Odd-frequency pairing in normal-metal/superconductor junctions

Abstract

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.

Symmetry protected topological superfluid3He-B

“…In Table 1, we summarize the four possible classes of Cooper pair amplitudes in bulk superconductors and superfluids, and the additional Cooper pairs induced by a symmetry breaking field [27,122,314,320,321,322]. In the case of spin-triplet superconductors and superfluids, ETO components f It has recently been demonstrated that in a bulk ETO superconductor and superfluid, OTE Cooper pair amplitudes, f OF µ (−k, r; ω n ) = −f OF µ (k, r; ω n ), are equivalent to the low-energy density of states originating from Andreev bound states that are bound to the surface or vortices [120,122,123,121].…”

Experimentally observable signatures of odd-frequency pairing in multiband superconductors

“…Notably, there are no zero-energy or subgap states, otherwise often associated with oddfrequency pairing. 43,48,49,[56][57][58][59] Recent works have pointed out that zero-energy states do not always accompany odd-frequency pairing, 41,54,60,61 and odd-frequency, oddinterband pairing provides another example when this happens.…”

“…s-, d-wave) or spin-triplet odd-parity (p-wave) superconductors, but it can also be even or odd under time, or equivalently frequency. [42][43][44] Examples of this are odd-frequency spin-triplet s-wave superconductivity giving rise to long-scale proximity effects in superconductorferromagnet systems 45,46 or odd-frequency spin-singlet p-wave pairing in non-magnetic junctions [47][48][49] . A recent example of the former is theoretically predicted odd-frequency pairing in MgB 2 under applied magnetic field.…”

Odd-Frequency Superconductivity in Sr2RuO4 Measured by Kerr Rotation

“…Odd-ω superconductivity is a dynamical phenomenon where the fermionic nature of the Cooper pair is preserved due to an oddness in frequency (or equivalently time) [13,14]. It is well established to exist in superconductor-ferromagnet junctions [15][16][17][18][19] and also predicted for superconductor-normal metal junctions [20][21][22]. However, odd-ω superconductivity has remained elusive in bulk materials without external magnetic fields.…”