Emergent Odd-Frequency Superconducting Order Parameter near Boundaries in Unconventional Superconductors

It was previously suggested that an odd-frequency pair amplitude exists in the vicinity of boundaries in unconventional superconductors. We develop this idea and quest for a novel superconducting order parameter with an odd-frequency dependence. For this purpose, we focus on $p$-wave superconductors and extend the quasi-classical theory to include the odd-frequency dependence in the order parameter. Both of the frequency and spatial dependences of the order parameter are determined self-consistently. Under a finite electron--phonon interaction, it is found that an odd-frequency order parameter is stabilized near the boundary and coexists with the even-frequency one. By analyzing the induced odd-frequency pair amplitude in terms of the superconducting quasi-particle wavefunction, it is found that the mid-gap bound state generates the emergent odd-frequency order parameter.