Determining the exact pairing symmetry of the superconducting order parameter in candidate unconventional superconductors remains an important challenge. Recently a new method, based on phase sensitive quasiparticle interference measurements, was developed to identify gap sign changes in isotropic multiband systems. Here we extend this approach to the single-band and multiband nodal $d$-wave superconducting cases relevant, respectively, for the cuprates and likely for the infinite-layer nickelate superconductors. Combining analytical and numerical calculations, we show that the antisymmetrized correction to the tunneling density of states due to nonmagnetic impurities in the Born and intermediate-scattering limits shows characteristic features for sign-changing and sign-preserving scattering wavevectors, as well as for the momentum-integrated quantity. Furthermore, using a realistic approach accounting for the Wannier orbitals, we model scanning tunneling microscopy data of $\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_{8+δ}$, which should allow the comparison of our theory with experimental data.
Superconductivity in the heavy-fermion compound CeCu2Si2 is a prototypical example of Cooper pairs formed by strongly correlated electrons. For more than 30 years, it has been believed to arise from nodal d-wave pairing mediated by a magnetic glue. Here, we report a detailed study of the specific heat and magnetization at low temperatures for a high-quality single crystal. Unexpectedly, the specific-heat measurements exhibit exponential decay with a two-gap feature in its temperature dependence, along with a linear dependence as a function of magnetic field and the absence of oscillations in t…
The temperature, field, and field-orientation dependences of the electronic specific heat Ce of the ironpnictide superconductor KFe2As2 have been investigated. Thermodynamic evidence of the presence of line nodes is obtained from the T and $\sqrt{H}$ linear dependences of Ce/T in the low-T and low-H region. Under a magnetic field rotated within the tetragonal ab plane, a fourfold oscillation is observed in Ce with a sign change at 0.08Tc. On the basis of the Doppler-shift analysis, the observed Ce minima in H // [100] at low T indicate the presence of line nodes somewhere on the Fermi surface …
By measuring point-contact Andreev reflection (PCAR) spectra in the newly discovered chromium-based quasi-one-dimensional superconductor RbCr$_{3}$As$_{3}$, we find clear evidence of two superconducting components, i.e., one having a gap value of about 1.8 meV and another with a gap value of about 5 meV. Since the current injection may have components in both the direction parallel and perpendicular to the [(Cr$_{3}$As$_{3}$)$^{-}$]$_\infty$ chains in the PCAR measurements, it naturally explains the two-component feature observed in this multi-band superconductor. Detailed analysis shows that …
Multiband superconductivity with dominant two-gap features are recently proposed to challenge the earlier accepted nodal $d$-wave pairing in the first unconventional superconductor CeCu$_{2}$Si$_{2}$. Here we obtain multiband Fermi-surface topology of CeCu$_{2}$Si$_{2}$ via first-principles calculations, and study the problem within an effective two hybridization band model including detailed band-structure. Within T-matrix approximation, our calculations reveal that different pairing candidates could yield qualitatively distinct features characterised by impurity-induced bound states. Except …
This paper gives a detailed numerical study of the superconducting electronic heat transport in the unconventional multiband superconductor Strontium Ruthenate Sr$_2$RuO$_4$. The study demostrates that a model with different nodal structures on different sheets of the Fermi surface is able to describe quantitatively experimental heat transport data. The contribution of the density of states DOS is given for each sheet of the Fermi surface and the total contribution is also calculated. Finally, a discussion of the universal character of the electronic heat transport in unconventional supercondu…
Contrary to the usual assumption, the electron Bloch states in crystals with spin-orbit coupling do not always transform under symmetry operations in the same way as the pure spin-1/2 states. This has profound consequences for the symmetry properties and nodal structure of superconductors, especially for the interband gap functions. Focusing on tetragonal superconductors, we show that the interband pairing in the conventional (s-wave) channel can have features which are traditionally associated with unconventional pairing, such as triplet components and odd parity, and can produce line nodes i…
We report 31P-NMR and specific heat measurements on an iron (Fe)-based superconductor SrFe2(As0.65P0.35)2 with Tc=26 K, which have revealed the development of antiferromagnetic correlations in the normal state and the unconventional superconductivity(SC) with nodal gap dominated by the gapless low-lying quasiparticle excitations. The results are consistently argued with an unconventional multiband SC state with the gap-size ratio of different bands being significantly large; the large full gaps in s\pm-wave state keep Tc high, whereas a small gap with a nodal-structure causes gapless feature u…
