We derive the current response to the linearly polarized electromagnetic field with finite frequency and wave vector incident normally on the specular surface of a clean nonconventional superconductor with orbital spontaneous magnetization parallel to the crystal axis and perpendicular to the crystal surface. The result includes the usual part known from the theory of conventional superconductivity and as well the magneto-optical term typical for the superconductors with spontaneous time reversal breaking. As an application of the basic current-field relation we consider the Kerr effect for the rotation of polarization of infrared light reflected from the superconductor surface.
UBe13 was the second discovered heavy fermion superconductor, and numerous pieces of evidence exist that imply that it is an unconventional (non-BCS, non-s wave pairing symmetry) superconductor. Exhibiting even more signs of unconventional superconductivity, Th-doped UBe13 is perhaps the most puzzling of any of the unconventional superconductors. This review considers both the parent, undoped compound as well as the more interesting Th-doped UBe13. After summarizing the rather thorough characterization, which because of the interest in these compounds has continued from their discovery in 1983…
We extend the circuit theory of superconductivity to cover transport and proximity effect in mesoscopic systems that contain unconventional superconductor junctions. The approach fully accounts for zero-energy Andreev bound states forming at the surface of unconventional superconductors. As a simple application, we investigate the transport properties of a diffusive normal metal in series with a d-wave superconductor junction. We reveal the competition between the formation of Andreev bound states and proximity effect, that depends on the crystal orientation of the junction interface.
Diamagnetically coupled magnetic field can be used as a probe of nodal positions in unconventional superconductors. The heat capacity depends on the angle $ϕ_0$ between the magnetic field and the nodal directions. We show that the anisotropy $C(ϕ_0)$ persists even in systems with strong paramagnetic coupling to the electrons's spins.
We report the observation of half-integer magnetic flux quantization in mesoscopic rings of superconducting $β$-Bi$_2$Pd thin films. The half-quantum fluxoid manifests itself as a $π$ phase shift in the quantum oscillation of the critical temperature. This result verifies unconventional superconductivity of $β$-Bi$_2$Pd, in accord with the expectation of a topological superconductor. We also discuss the strong indication that $β$-Bi$_2$Pd is a spin-triplet superconductor.
We have derived new boundary conditions on wave function at the normal metal / superconductor (NS) interface beyond effective mass approximation. These conditions are based on tight-binding approach and enable one to formulate quantitative model for tunneling spectroscopy of superconductors with complex non-parabolic energy spectra. The model is applied to superconductors with unconventional pairing and with multiband electronic structure. In the case of single band unconventional superconductors this model provides known conductance formula (Phys. Rev. Lett. 74 3451 1995), but with generalize…
The discovery of new families of unconventional superconductors is important both experimentally and theoretically, especially if it challenges current models and thinking. By using muon spin relaxation in zero-field, time-reversal symmetry breaking has been observed in Re6Hf. Moreover, the temperature dependence of the superfluid density exhibits s-wave superconductivity with an enhanced electron-phonon coupling. This, coupled with the results from isostructural Re6Zr, shows that the Re6X family are indeed a new and important group of unconventional superconductors.
We study tunneling in ferromagnet/unconventional superconductor (F/S) junctions. We include the effects of spin polarization, interfacial resistance, and Fermi wavevector mismatch (FWM) between the F and S regions. Andreev reflection (AR) at the F/S interface, governing tunneling at low bias voltage, is strongly modified by these parameters. The conductance exhibits a very wide variety of features as a function of applied voltage.
Nine years ago, superconductors based on the magnetic element iron were discovered. A flurry of research activity has revealed an unprecedented diversity of chemical structures and physical properties. Similarly to other unconventional superconductors, stripe-type antiferromagnetism seems to play an important role. Particularly interesting is its strong coupling to the crystal lattice. The systematic comparison of different compounds may allow to identify key elements in the mechanism of superconductivity and provide guidelines for the search for new and better superconductors.
