Horizontal Line Nodes in Sr2RuO4 Proved by Spin Resonance
K. Iida, M. Kofu, K. Suzuki, N. Murai, S. Ohira-Kawamura, R. Kajimoto, Y. Inamura, M. Ishikado, S. Hasegawa, T. Masuda, Y. Yoshida, K. Kakurai, K. Machida, S. -H. Lee
备注：
arXiv admin note: substantial text overlap with arXiv:1904.01234
We investigated the low-energy incommensurate (IC) magnetic fluctuations in Sr$_2$RuO$_4$ by the high-resolution inelastic neutron scattering measurements and random phase approximation (RPA) calculations. We observed a spin resonance with energy of $\hbarω_\text{res}=0.56$~meV centered at a characteristic wavevector $\mathbf{Q}_\text{res}=(0.3, 0.3, 0.5)$. The resonance energy corresponds well to the superconducting gap $2Δ=0.56$~meV estimated by the tunneling spectroscopy. The spin resonance shows the $L$ modulation with a maximum at around $L = 0.5$. The $L$ modulated intensity of the spin resonance and our RPA calculations indicate that the superconducting gaps regarding the quasi-one-dimensional $α$ and $β$ sheets at the Fermi surfaces have the horizontal line nodes. These results may set a strong constraint on the pairing symmetry of Sr$_2$RuO$_4$. We also discuss the implications on possible superconducting order parameters.
A possible mechanism for the horizontal line nodes in triplet superconductor, Sr$_2$RuO$_4$, is proposed. We consider the interlayer Coulomb interaction, as well as the on-site Coulomb repulsion, between electrons in different bands. In the second order perturbation in the interband interaction, the effective interaction becomes dependent on $\cos \frac{q_z}{2}$, resulting in the horizontal line nodes.
We calculate the tunneling conductance spectra of a normal-metal/insulator/triplet superconductor using the Blonder-Tinkham-Klapwijk (BTK) formulation. Possible states for the superconductor are considered with horizontal lines of nodes, breaking the time reversal symmetry. These results would be useful to discriminate between pairing states in superonductor Sr$_2$RuO$_4$ and also in UPt$_3$.
On the basis of the microscopic quasi-classical Eilenberger theory, we analyze the recent angle-resolved specific heat experiment carried out at low temperature for Sr$_2$RuO$_{4}$ to identify the superconducting gap symmetry, comprising either horizontal or vertical line nodes relative to the tetragonal crystal symmetry. Several characteristics, in particular, the landscape of the in-plane oscillation amplitude $A_4(B, T)$ with a definite sign for almost the entire $B$-$T$ plane are best explained by the horizontal line node symmetry, especially when the multiband effect and Pauli paramagneti…
The anisotropy and temperature dependence of the magnetic field penetration depth in the iron-based superconductor KFe$_2$As$_2$ have been measured down to 20 mK by means of muon spin rotation. We have observed that the flux-line lattice forms triangular symmetry when we applied the field parallel to c-axis, suggesting the interaction between vortices should be isotropic. The temperature dependence of the penetration depth observd with an applied field parallel and perpendicular to c-axis are different. These results can be accounted for by a superconducting gap function with a horizontal line…
Dipole interaction in the triplet superconductivity is studied. If the Cooper pairs are formed by the electrons at $\mib{r}$ and $\mib{r}+(\pm\frac{a}{2},\pm\frac{a}{2},\pm\frac{c}{2})$ in the body centered tetragonal lattice, (in this case the line nodes run horizontally on the cylindrical Fermi surface), the dipole energy is low when the ${\mib{d}}$-vector is perpendicular to the direction of the angular momentum of the Cooper pairs ($\mib{l}$-vector). This result is in contrast with the dipole energies in the ABM state of superfluid $^3$He, where $\mib{d}$-vector is forced to be parallel or…
We analyze the possibilities of triplet pairing in Sr2RuO4 based upon an idea of interlayer coupling. We have considered two models differing by the effective interactions. In one model the quasi-particle spectra have horizontal line nodes on all three Fermi surface sheets, while in the other the spectra have line or point nodes on the alpha and beta sheets and no nodes on the gamma sheet. Both models reproduce the experimental heat capacity and penetration depth results, but the calculated specific heat is sightly closer to experiment in the second solution with nodes only on the alpha and be…
Iron-based compounds (IBS) display a surprising variety of superconducting properties that seems to arise from the strong sensitivity of these systems to tiny details of the lattice structure. In this respect, systems that become superconducting under pressure, like CaFe$_2$As$_2$, are of particular interest. Here we report on the first directional point-contact Andreev-reflection spectroscopy (PCARS) measurements on CaFe$_2$As$_2$ crystals under quasi-hydrostatic pressure, and on the interpretation of the results using a 3D model for Andreev reflection combined with ab-initio calculations of …