The pseudogap (PG) excitations in the framework of multiband superconductivity are analysed. Calculations for the doping phase diagram of electron-doped cuprate superconductors have been made. A nonrigid multiband model has been used. The interband pairing acts between the upper Hubbard band (UHB) and midgap states created (and shifted upwards) by doping. The appearance of two PG and two phase diagram critical points (CP) is suggested. The larger antinodal PG reflects the destruction of the long-range antiferromagnetic (AF) order. The second, smaller PG is a near-nodal event with exposed dopin…
Recently, a family of nonmagnetic kagome metals \textit{A}$\mathrm{V_3Sb_5}$ (\textit{A}=K, Rb, and Cs) has attracted significant attention for realizing the intertwining of quantum order and nontrivial topology. However, these compounds have been identified to host complex band structures. Therefore, it is desirable to design and synthesize novel kagome materials with a simple band topology and good transport properties. In this study, using first-principles calculations, we present the electronic properties and the intrinsic spin Hall effect of V$_6$Sb$_4$, the latest experimentally synthesi…
We study three-dimensional time-reversal-invariant topological superconductivity in noncentrosymmetric materials such as RhSi, CoSi, and AlPt which host coupled multifold nodes energetically split by the spin-orbit coupling at the same time-reversal-invariant momentum (TRIM). The topological superconductivity arises from the $s_{+} \oplus s_{-}$ gap function, which is $\boldsymbol{k}$ independent, but with opposite signs for the two nodes split at the same TRIM. We consider various electron-electron interactions in the tight-binding model for RhSi and find that the topological superconducting …
We calculated the critical temperature in the framework of s+-+d-wave multiband Eliashberg theory. We have solved these equations numerically to see at what values of the input parameters there is a solution with a non-zero critical temperature and what is the symmetry of the order parameter of this solution. For our model we consider the pnictide case with simplifications that allow us to obtain the most general possible information. For selected and representative cases in which the order parameter has s+-+d symmetry, we calculated the superconducting density of states, the temperature depen…
We report measurements of the London penetration depth $Δλ(T)$ and the electronic specific heat $C_e(T)$ on high-quality single crystals of the filled-skutterudite superconductor PrPt$_4$Ge$_{12}$ ($T_c\simeq$8K). Both quantities show a weak temperature dependence at $T\ll T_c$, following $Δλ\sim T^n$ ($n\simeq 3.2$) and $C_e/T\sim T^{2.8}$. Such temperature dependences deviate from both conventional s-wave type and nodal superconductivity. A detailed analysis indicates that the superfluid density $ρ_s(T)$, derived from the penetration depth, as well as the electronic specific heat can be cons…
We find evidence that the newly discovered Fe-based superconductor KCa$_2$Fe$_4$As$_4$F$_2$ ($T_c~=~33.36(7)$~K) displays multigap superconductivity with line nodes. Transverse field muon spin rotation ($μ$SR) measurements show that the temperature dependence of the superfluid density does not have the expected behavior of a fully-gapped superconductor, due to the lack of saturation at low temperatures. Moreover, the data cannot be well fitted using either single band models or a multiband $s$-wave model, yet are well described by two-gap models with line nodes on either one or both of the gap…
CeCoIn5 is a heavy fermion superconductor with strong similarities to the high-Tc cuprates, including quasi-two-dimensionality, proximity to antiferromagnetism, and probable d-wave pairing arising from a non-Fermi-liquid normal state. Experiments allowing detailed comparisons of their electronic properties are of particular interest, but in most cases are difficult to realize, due to their very different transition temperatures. Here we use low temperature microwave spectroscopy to study the charge dynamics of the CeCoIn5 superconducting state. The similarities to cuprates, in particular to ul…
Recent experimental findings show that the pairing interaction in hole-doped cuprates resides in the nodal (FS arcs) region accompanied by the separate antinodal pseudogap. A corresponding multiband model of cuprate superconductivity is developed. It is based on the electronic spectrum evolving with doping and extends authors earlier approaches. The leading pair-transfer interaction is supposed between the itinerant (mainly oxygen) band and a nodal defect (polaron) band created by doping. These components are overlapping. The defect subband created in the antinodal region of the momentum space…
This work presents a quantitative theoretical study of the sound attenuation in the unconventional multiband superconductor Sr2RuO4 below the superconducting transition temperature Tc. Sound attenuation in this material is shown to have the remarkable property of being able to identify different nodal structures on different bands. The nodal structures on the γband on the one hand, and on the αand βbands on the other, are both found to be characterized by the existence of point nodes, but are significantly different in their quantitative aspects.
