chiral kz(kx + iky) symmetry.

Nodal gap detection through polar angle-resolved density of states measurements in uniaxial superconductors

We propose a spectroscopic method to identify the nodal gap structure in unconventional superconductors. This method best suits for locating the horizontal line node and for pinpointing the isolated point nodes by measuring polar angle ($\theta$) resolved zero energy density of states $N(\theta)$. This is measured by specific heat or thermal conductivity at low temperatures under a magnetic field. We examine a variety of uniaxially symmetric nodal structure, including point and/or line nodes with linear and quadratic dispersions, by solving Eilenberger equation in vortex states. It is found that (A) the maxima of $N(\theta)$ continuously shift from the anti-nodal to the nodal direction ($\theta_{\rm n}$) as a field increases accompanying the oscillation pattern reversal at low and high fields. Furthermore, (B) local minima emerge next to $\theta_{\rm n}$ on both sides except for the case of linear point node. These features are robust and detectable experimentally. Experimental results of $N(\theta)$ performed on several superconductors, UPd$_2$Al$_3$, URu$_2$Si$_2$, Cu$_x$Bi$_2$Se$_3$, and UPt$_3$, are examined and commented in light of the present theory.

Doping Evolution of the Superconducting Gap Structure in Heavily Hole-Doped Ba1−xKxFe2As2: a Heat Transport Study**Supported by the National Basic Research Program under Grant Nos 2012CB821402 and 2015CB921401, the National Natural Science Foundation of China, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (No 15XD1500200).

We perform systematic thermal conductivity measurements on heavily hole-doped Ba1−xKxFe2As2 single crystals with 0.747 ≤ x ≤ 0.974. At x=0.747, the κ0/T is negligible, indicating a nodeless superconducting gap. A small residual linear term κ0/T (≈0.035 mW·K−2cm−1) appears at x=0.826, and it increases slowly up to x=0.974, followed by a substantial increase of more than 20 times to the pure KFe2As2 (x=1.0). This doping dependence of κ0/T clearly shows that the nodal gap appears near x = 0.8, possibly associated with the change of Fermi surface topology. The small values of κ0/T from x=0.826 to 0.974 are consistent with the ϒ-shaped nodal s-wave gap recently revealed by angle-resolved photoemission spectroscopy experiments at x=0.9. Furthermore, the substantial increase of κ0/T from x=0.974 to 1.0 is inconsistent with a symmetry-imposed d-wave gap in KFe2As2, and a possible nodal gap structure in KFe2As2 is discussed.

Heat transport inRbFe2As2single crystals: Evidence for nodal superconducting gap

The in-plane thermal conductivity of iron-based superconductor RbFe$_2$As$_2$ single crystal ($T_c \approx$ 2.1 K) was measured down to 100 mK. In zero field, the observation of a significant residual linear term $\kappa_0/T$ = 0.65 mW K$^{-2}$ cm$^{-1}$ provides clear evidence for nodal superdonducting gap. The field dependence of $\kappa_0/T$ is similar to that of its sister compound CsFe$_2$As$_2$ with comparable residual resistivity $\rho_0$, and lies between the dirty and clean KFe$_2$As$_2$. These results suggest that the (K,Rb,Cs)Fe$_2$As$_2$ serial superconductors have a common nodal gap structure.

Nodal gap structure in Fe-based superconductors due to the competition between orbital and spin fluctuations

To understand the origin of the nodal gap structure realized in BaFe$_2$(As,P)$_2$, we study the three-dimensional gap structure based on the three-dimensional ten-orbital Hubbard model with quadrupole interaction. In this model, strong spin and orbital fluctuations develop by using the random-phase-approximation. By solving the Eliashberg gap equation, we obtain the fully-gapped s-wave state with (without) sign reversal between hole-like and electron-like Fermi surfaces due to strong spin (orbital) fluctuations, so called the $s_\pm$-wave ($s_{++}$-wave) state. When both spin and orbital fluctuations strongly develop, which will be realized near the orthorhombic phase, we obtain the nodal s-wave state in the crossover region between $s_{++}$-wave and $s_\pm$-wave states. The obtained nodal s-wave state possesses the loop-shape nodes on electron-like Fermi surfaces, due to the competition between attractive and repulsive interactions in k-space. In contrast, the SC gaps on the hole-like Fermi surfaces are fully-gapped due to orbital fluctuations. The present study explains the main characters of the anisotropic gap structure in BaFe$_2$(As,P)$_2$ observed experimentally.

NMR study of hole-doped iron-pnictide superconductor Ba1−xKxFe2As2 (x = 0.27−1)

We performed the 75As nuclear magnetic resonance measurements of Ba1−xKxFe2As2 (BKFA) for x = 0.27, 0.39, 0.58, 0.64, 0.69 and 1 to understand gap structure in this system. Temperature dependence of spin lattice relaxation rate (1/T1) below superconducting (SC) transition temperature Tc changes gradually from x = 0.39 to 1.0. This result indicates that the SC gap structure changes smoothly from full gap state with maximum Tc into nodal-line structure state for x = 1 (KFe2As2). BKFA does not have distinct SC symmetry change. The nodal gap structure appears continuously and develops gradually with increasing x.

