### Abstract

We theoretically study the induced odd-frequency pairing states in ballistic normal-metal/superconductor (N/S) 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 dxy-wave state), 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 px wave) 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 (LDOS). In px- and dxy-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 (MARS) 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.

Original language | Undefined |
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Pages (from-to) | 054522- |

Journal | Physical review B: Condensed matter and materials physics |

Volume | 76 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2007 |

### Keywords

- METIS-242709
- IR-59161

## Cite this

Tanaka, Y., Tanuma, Y., & Golubov, A. A. (2007). Odd-frequency pairing in normal-metal/superconductor junctions.

*Physical review B: Condensed matter and materials physics*,*76*(5), 054522-. https://doi.org/10.1103/PhysRevB.76.054522