The elementary vortex pinning potential is studied in unconventional superconductors within the framework of the quasiclassical theory of superconductivity. Numerical results are presented for d-, anisotropic s-, and isotropic s-wave superconductors to show explicitly that in unconventional superconductors the vortex pinning potential is determined mainly by the loss of the condensation energy in bulk due to the presence of the pinning center, i.e., by the breakdown of the Anderson's theorem. It is found that the vortex pinning energy in the d-wave pairing case is 4 -- 13 times larger than tho…
The first integrals of the Ginzburg-Landau equations for a vortex-free state of superconductors with different mixed symmetries of the order parameter are found. The general boundary conditions for the order parameter at the ideal interface between the superconductor and vacuum are derived. Based on these integrals and boundary conditions, we analyze the stability criteria for vortex-free state in unconventional superconductors. The threshold field above which the Abrikosov vortices can enter the superconductor is found to be higher or equal to the thermodynamic critical field for all states u…
In this research, we derive a simple expression for the ratio of normal state superconducting state of spin lattice relaxation rate of unconventional superconductors from the BCS weak-coupling equation. The unconventional superconductors we consider have three types of order parameters as d-wave, 3He A-phase and p-wave 3-dimensional order parameter that had been done before by Parker and Haas[7].After using some numerical approximations and some boundary conditions, we can find the ratio of normal state to superconducting state of spin lattice relaxation rate in power series of temperature dep…
Using Ginzburg-Landau formalism, we theoretically study the isothermal shear modulus anomaly of an unconventional superconductor with a multicomponent order parameter, when the superconducting transition is split by a symmetry-breaking field. Experimental signatures are proposed for both chiral and nematic superconductors. Particularly striking is the vanishing of $C_{66}$ across the lower transition to a nematic superconducting state. Our findings can guide future experiments and shed new lights on materials such as Sr$_2$RuO$_4$ and $M_x$Bi$_2$Se$_3$.
We comment on a recent letter by Schopohl and Dolgov (SD) which raised the possibility that a d-wave superconductor might be thermodynamically unstable in the limit $T\to 0$. The point of our comment is that this instability can be inevitably prevented by the non-local effects.
We report the sulfur isotope effect on transition temperature in a BiS2-based superconductor Bi4O4S3. Polycrystalline samples of Bi4O4S3 were prepared using 32S and 34S isotope chemicals. From magnetization analyses, the isotope exponent (aS) was estimated as -0.1 < aS < 0.1. Although the Tc estimated from electrical resistivity was scattered as compared to those estimated from the magnetization, we observed no clear correlation between Tc and the isotope mass. The present results suggest that unconventional paring states are essential in Bi4O4S3.
Cooper pairs in two-dimensional unconventional superconductors with broken inversion symmetry are in a mixture of an even-parity spin-singlet pairing state with an odd-parity spin-triplet pairing state. We study the magnetic properties of the impurity bound states in such superconductors and find striking signatures in their spin polarization which allow to unambiguously discriminate a non-topological superconducting phase from a topological one. Moreover, we show how these properties, which could be measured using spin-polarized scanning tunneling microscopy (STM), also enable to determine th…
We study the influence of non-magnetic impurity scatterings on the tunneling conductance of a junction consisting of a normal metal and a disordered unconventional superconductor by solving the quasiclassical Eilenberger equation self-consistently. We find that the impurity scatterings in both the Born and unitary limits affect the formation of the Andreev bound states and modify strongly the tunneling spectra around zero bias. Our results are interpreted well by the appearance of odd-frequency Cooper pairs near the interface and by the divergent behavior of the impurity self-energy. The prese…
The optical conductivity measurements give a powerful tool to investigate the nature of the superconducting gap for conventional and unconventional superconductors. In this article, first, general analyses of the optical conductivity are given stemmed from the Mattis-Bardeen formula for conventional BCS superconductors to unconventional anisotropic superconductors. Second, we discuss the reflectance-transmittance (R-T) method which has been proposed to measure far-infrared spectroscopy. The R-T method provides us precise measurements of the frequency-dependent conductivity. Third, the optical …
The effect of weak anisotropic (momentum-dependent) impurity scattering in unconventional superconductors has been investigated. It is shown that the anisotropic scattering can lead either to a small reduction or a small enhancement of the isotropic pair-breaking effect. The influence of the anisotropy of the scattering potential becomes significant for the order parameters with large Fermi surface average values. In that case an unexpected enhancement (up to 10%) of the critical temperature over the critical temperature in the absence of impurities is predicted for a small impurity concentrat…
This topical review summarizes various features of magnetic penetration depth in unconventional superconductors. Precise measurements of the penetration depth as a function of temperature, magnetic field and crystal orientation can provide detailed information about the pairing state. Examples are given of unconventional pairing in hole- and electron-doped cuprates, organic and heavy fermion superconductors. The ability to apply an external magnetic field adds a new dimension to penetration depth measurements. We discuss how field dependent measurements can be used to study surface Andreev bou…
It is usually believed that a coherence peak just below T$_{c}$ in the nuclear spin lattice relaxation rate T$_{1}^{-1}$ in superconducting materials is a signature of conventional s-wave pairing. In this paper we demonstrate that any unconventional superconductor obeying BCS pure-case weak-coupling theory should show a small T$_{1}^{-1}$ coherence peak near T$_{c}$, generally with a height between 3 and 15 percent greater than the normal state T$_{1}^{-1}$ at T$_{c}$. It is largely due to impurity effects that this peak has not commonly been observed.