We calculate the thermal conductivity kappa in magnetic fields near H_c2 for spin-triplet superconducting states with line nodes vertical and horizontal relative to the RuO_2-planes. The method for calculating the Green's functions takes into account the spatial variation of the order parameter and superconducting flow for the Abrikosov vortex lattice. For in-plane magnetic field we obtain variations of the in-plane kappa with two-fold symmetry as a function of rotation angle where the minima and maxima occur for field directions parallel and perpendicular to the heat flow. The amplitude of th…
We present the group-thoretical classification of gap functions in superconductors coexisting with some magnetic order in non-symmorphic magnetic space groups. Based on the weak-coupling BCS theory, we show that UCoGe-type ferromagnetic superconductors must have horizontal line nodes on either $k_z=0$ or $\pmπ/c$ plane. Moreover, it is likely that additional Weyl point nodes exist at the axial point. On the other hand, in UPd$_2$Al$_3$-type antiferromagnetic superconductors, gap functions with $A_g$ symmetry possess horizontal line nodes in antiferromagnetic Brillouin zone boundary perpendicul…
We calculate the ultrasonic attenuation in magnetic fields for superconducting states with line nodes vertical or horizontal relative to the RuO_2 planes. This theory, which is valid for fields near Hc2 and not too low temperatures, takes into account the effects of supercurrent flow and Andreev scattering by the Abrikosov vortex lattice. For rotating in-plane field H(theta) the attenuation alpha(theta)exhibits variations of fourfold symmetry in the rotation angle theta. In the case of vertical nodes, the transverse T100 sound mode yields the weakest(linear)H and T dependence of alpha, while t…
The magneto-thermal conductivity kappa is calculated for the vortex state of UPd_2Al_3 by assuming horizontal gap nodes. The Green's function method we employed takes into account the effects of supercurrent flow and Andreev scattering on the quasiparticles due to Abrikosov's vortex lattice order parameter. The calculated angular dependence of kappa_{yy} for field rotation theta_0 in the ac-plane depends strongly on field strength H, impurity scattering, anisotropy of the Fermi velocity, and temperature. For finite temperatures and the clean unitary scattering limit we get qualitative agreemen…
Several possible odd-parity states are listed up group-theoretically and examined in light of recent experiments on Sr$_2$RuO$_4$. Those include some of the $f$-wave pairing states, ${\mib d}({\mib k})\propto{\hat{\mib z}} k_xk_y(k_x + {\rm i}k_y)$ and ${\hat{\mib z}} (k_x^2-k_y^2)(k_x + {\rm i}k_y)$ and other ${\hat{\mib z}} (k_x + {\rm i}k_y)\cos ck_z$ ($c$ is the $c$-axis lattice constant) as most plausible candidates. These are time-reversal symmetry broken states and have line nodes running either vertically (the former two) or horizontally (the latter), consistent with experiments. Chara…
We consider Fe-based superconductors with s^{+-} gap with accidental nodes on electron pockets. We analyze how the gap structure changes if we include into the consideration the hybridization between the two electron pockets (the inter-pocket hopping term with momentum (pi,pi,pi). We derive the hybridization term and relate it to the absence of inversion symmetry in the Fe-plane because of two non-equivalent locations of pnictogen (chalcogen) above and below the plane. We find that the hybridization tends to eliminate the nodes -- as it increases, the pairs of neighboring nodes approach each o…
We performed inelastic neutron scattering on powder sample of the P-doped iron-based superconductor BaFe2(As0.65P0.35)2 with Tc = 30K, whose superconducting (SC) order parameter is expected to have line nodes. We have observed spin resonance at Q $\sim$ 1.2Å^{-1} and E=12 meV in the SC state. The resonance enhancement, which can be a measure of the area of sign reversal between the hole and electron Fermi surfaces (FSs), is comparable to those of other iron-based superconductors without line nodes. This fact indicates that the sign reversal between the FSs is still dominant in this system, and…
To clarify the superconducting gap structure of the spin-triplet superconductor Sr_2RuO_4, the in-plane thermal conductivity has been measured as a function of relative orientations of the thermal flow, the crystal axes, and a magnetic field rotating within the 2D RuO_2 planes. The in-plane variation of the thermal conductivity is incompatible with any model with line nodes vertical to the 2D planes and indicates the existence of horizontal nodes. These results place strong constraints on models that attempt to explain the mechanism of the triplet superconductivity.