The emergence of the coherent quasiparticle peak and the development of the peak-dip-hump structure in the anti-nodal region below $T_{c}$ is the most prominent non-BCS signature of the underdoped high-$T_{c}$ cuprates, in which no coherent quasiparticle can be defined in the anti-nodal region above $T_{c}$. The peak-dip-hump structure has been commonly interpreted as the result of the coupling of the electron to some Bosonic mode. However, such an electron-Boson coupling picture does not answer the question of \textit{why the quasiparticle dispersion is so flat in the anti-nodal region}, a be…
In this work we study thermodynamic manifestations of the quantum criticality in multiband unconventional superconductors. As a guiding example we consider the scenario of magnetic quantum critical point in the model that captures superconductivity coexistence with the spin-density wave. We show that in situations when the superconducting order parameter has incidental nodes at isolated points, quantum magnetic fluctuations lead to the renormalization of the relative $T$-linear slope of the London penetration depth. This leads to the nonmonotonic dependence of the penetration depth as a functi…
In this letter, we have examined the superconducting ground state of the HfV$_2$Ga$_4$ compound using resistivity, magnetization, zero-field (ZF) and transverse-field (TF) muon-spin relaxation and rotation ($μ$SR) measurements. Resistivity and magnetization unveil the onset of bulk superconductivity with $T_{\bf c}\sim$ 3.9~K, while TF-$μ$SR measurements show that the temperature dependence of the superfluid density is well described by a nodal two-gap $s$+$d$-wave order parameter model. In addition, ZF muon relaxation rate increases with decreasing temperature below 4.6 K, indicating the pres…
Although the pairing mechanism of the Fe-based superconductors (FeSCs) has not yet been settled with a consensus, as to the pairing symmetry and the superconducting (SC) gap function, the abundant majority of experiments are supporting for the spin-singlet sign-changing s-wave SC gaps on multibands ($s^{\pm}$-wave state). This multiband $s^{\pm}$-wave state is a very unique gap state {\it per se} and displays numerous unexpected novel SC properties such as a strong reduction of the coherence peak, non-trivial impurity effects, nodal-gap-like nuclear magnetic resonance (NMR) signals, various Vo…
Multiband superconductivity, involving resonant pair scattering between different bands, has emerged as a possible explanation of some of the main characteristics of the recently discovered iron pnictides. A key feature of such interband pairing mechanism is that it can generate or enhance superconducting pairing irrespective of whether it is attractive or repulsive. The latter case typically leads to the superconducting gap switching its sign among different sections of the Fermi surface. In iron pnictides, the natural scenario is that the gap changes sign between the hole and the electron Fe…
We study the spin resonance in superconducting state of iron-based materials within multiband models with two unequal gaps, $Δ_L$ and $Δ_S$, on different Fermi surface pockets. We show that due to the indirect nature of the gap entering the spin susceptibility at the nesting wave vector $\mathbf{Q}$ the total gap $\tildeΔ$ in the bare susceptibility is determined by the sum of gaps on two different Fermi surface sheets connected by $\mathbf{Q}$. For the Fermi surface geometry characteristic to the most of iron pnictides and chalcogenides, the indirect gap is either $\tildeΔ= Δ_L + Δ_S$ or $\ti…