We investigate equilibrium flux lattice structures in superconductors with unconventional order parameters, such as high-$T_c$ cuprates, using a generalized London model with non-local electrodynamics derived from a simple microscopic model. We find a rich phase diagram containing triangular, centered rectangular and square lattices with various orientations to the ionic lattice, as a function of magnetic field and temperature.
Unconventional mixed state in the nematic superconductor LiFeAs
CNR-SPIN（意大利国家研究委员会自旋物理研究所）；Division of Mathematics, University of Dundee（邓迪大学数学系）；Maxwell Institute, University of Edinburgh（爱丁堡大学麦克斯韦研究所）；School of Mathematics, University of Leeds（利兹大学数学学院）；Leibniz-Institute for Solid State and Materials Research, IFW-Dresden（德累斯顿IFW莱布尼茨固体与材料研究所）；PSI Center for Neutron and Muon Sciences CNM（瑞士保罗谢尔研究所中子与缪子科学中心）；Laboratorium für Festkörperphysik, ETH Zürich（苏黎世联邦理工学院凝聚态物理实验室）
We investigate the effects of Andreev bound states due to the unconventional pairing on the inverse proximity effect of ferromagnet/superconductor junctions. Utilizing quasiclassical Eilenberger theory, we obtain the magnetization penetrating into the superconductor. We show that in a wide parameter range the direction of the induced magnetization is determined by two factors: whether Andreev bound states are present at the junction interface and the sign of the spin-mixing angle. In particular, when Andreev bound states appear at the interface, the direction of the induced magnetization is op…
We study the effect of the superconducting gap nodes on the vortex lattice properties of high temperature superconductors at very low temperatures. The nonlinear, nonlocal and nonanalytic nature of this effect is shown to have measurable consequences for the vortex lattice geometry and the effective penetration depth in the mixed state as measured by muon-spin-rotation experiments.
Vortex matter in mesoscopic superconductors is known to be strongly affected by the geometry of the sample. Here we show that in nanoscale superconductors with coherence length comparable to the Fermi wavelength the shape resonances of the order parameter results in an additional contribution to the quantum topological confinement - leading to unconventional vortex configurations. Our Bogoliubov-de Gennes calculations in a square geometry reveal a plethora of asymmetric, giant multi-vortex, and vortex-antivortex structures, stable over a wide range of parameters and which are very different fr…
Higgs spectroscopy is a new field in which Higgs modes in nonequilibrium superconductors are analyzed to gain information about the ground state. One experimental setup in which the Higgs mode in s-wave superconductors was observed is periodic driving with THz light, which shows resonances in the third-harmonic generation (THG) signal if twice the driving frequency matches the energy of the Higgs mode. We derive expressions of the driven gap oscillations for arbitrary gap symmetry and calculate the THG response. We demonstrate that the possible Higgs modes for superconductors with non-trivial …
We study the renormalization of sound attenuation and sound velocity by fluctuation Cooper pairs in layered superconductors. We consider the influence of s- and d-wave symmetry of the fluctuating order parameter, on both longitudinal and transverse phonon modes. We show that both unconventional order parameter symmetry and transverse sound polarization suppress the AL and MT terms, while the DOS contribution is the least affected. The combination of these effects can change the sign of the overall fluctuation corrections above Tc. We also compare the results obtained using the Ginzburg-Landau …
We investigate the field-angle-dependent zero-energy density of states for YNi2B2C with using realistic Fermi surfaces obtained by band calculations. Both the 17th and 18th bands are taken into account. For calculating the oscillating density of states, we adopt the Kramer-Pesch approximation, which is found to improve accuracy in the oscillation amplitude. We show that superconducting gap structure determined by analyzing STM experiments is consistent with thermal transport and heat capacity measurements.
