We theoretically study the induced odd-frequency pairing states in ballistic normal-metal/superconductor ͑N/S͒ junctions where a superconductor has even-frequency symmetry in the bulk and a normal-metal layer has an arbitrary length. Using the quasiclassical Green's function formalism, we demonstrate that, quite generally, the pair amplitude in the junction has an admixture of an odd-frequency component due to the breakdown of translational invariance near the N/S interface where the pair potential acquires spatial dependence. If a superconductor has an even-parity pair potential ͑spin-singlet s-wave or spin-singlet d xy -wave state͒, the odd-frequency pairing component with odd parity is induced near the N/S interface, while in the case of an odd-parity pair potential ͑spin-triplet p x wave͒ the odd-frequency component with even parity is generated. We show that in conventional s-wave junctions, the amplitude of the odd-frequency pairing state is strongest in the case of a full-transparency N/S interface and is enhanced at energies corresponding to the peaks in the local density of states ͑LDOS͒. In p x -and d xy -wave junctions, the amplitude of the odd-frequency component on the S side of the N/S interface is enhanced at zero energy where the midgap Andreev resonant state ͑MARS͒ appears due to the sign change of the pair potential. The odd-frequency component extends into the N region and exceeds the even-frequency component at energies corresponding to the LDOS peak positions, including the MARS. At the edge of the N region the odd-frequency component is nonzero while the even-frequency one vanishes. We show that the concept of the odd-frequency pairing state plays a pivotal role to interpret a number of phenomena in nonuniform superconducting systems, like McMillan-Rowell and midgap Andreev resonance states.

The spatial variations of the local density of states ͑LDOS͒ on the surface of d x 2 Ϫy 2-wave superconductors are calculated using an extended Hubbard model. Our model has the ability to treat the surface geometry with atomic-scale roughness, and gives realistic results beyond those based on the quasiclassical theory. The spatial variation of the LDOS is obtained for various types of surface geometries using a self-consistently determined pair potential. As surface bound states are peculiar to d-wave superconductors, we pay attention to the appearance of zero-energy states ͑ZES's͒. For a flat surface, the formation of a ZES on a ͓110͔ surface and the lack of one on a ͓100͔ surface is obtained, which is consistent with the results of the quasiclassical theory. However, the LDOS shows abundant spatial dependence when the surfaces have roughness. The appearance of a ZES on nominal ͓100͔ surfaces and its disappearance on nominal ͓110͔ surfaces are obtained depending on geometry. The calculations explain the origin of various spectra observed by tunneling spectroscopy on high-T c superconductors. ͓S0163-1829͑98͒04113-7͔

Tunneling conductance spectra of normal metal/insulator/triplet superconductor junctions are investigated theoretically. As triplet paring states we select several types of symmetries that are promising candidates for the superconducting states in UPt 3 and in Sr 2 RuO 4 . The calculated conductance spectra are sensitive to the orientation of the junction which reflects the anisotropy of the pairing states. They show either zero-bias conductance peaks or gap-like structures depending on the orientation of the junctions. The existence of a residual density of states, peculiar to nonunitary states, is shown to have a significant influence on the properties of the conductance spectra. Present results serve as a guide for the experimental determination of the symmetry of the pair potentials in UPt 3 and Sr 2 RuO 4 .KEYWORDS: triplet superconductor, nonunitary pair potential, zero-bias conductance peak, §1. IntroductionSince the discovery of superconductivity in heavy fermion compounds the determination of the symmetry of the pair potential in these materials has been an important issue. 1) Among the heavy fermion superconductors, so far properties of UPt 3 have been studied most extensively. Based on NMR experiments 2) the possibility of odd parity pairing states, i.e., triplet pairing states, has been suggested. Theoretically, several papers propose two-dimensional even parity states, i.e. singlet pairing states, 3,4,5) while others suggest triplet pairing states belonging to the onedimensional representation (A u ) 6, 7) and to the two-dimensional representation (E u ). 8, 9) Although theoretical 10,11,12) and experimental 13,14,15,16,17) spectroscopic studies of the thermal conduction and transverse sound attenuation of UPt 3 have been performed, the symmetry of the pair potentialcould not yet been determined.Recently superconductivity has been discovered in Sr 2 RuO 4 , 18) which is the first example of a noncuprate layered perovskite superconductor. Since this compound is isostructual to the cuprate superconductors the electronic properties in the normal state 19) Phase-sensitive measurements provide the most useful information for the determination of the symmetry of the pair potentials. 27) Recently it was shown that tunneling spectroscopy has the ability to detect the phase of the pair potential, 28,29,30) as follows: In anisotropic superconductors quasiparticles feel different signs of the pair potentials depending on the directions of their motions. 31) At the normal metal/superconductor interface the anisotropy of the pair potential significantly influences the properties of the Andreev reflections. 32, 33) As a result tunneling conductance spectra of the normal metal/insulator/anisotropic superconductor junctions are modified due to the anisotropy of pair potential. 28,29) The most remarkable feature is the existence of zerobias conductance peaks (ZBCP) in the tunneling spectra for d-wave symmetry. The origin of these peaks is the localized zero energy states (ZES) 34) due to the change of sign of the p...

The local density of states is studied theoretically in terms of the odd-frequency (odd-omega) Cooper pairing induced around a vortex core. We find that a zero energy peak in the density of states at the vortex center is robust against nonmagnetic impurities in a chiral p-wave superconductor owing to an odd-omega s-wave pair amplitude. We suggest how to discriminate a spin-triplet pairing symmetry and spatial chiral-domain structure by scanning tunneling spectroscopy via odd-omega pair amplitudes inside vortex cores.

Spatial dependencies of the pair potential and the local density of states near the surfaces of d x 2 −y 2 -wave superconductors are studied theoretically. The calculation is based on the t-J model within a mean-field theory with Gutzwiller approximation. Various types of surface geometries are considered. Similar to our result in the extended Hubbard model, it is found that the formation of zero-energy states strongly depends on the surface geometry. In addition to this feature, the zero-energy states give peak splitting for the (110) surfaces when the superexchange interaction J is large. This is due to the induced s-wave component near the surface.The present result explains the microscopic origin of the spontaneous time-reversal symmetry breaking at the surfaces of high-T c superconductors.KEYWORDS: t-J model, d-wave superconductor, Gutzwiller approximation, scanning tunneling spectroscopy, zero-energy peaks, spontaneous time-reversal symmetry breakingThe t-J model 1) is believed to explain the phase diagram of high-T c materials including the socalled pseudogap above T c and thus is a realistic model for high-T c superconductors. 2) Although there are many works regarding this model, the quasiparticle properties in nonuniform systems including surfaces or interfaces are not clear at this stage except for those in the case with a vortex. 3) Since d x 2 −y 2 -wave pair potential is the most promising symmetry in the t-J model for actual doping concentrations, we expect an interference effect of the quasiparticle due to the signchange of the pair potential 4) at the surface.Recent theoretical works within quasiclassical approximation revealed a novel interference effect peculiar to the unconventional superconductivity. 5,6,7,8,9,10,11) One of the interesting interference effects is the appearance of zero-energy states (ZES) at the surface, which is due to the sign-change of pair potential when the quasiparticles are reflected at the surface. 7) This localized ZES manifests itself as a zero-bias conductance peak (ZBCP) in tunneling experiments. 5, 12) A recent experiment 1