We discuss a strong relationship between Majorana fermions and odd-frequency Cooper pairs which appear at a disordered normal (N) nanowire attached to a topologically nontrivial superconducting (S) one. The transport properties in superconducting nanowire junctions show universal behaviors irrespective of the degree of disorder: the quantized zero-bias differential conductance at 2e 2 /h in NS junction and the fractional current-phase (J -ϕ) relationship of the Josephson effect in SNS junction J ∝ sin(ϕ/2). Such behaviors are exactly the same as those found in the anomalous proximity effect of odd-parity spin-triplet superconductors. We show that the odd-frequency pairs exist wherever the Majorana fermions stay.

Motivated by a recent experiment [Keizer et al., Nature (London) 439, 825 (2006)], we study the Josephson effect in superconductor/diffusive half metal/superconductor junctions using the recursive Green function method. The spin-flip scattering at the junction interfaces opens the Josephson channel of the odd-frequency spin-triplet Cooper pairs. As a consequence, the local density of states in a half metal has a large peak at the Fermi energy. Therefore the odd-frequency pairs can be detected experimentally by using the scanning tunneling spectroscopy.

We present a theory of tunneling spectroscopy for normal metal/Larkin-Ovchinnikov state junctions in which the spatial periodic modulation in the pair potential amplitude is taken into account. The tunneling spectra show the characteristic line shapes reflecting the minigap structures under the periodic pair potentials depending on the boundary condition of the pair potentials at the junction interface. These features are qualitatively different from the tunneling spectra of the Fulde-Ferrell state. We propose an experimental setup which identifies the superconducting state of CeCoIn5.

We discuss the appearance of odd-frequency Cooper pairs in two-band superconductors by solving the Gor'kov equation analytically. We introduce the equal-time s-wave pair potentials as realized in MgB2 and iron pnictides. Although the order parameter symmetry is conventional, the band degree of freedom enriches the symmetry variety of pairing correlations. The hybridization and the asymmetry between the two conduction bands induce odd-frequency pairs as a subdominant pairing correlation in the uniform ground state. To study the magnetic response of odd-frequency Cooper pairs, we analyze the Meissner kernel represented by the Gor'kov Green function. In contrast to the even-frequency pairs linked to the pair potential, the induced odd-frequency Cooper pairs indicate a paramagnetic property. We also discuss the relation between the amplitude of the odd-frequency pairing correlation and the stability of superconducting states in terms of the self-consistent equation for the pair potential.

We study Josephson current in superconductor/diffusive ferromagnet/superconductor junctions by using the recursive Green function method. When the exchange potential in a ferromagnet is sufficiently large compared to the pair potential in a superconductor, an ensemble average of Josephson current is much smaller than its mesoscopic fluctuations. The Josephson current vanishes when the exchange potential is extremely large so that a ferromagnet is half-metallic. Spin-flip scattering at junction interfaces drastically changes the characteristic behavior of Josephson current. In addition to spin-singlet Cooper pairs, equal-spin triplet pairs penetrate into a half metal. Such equal-spin pairs have an unusual symmetry property called odd-frequency symmetry and carry the Josephson current through a half metal. The penetration of odd-frequency pairs into a half metal enhances the low energy quasiparticle density of states, which could be detected experimentally by scanning tunneling spectroscopy. We will also show that odd-frequency pairs in a half metal cause a nonmonotonic temperature dependence of the critical Josephson current.

A formula for the Josephson current between two superconductors with anisotropic pairing symmetries is derived based on the mean-field theory of superconductivity. Zero-energy states formed at the junction interfaces is one of basic phenomena in anisotropic superconductor junctions. In the obtained formula, effects of the zero-energy states on the Josephson current are taken into account through the Andreev reflection coefficients of a quasiparticle. In low temperature regimes, the formula can describe an anomaly in the Josephson current which is a direct consequence of the exsitence of zero-energy states. It is possible to apply the formula to junctions consist of superconductors with spin-singlet Cooper pairs and those with spin-triplet Cooper pairs.

The Josephson effect in p-wave superconductor/diffusive normal metal/p-wave superconductor junctions is studied theoretically. Amplitudes of Josephson currents are several orders of magnitude larger than those in s-wave junctions. Current-phase (J-') relations in low temperatures are close to those in ballistic junctions such as J / sin '=2 and J / ' even in the presence of random impurity potentials. A cooperative effect between the midgap Andreev resonant states and the proximity effect causes such anomalous properties and is a character of the spin-triplet superconductor junctions. DOI: 10.1103/PhysRevLett.96.097007 PACS numbers: 74.50.+r, 74.25.Fy, 74.70.Tx The internal -phase shift (sign change) of pair potentials is essential for unconventional superconductivity and is the source of the midgap Andreev resonant state (MARS) [1][2][3]. It is now known that the MARS is responsible for anomalous transport properties in superconducting junctions [4]. In normal metal/superconductor junctions, transport properties are affected also by the proximity effect which is interpreted in terms of diffusion of Cooper pairs into normal metals. In what follows, we assume that normal metals are in the diffusive transport regime due to impurity scatterings. Recent theoretical studies have revealed sensitivity of the proximity effect to the internal phase of pair potentials [5,6]. In normal metals attached to unconventional superconductors, Cooper pairs have a sign degree of freedom reflecting the -phase shift of pair potentials. Suppression of the proximity effect is usually expected because the wave function of a Cooper pair originated from the positive part of pair potentials cancel that originated from the negative part [5,6]. Two of us, however, discussed anomalous enhancement of the zero-bias tunneling conductance due to the proximity effect in a presence of the MARS [7,8].In superconductor/normal metal/superconductor junctions, another phase degree of freedom affects quantum transport. Namely, the external phase difference across the junctions ' drives Josephson currents. An importance of studying the Josephson effect is growing these days because quantum interference devices consisting of Josephson junctions can be basis of future technologies. In fact, a recent experiment has tried to apply high-T c superconductors to coherent devices [9]. In unconventional junctions, the MARS is considered to have the phase degree of freedom. When MARSs are formed at the two junction interfaces, the external phase may modify interference effects between the two MARSs and Josephson currents. The research in this direction can shed new light on quantum transport in unconventional superconductors.In this Letter, we theoretically study Josephson currents between two p-wave superconductors through normal metals by solving the Bogoliubov-de Gennes equation using the recursive Green function method [10,11]. We show that amplitudes of Josephson currents in the p-wave junctions are much larger than those in the s-wave junctions when transm...

We propose a novel experiment to identify the symmetry of superconductivity on the basis of theoretical results for differential conductance of a normal metal connected to a superconductor. The proximity effect from the superconductor modifies the conductance of the remote current depending remarkably on the pairing symmetry: spin singlet or spin triplet. The clear-cut difference in the conductance is explained by symmetry of Cooper pairs in a normal metal with respect to frequency. In the spin-triplet case, the anomalous transport is realized due to an odd-frequency symmetry of Cooper pairs.