Thermal conductivity and specific heat were measured in the superconducting state of the heavy fermion material Ce_{1-x}La_{x}CoIn_{5}. With increasing impurity concentration x, the suppression of T_{c} is accompanied by the increase in the residual electronic specific heat expected of a d-wave superconductor, but it occurs in parallel with a decrease in residual electronic thermal conductivity. This contrasting behavior reveals the presence of uncondensed electrons coexisting with nodal quasiparticles. An extreme multiband scenario is proposed, with a d-wave superconducting gap on the heavy-e…
Fermi surface and topology of multiband superconductor BeAu
Leibniz Institute for Solid State and Materials Research (IFW) Dresden（德累斯顿莱布尼茨固体与材料研究所）；Würzburg-Dresden Cluster of Excellence ct.qmat（维尔茨堡-德累斯顿卓越集群 ct.qmat）；Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics（布达佩斯技术与经济大学物理学院理论物理系）；Regroupement Québécois sur les Matériaux de Pointe (RQMP)（魁北克尖端材料研究组）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）；Département de Physique et Institut Quantique, Université de Sherbrooke（舍布鲁克大学物理系与量子研究所）；Institute of Theoretical Physics, Technische Universität Dresden（德累斯顿工业大学理论物理研究所）
We develop the symmetry classification of superconducting gap functions in electron bands that do not transform under the crystal point group operations like the pure spin-1/2 states. The Bloch state bases in twofold degenerate bands with spin-orbit coupling are defined across the Brillouin zone in the way which satisfies the symmetry and continuity requirements. These bases are used to construct general multiband pairing Hamiltonians in centrosymmetric crystals. Focusing on single-band pairing, four exceptional cases are identified in which the triplet gap function does not transform under th…
Two topics of high current interest in the field of unconventional superconductivity are non-centrosymmetric superconductors and multiband superconductivity. Half-Heusler superconductors such as YPtBi exemplify both. In this paper, we study bulk and surface states in nodal superconducting phases of the half-Heusler compounds, belonging to the $A_1$ ($s+p$-like) and $T_2$ ($k_zk_x+ik_zk_y$-like) irreducible representations of the point group. These two phases preserve and break time-reversal symmetry, respectively. For the $A_1$ case, we find that flat surface bands persist in the multiband sys…
Superconductors involving electrons with internal degrees of freedom beyond spin can have internally anisotropic pairing states that are impossible in single-band superconductors. As a case in point, in even-parity multiband superconductors that break time-reversal symmetry, nodes of the superconducting gap are generically inflated into two-dimensional Bogoliubov Fermi surfaces. The detection and characterization of these quasiparticle Fermi surfaces requires the understanding of their experimental consequences. In this paper, we derive the low-energy density of states for a broad range of pos…
Thermal conductivity measurements were performed on single crystal samples of the superconducting filled skutterudite compounds PrOs$_4$Sb$_{12}$ and PrRu$_4$Sb$_{12}$ both as a function of temperature and magnetic field applied perpendicular to the heat current. In zero magnetic field, the low temperature electronic thermal conductivity of PrRu$_4$Sb$_{12}$ is vanishingly small, consistent with a fully-gapped Fermi surface. For PrOs$_4$Sb$_{12}$, however, we find clear evidence for residual electronic conduction as the temperature tends to zero Kelvin which is consistent with the presence of …
Sr$_2$RuO$_4$ has long been the focus of intense research interest because of conjectures that it is a correlated topological superconductor. It is the momentum space (k-space) structure of the superconducting energy gap $Δ_i(\mathbf{k})$ on each band $i$ that encodes its unknown superconducting order-parameter. But, because the energy scales are so low, it has never been possible to directly measure the $Δ_i(\mathbf{k})$ of Sr$_2$RuO$_4$. Here we implement Bogoliubov quasiparticle interference (BQPI) imaging, a technique capable of high-precision measurement of multiband $Δ_i(\mathbf{k})$. At…