Unconventional Pairing State in Weakly Correlated Non-centrosymmetric Superconductors

We briefly review the recent progress on the study of weakly correlated non-centrosymmetric (NCS) superconductors, with an emphasis on their novel pairing symmetry and the upper critical field arising from the absence of inversion symmetry. Measurements of penetration depth, specific heat, NMR, etc., have shown that non-BCS-like superconductivity with a nodal gap structure may appear in simple metallic NCS superconductors, which allows the admixture of spin-singlet and spin-triplet pairing states attributed to the antisymmetric spin-orbital coupling (ASOC). Furthermore, large upper critical field with a value exceeding the Pauli paramagnetic limit was also experimentally observed in NCS superconductors.

Thermodynamic Properties of κ-(BEDT-TTF)2X Salts: Electron Correlations and Superconductivity

Heat capacity measurements of κ-(BEDT-TTF)2X (BEDT-TTF: Bis(ethylendithio) tetrathiafulvalene, X: counteranions) which are classified as two-dimensional (2D) dimer-Mott system are reported. At first, we explain structural and electronic features originated from rigid dimerization in donor arrangement in 2D layers. The antiferromagnetic Mott insulating phase located at low-pressure region in the phase diagram shows vanishing γ electronic heat capacity coefficient in the heat capacity, which claims opening of a charge-gap in this insulating state. Then, a systematic change of the γ around the Mott boundary region is reported in relation to the glass freezing of ethylene dynamics. The thermodynamic parameters determined by ∆Cp/γTc of 10 K class superconductors, κ-(BEDT-TTF)2Cu(NCS)2 and κ-(BEDT-TTF)2Cu[N(CN)2]Br demonstrate that a rather large gap with a strong coupling character appears around the Fermi-surface. On the other hand, the low temperature heat capacity clearly shows a picture of nodal-gap structure due to an anisotropic pairing. The comparison with lower Tc compounds in the κ-type structure is also performed so as to discuss overall features of the κ-type superconductors. The heat capacity measurements of hole-doped systems containing mercury in the counteranions show an anomalous enhancement of γ, which is consistent with the T1−1 of NMR experiments etc. The results of heat capacity measurements under high pressures are also reported.

Nodal superconducting order parameter and thermodynamic phase diagram of (TMTSF)2ClO4

The organic materials (TMTSF)${}_{2}$$X$ are unique unconventional superconductors with archetypal quasi-one-dimensional (Q1D) electronic structures. Here, based on our comprehensive field-angle-resolved calorimetry of (TMTSF)${}_{2}$ClO${}_{4}$, we succeeded in mapping the nodal gap structure by discriminating between the Fermi wave vectors and Fermi velocities. In addition, the thermodynamic phase diagrams of (TMTSF)${}_{2}$ClO${}_{4}$ for all principal field directions are obtained. These findings, providing strong evidence of nodal spin-singlet superconductivity, serves as solid bases for the further elucidation of anomalous superconducting phenomena in (TMTSF)${}_{2}$$X$.

Heat-Capacity Measurements of Energy-Gap Nodes of the Heavy-Fermion SuperconductorCeIrIn5Deep inside the Pressure-Dependent Dome Structure of Its Superconducting Phase Diagram

We use heat-capacity measurements as a function of field rotation to identify the nodal gap structure of CeIrIn(5) at pressures to 2.05 GPa, deep inside its superconducting dome. A fourfold oscillation in the heat capacity at 0.3 K is observed for all pressures, but with its sign reversed between 1.50 and 0.90 GPa. On the basis of recent theoretical models for the field-angle-dependent specific heat, all data, including the sign reversal, imply a d(x(2)-y(2)) order parameter with nodes along [110], which constrains theoretical models of the pairing mechanism in CeIrIn(5).

ANALYSIS OF MAGNETIC CRITICAL FIELDS IN IRON-BASED SmFeAsO0.85 HIGH-Tc SUPERCONDUCTOR

The magnetic properties of the newly discovered iron-oxypnictide SmFeAsO 0.85 high-Tc superconductor with a Tc of around 55 K were investigated. Bulk SmFeAsO 0.85 was prepared by a method for high-pressure synthesis. The lower critical field H c1 was estimated from the magnetization at low fields; H c1 (0) was measured to be 212 Oe. A linear temperature dependence instead of saturation at low temperatures in H c1 (T) revealed unconventional superconductivity with a nodal gap structure in our SmFeAsO 0.85 superconductor. The results showed that the well-known secondary peak in the temperature dependence of the critical current density Jc is absent in the SmFeAsO 0.85 high-Tc superconductor. The irreversibility line B irr was fitted well by the power law dependence (1 - T/Tc)n with n ~ 1.5. This is indicative of the flux creep phenomena in the SmFeAsO 0.85 high-Tc superconductor. In addition, within the range of measurement temperatures in this study, no crossover was observed in the temperature dependence of the irreversibility line B irr which may be due to low anisotropy in our SmFeAsO 0.85 superconductor.

Nesting of electron and hole Fermi surfaces in nonsuperconducting BaFe2P2

We report detailed measurements of the de Haas-van Alphen (dHvA) effect in BaFe_2P_2 which is the end member of the superconducting series, BaFe_2(As_{1-x}P_x)_2. Using high purity samples we are able to observe dHvA oscillations from all the sheets of Fermi surface and hence build up a complete detailed picture of its structure. The results show the existence of a highly warped section of hole surface which may be the origin of the nodal gap structure in this series. Importantly, we find that, even for this non-superconducting end member, one of the hole surfaces almost exactly nests with the inner electron surface. This suggests that improved geometric nesting does not drive the increase in pairing strength with decreasing x in this system.