We study large arrays of mesoscopic junctions between gapless superconductors where the tunneling processes of both, particle-hole and Cooper, pairs give rise to a strongly retarded effective action which, contrary to the standard case, can not be readily characterized in terms of a local Josephson energy. This complexity is expected to arise in, e.g., the grain boundary and c-axis junctions in layered high-T_c superconductors. A new representation for describing collective phenomena in this system is introduced, and its phase diagram is discussed, alongside the electrical conductivity.
We show that the clean superconductor with line of gap nodes is not in conflict with the Nernst theorem. The answer to the question in the title of the Schopohl-Dolgov paper in Phys. Rev. Lett. 80 (1998) 4761 (cond-mat/9802264) is yes.
We theoretically discuss the magnetic-field-angle dependence of the zero-energy density of states (ZEDOS) in superconductors. Point-node and line-node superconducting gaps on spherical and cylindrical Fermi surfaces are considered. The Doppler-shift (DS) method and the Kramer-Pesch approximation (KPA) are used to calculate the ZEDOS. Numerical results show that consequences of the DS method are corrected by the KPA.
We present a general theory of the proximity effect in junctions between diffusive normal metals (DN) and unconventional superconductors in the framework of the quasiclassical Green's function formalism. Various possible symmetry classes in a superconductor are considered which are consistent with the Pauli principle: even-frequency spin-singlet even-parity (ESE) state, even-frequency spin-triplet odd-parity (ETO) state, odd-frequency spin-triplet even-parity (OTE) state and odd-frequency spin-singlet odd-parity (OSO) state. For each of the above four cases symmetry and spectral properties of …
For certain orientations of Josephson junctions between two p_x-wave or two d-wave superconductors, the subgap Andreev bound states produce a 4pi-periodic relation between the Josephson current I and the phase difference phi: I ~ sin(phi/2). Consequently, the ac Josephson current has the fractional frequency eV/h, where V is the dc voltage. In the tunneling limit, the Josephson current is proportional to the first power (not square) of the electron tunneling amplitude. Thus, the Josephson current between unconventional superconductors is carried by single electrons, rather than by Cooper pairs…
In this article, we wish to give a historical and personal account of the advent of our tricrystal phase-sensitive symmetry experiments using scanning SQUID microscope for establishing d-wave pairing symmetry in the cuprate superconductors. On the occasion of commemorating the 100th anniversary of the discovery of superconductivity by Kamerlingh Onnes, we are honored to present this article as part of the Jubilee Celebration.
We perform large-scale numerical calculations self-consistently solving the Bogoliubov-de Gennes (BdG) equations in the magnetic field together with random impurities to directly demonstrate the typical quasi-particle interference (QPI) in the presence of vortices as observed by scanning tunneling microscopy/spectroscopy experiments in unconventional superconductors. The calculations reveal that vortex itself never works as a scatter causing the QPI pattern but vortex core containing impurity brings about the enhancement of the sign-preserving QPI peaks. Its origin is Andreev bound-states dist…
We consider a thin-film normal metal/superconductor junction in the presence of an externally applied in-plane magnetic field for several symmetries of the superconducting order parameter. For p-wave superconductors, a strongly spin-polarized current emerges due to an interplay between the nodal structure of the superconducting order parameter, the existence or non-existence of zero-energy surface states, and the Zeeman-splitting of the bands which form superconductivity. Thus, the polarization depends strongly on the orbital symmetry of the superconducting state. Our findings suggest a mechan…
Antiferromagnetic (AFM) (or spin-density wave) quantum critical fluctuation enhanced just below H_c2(0) have been often observed in d-wave superconductors with a strong Pauli paramagnetic depairing (PD) including CeCoIn_5. It is shown here that such a tendency of field-induced AFM ordering is a consequence of strong PD and should appear particularly in superconductors with a gap node along the AFM modulation. Two phenomena seen in CeCoIn_5, the anomalous vortex lattice form factor and the AFM order in the Fulde-Ferrell-Larkin-Ovchinnikov state, are explained based on this peculiar PD effect.
