Abstract
The Meissner effect is one of the defining properties of superconductivity, with a conventional superconductor completely repelling an external magnetic field. In contrast to this diamagnetic behavior, odd-frequency superconducting pairing has often been seen to produce a paramagnetic Meissner effect, which instead makes the superconductor unstable due to the attraction of magnetic field. In this work we study how both even- and odd-frequency superconducting pairing contributes to the Meissner effect in a generic two-orbital superconductor with a tunable odd-frequency pairing component. By dividing the contributions to the Meissner effect into intra- and inter-band processes, we find that the odd-frequency pairing actually generates both dia- and paramagnetic Meissner responses, determined by the normal-state band structure. More specifically, for materials with two electron-like (hole-like) low-energy bands we find that the odd-frequency inter-band contribution is paramagnetic but nearly canceled by a diamagnetic odd-frequency intra-band contribution. Combined with a diamagnetic even-frequency contribution, such superconductors thus always display a large diamagnetic Meissner response to an external magnetic field, even in the presence of large odd-frequency pairing. For materials with an inverted, or topological, band structure, we find the odd-frequency inter-band contribution to instead be diamagnetic and even the dominating contribution to the Meissner effect in the near-metallic regime. Taken together, our results show that odd-frequency pairing in multi-orbital superconductors does not generate a destabilizing paramagnetic Meissner effect and can even generate a diamagnetic response in topological materials.
Highlights
Odd-frequency superconductivity is an unusual superconducting state where the two electrons forming a Cooper pair join each other at different times, such that the resulting pair amplitude is odd under the interchange of the internal time coordinate or, equivalently, has an odd frequency dependence [1,2,3,4]
Odd-frequency pairing has often been discussed to give a paramagnetic Meissner effect, with only conventional evenfrequency pairing assume to give the diamagnetic Meissner effect necessary to stabilize a superconductor in an external magnetic field [23,24,25,26,27,28,29,30]
We disprove this simplistic picture for multiorbital superconductors where odd-frequency bulk superconducting pairing is ubiquitous
Summary
Odd-frequency superconductivity is an unusual superconducting state where the two electrons forming a Cooper pair join each other at different times, such that the resulting pair amplitude is odd under the interchange of the internal time coordinate or, equivalently, has an odd frequency dependence [1,2,3,4]. In contrast to this conventional diamagnetic response, odd-frequency pairing has instead in many circumstances been predicted to cause a paramagnetic Meissner effect, where the superconductor instead attracts a magnetic field, which destabilizes the whole superconducting state by enhancing the magnetic field inside the superconductor [23,24,25,26,27,28,29,30] This paramagnetic response has recently been experimentally confirmed in regions where odd-frequency superconductivity is proximity-induced in various heterostructures [27,30]. We have shown that for a very unconventional superconductor, the (Cu-)doped Bi2Se3 topological insulator, odd-frequency pairing can produce a small, but still, diamagnetic Meissner response [9] that produces a counterexample of the expectation of paramagnetic Meissner effect from oddfrequency pairing This is a very special situation, as doped Bi2Se3 has a linear band dispersion with exceptionally strong spin-orbit coupling and a highly unconventional superconducting state consisting of spin-triplet interorbital pairing that generates a nematic superconducting state. We provide some more details of the calculations in Appendixes A and B
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