Low-energy incommensurate (IC) magnetic fluctuations in the multiband superconductor Sr$_2$RuO$_4$ is investigated by high-resolution inelastic neutron scattering measurements and random phase approximation (RPA) calculations. Below $T_\text{c}$, the substantial spin gap is observed at $\mathbf{Q}_\text{IC}=(0.3, 0.3, L)$ where the quasi-one-dimensional $α$ and $β$ sheets consisting of the Fermi surfaces are in good nesting conditions. $L$ modulated intensity of the low-energy IC magnetic fluctuations and our RPA calculations indicate that the superconducting gaps regarding the $α$ and $β$ she…
The anisotropy of the electron scattering rate and lifetime $Γ_\mathbf{p}=1/τ_\mathbf{p}$ observed by Angle Resolved Photoemission Spectroscopy (ARPES) is evaluated using $s$-$d$ Kondo-Zener exchange Hamiltonian used previously to describe superconducting properties of high-$T_c$ cuprates; for correlation between critical temperature $T_c$ and BCS coupling constant, for example. The performed qualitative analysis reveals that "cold spots" correspond to nodal regions of the superconducting phase where the superconducting gap is zero, because the exchange interaction is annulled. Vice versa, "ho…
The recent ARPES results for the imaginary part of the self-energy obtained on a number of HTSC bismuthates \cite{Kordyuk} are analyzed. By accepting the ''Fermi-Bose'' {\it division-procedure} of the self-energy into the Fermi-liquid and bosonic parts - which is proposed in \cite{Kordyuk}, one obtains very small bosonic coupling constant $λ_{B, Im} <0.2$. If this procedure would be correct then the standard Eliashberg theory makes any bosonic mechanism of pairing irrelevant. As a consequence we are confronted with a trilemma: (1) to abandon the ``Fermi-Bose'' division-procedure \cite{Kordyuk}…
It is commonly believed that in the absence of disorder or an external magnetic field, there are three possible types of superconducting excitation gaps: the gap is nodeless, it has point nodes, or it has line nodes. Here, we show that for an even-parity nodal superconducting state which spontaneously breaks time-reversal symmetry, the low-energy excitation spectrum generally does not belong to any of these categories, instead it has extended Bogoliubov Fermi surfaces. These Fermi surfaces can be visualized as two-dimensional surfaces generated by "inflating" point or line nodes into spheroids…
We study thermal transport in a two-dimensional system with coexisting $s$- or $d$-wave Superconducting (SC) and Spin Density Wave (SDW) orders. We analyse the nature of coexistence phase in a tight-binding square lattice with $\mathbf{Q}=(π,π)$ SDW ordering. The electronic thermal conductivity is computed within the framework of the Boltzmann kinetic theory, using Born approximation for the impurity scattering collision integral. We describe the influence of the Fermi surface (FS) topology, the competition between the SC and SDW order parameters, the presence or absence of zero energy excitat…
Iron-Based Superconductors: A Decade of Materials, Magnetism, and Mechanisms
Beijing Normal University（北京师范大学）；Renmin University of China（中国人民大学）；Fudan University（复旦大学）；Institute of Science Tokyo（东京科学大学）；Korea University（高丽大学）；Rice University（莱斯大学）
We have investigated the superconducting ground state of the newly discovered superconductor ThFeAsN with a tetragonal layered crystal structure using resistivity, magnetization, heat capacity and transverse-field (TF) muon-spin rotation ($μ$SR) measurements. Our resistivity and magnetization measurements reveal an onset of bulk superconductivity with $T_{\bf c}\sim$ 30 K. The heat capacity results show a very small anomaly, $Δ$C$_{ele}$$\sim$0.214 (J/mol-K) at $T_{\bf c}\sim$ 30 K and exhibits exponential behavior below $T_{\bf c}$, which fits better to two superconducting gaps rather than a …