On the basis of the quasiclassical theory of superconductivity, we obtain a formula for the local density of states (LDOS) around a vortex core of superconductors with anisotropic pair-potential and Fermi surface in arbitrary directions of magnetic fields. Earlier results on the LDOS of d-wave superconductors and NbSe$_2$ are naturally interpreted within our theory geometrically; the region with high intensity of the LDOS observed in numerical calculations turns out to the enveloping curve of the trajectory of Andreev bound states. We discuss experimental results on YNi$_2$B$_2$C within the qu…
We expose an analysis of the magnetic field distribution in the Abrikosov lattice of high-kappa superconductors with d-wave pairing in the case where the critical field is mainly determined by the Pauli limit and the superfluid currents mainly come from the paramagnetic interaction of electron spins with the local magnetic field. The result found in frame of the generalized Clem variational approach is compatible with the recent observation that the form factor in CeCoIn_5 increases with increasing field and then decreases at the approach of H_c2. (A. D. Bianchi et al., Science vol. 319, 177 (…
For the iron-based high $T_c$ superconductor LaFeAsO$_{1-x}$F$_x$, we construct a minimal model, where all of the five Fe $d$ bands turn out to be involved. We then investigate the origin of superconductivity with a five-band random-phase approximation by solving the Eliashberg equation. We conclude that the spin fluctuation modes arising from the nesting between the disconnected Fermi pockets realise, basically, an extended s-wave pairing, where the gap changes sign across the nesting vector.
We present analytic and numerical results for the thermoelectric effect in unconventional superconductors with a dilute random distribution of impurities, each scattering isotropically but with a phase shift intermediate between the Born and unitary limits. The thermoelectric response function has a linear temperature dependence at low temperatures, with a slope that depends on the impurity concentration and phase shift. Although the thermoelectric effect vanishes identically in the strict Born and unitary limits, even a small deviation of the phase shift from these limits leads to a large res…
We investigate the electronic transport coefficients in unconventional superconductors at low temperatures, where charge and heat transport are dominated by electron scattering from random lattice defects. We discuss the features of the pairing symmetry, Fermi surface, and excitation spectrum which are reflected in the low temperature heat transport. For temperatures $k_B T \la γ\ll Δ_0$, where $γ$ is the bandwidth of impurity induced Andreev states, certain eigenvalues become {\it universal}, i.e., independent of the impurity concentration and phase shift. Deep in the superconducting phase ($…
A theory of the upper critical field in a trigonal superconductor with a two-component order-parameter is developed. We demonstrate the existence of sixfold modulations of the upper critical field in the basal plane. The form of the angular dependence of H_c2 in the plane of the main symmetry axis is studied in the whole range of phenomenological rigidity coefficients. A qualitative dependence of these coefficients on the form of the Fermi surface is established. The results obtained are discussed in connection with the measurements of the direction dependence of H_c2 in the heavy fermion supe…
Orientational field-dependence of the zero energy density of states (ZEDOS) is calculated for superconductors with the polar state (line node), axial state (point node) and 3D d-wave state. Depending on the gap topology and relative field direction the field dependencies of ZEDOS sensitively differ, providing us a useful and practical method to identify the gap topology. It is also demonstrated that for d-wave state the field rotation in the basal plane shows a sizable oscillation ~3% of ZEDOS. This is directly measurable in low-T specific heat experiment in the mixed state.