The apparent $T^2$-temperature dependence of the London penetration depth in Sr$_2$RuO$_4$ is discussed on the basis of a multi-gap model with horizontal line nodes. The influence of the nodes in combination with nonlocal electromagnetic response leads to low-temperature behaviors as $-T^2\ln T$ in the London limit and as $T^2$ in the Pippard limit. These behaviors appear only at very low temperature. On the other hand, the interplay of the superconductivity of different bands is responsible for the observed $T^2$-like behavior over the wide temperature range. The experimental data can be fitt…
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 ($θ$) resolved zero energy density of states $N(θ)$. 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…
There is strong experimental evidence that the superconductor Sr2RuO4 has a chiral p-wave order parameter. This symmetry does not require that the associated gap has nodes, yet specific heat, ultrasound and thermal conductivity measurements indicate the presence of nodes in the superconducting gap structure of Sr2RuO4. Theoretical scenarios have been proposed to account for the existence of accidental nodes or deep accidental minima within a p-wave state. To elucidate the nodal structure of the gap, it is essential to know whether the lines of nodes (or minima) are vertical (parallel to the te…
There is considerable evidence that the superconducting state of Sr$_2$RuO$_4$ breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking its onset temperature, $T_\text{TRSB}$, is generally found to match the critical temperature, $T_\text{c}$, within resolution. In combination with evidence for even parity, this result has led to consideration of a $d_{xz} \pm id_{yz}$ order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding $T_\text{TRSB} = T_\text{c}$, but it has a hard-to-explain horizontal line node at…
The gap structure of Sr$_2$RuO$_4$, which is a longstanding candidate for a chiral p-wave superconductor, has been investigated from the perspective of the dependence of its specific heat on magnetic field angles at temperatures as low as 0.06 K ($\sim 0.04T_{\rm c}$). Except near $H_{\rm c2}$, its fourfold specific-heat oscillation under an in-plane rotating magnetic field is unlikely to change its sign down to the lowest temperature of 0.06 K. This feature is qualitatively different from nodal quasiparticle excitations of a quasi-two-dimensional superconductor possessing vertical lines of ga…
We calculate the energy dispersion of the surface Andreev states and their contribution to tunneling conductance for the order parameters with horizontal and vertical lines of nodes proposed for superconducting Sr2RuO4. For vertical lines, we find double peaks in tunneling spectra reflecting the van Hove singularities in the density of surface states originating from the turning points in their energy dispersion. For horizontal lines, we find a single cusp-like peak at zero bias, which agrees very well with the experimental data on tunneling in Sr2RuO4.
We analyze the behavior of the thermal conductivity in quasi-two dimensional superconductors with line nodes. Motivated by measurements of the anisotropy between the interplane and intraplane thermal transport in CeIrIn_5 we show that a simple model of the open Fermi surface with vertical line nodes is insufficient to describe the data. We propose two possible extensions of the model taking into account a) additional modulation of the gap along the axial direction of the open Fermi surface; and b) dependence of the interplane tunneling on the direction of the in-plane momentum. We discuss the …