Effect of surface pair breaking, entirely neglected by M. Samanta and S. Datta [Phys. Rev. B v. 57, 10972 (1998)], is quite important in considering surface (or interface) quasiparticle bound states and associated characteristics of junctions involving unconventional superconductors. The whole class of bound states with nonzero energy is simply omitted within the framework of the approach, using uniform spatial profile of the order parameter up to the interface. The contribution of these bound states (as well as midgap states) to current-voltage characteristics of the SIS' tunnel junctions wer…
We derive the Ginzburg-Landau theory of unconventional singlet superconductors in the presence of a Zeeman field and impurities, in order to examine the resulting Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases. We show that the behavior of the FFLO phases in unconventional superconductors in the presence of impurities is qualitatively different from that found for s-wave superconductors.
We calculate the wave-vector dependent electronic spin susceptibility $χ_{αβ}(\mathbf q, \mathbf H_0)$ of a d-wave superconductor in uniform magnetic field $\mathbf{H}_0$ with Pauli pair-breaking. We find that the transverse component of the susceptibility tensor can be greater than its normal state value; the longitudinal component also slightly increases but in a very limited range of $q$s. We identify several wave vectors $\{\mathbf q_\perp,\mathbf q_\parallel\}$, that correspond to the maxima of either $χ_\perp$ or $χ_\parallel$. We compare our results with available data on the high-field…
We present the results of a study of the vortex lattice (VL) of the nickel chalcogenide superconductor TlNi2Se2, using small angle neutron scattering. This superconductor has the same crystal symmetry as the iron arsenide materials. Previous work points to it being a two-gap superconductor, with an unknown pairing mechanism. No structural transitions in the vortex lattice are seen in the phase diagram, arguing against d-wave gap symmetry. Empirical fits of the temperature-dependence of the form factor and penetration depth rule out a simple s-wave model, supporting the presence of nodes in the…
It has long been a challenge to describe the origin of unconventional superconductivity. The two known examples with high Tc, based on iron and copper, have very different electronic structures, while other materials with similar electronic structure may not show superconductivity at all. In this paper, the authors show that by using high accuracy diffusion Monte Carlo calculations, the unconventional superconductors of both high Tc types form a cluster at intermediate spin-charge coupling. The spin-charge coupling may serve as a normal state marker for unconventional superconductivity, and pr…
Recent experiments have shown rotation of the plane of polarization of light reflected from the surface of some superconductors. This indicates that time reversal and certain mirror symmetries are broken in the ordered phase. The photon energy exceeds the electronic bandwidth, so that completely filled or completely empty bands must play a role. We show that in strong-coupling theory a Coulomb interaction may produce an order parameter in the unoccupied band that explains the observations. The theory puts tight constraints on the form of the order parameter in different bands. We propose that …
We demonstrate that superconductors which break time-reversal symmetry can exhibit thermoelectric properties, which are entirely different from the Ginzburg mechanism. As an example, we show that in the $s+is$ superconducting state there is a reversible contribution to thermally induced supercurrent, whose direction is not invariant under time-reversal operation. Moreover in contrast to Ginzburg's mechanism it has a singular behavior near the time-reversal symmetry breaking phase transition. The effect can be used to confirm or rule out the $s+is$ state, which is widely expected to be realized…
We will extend the concept of electron band Berry curvatures to superconducting materials. We show that this can be defined for the Bogoliubov-de Gennes equation describing the superconducting state in a periodic crystal. In addition, the concept is exploited to understand the driving mechanism for the optical Kerr effect in time reversal symmetry breaking superconductors. Finally, we establish a sum rule analogue to the normal state Hall sum rule making quantitative contact between the imaginary part of the optical conductivity and the Berry curvature. The general theory will be applied and t…
In the framework of London theory we study the novel magnetic state in two-component superconductors with finite density of fractional flux vortices stabilized near the surface. We show that the process of vortex entry into the two-component superconductor consists of several steps, while the external magnetic field increases from zero. At the first stage only vortices in one of the order parameter components penetrate and sit at the equilibrium position near the surface. When the magnetic field is increased further vortices in the second order parameter component eventually enter the supercon…
