Odd-frequency Cooper pair around a magnetic impurity

The Yu-Shiba-Rusinov (YSR) state appears as a bound state of a quasiparticle at a magnetic atom embedded in a superconductor. We discuss why the YSR state has energy below the superconducting gap and why the pair potential changes the sign at the magnetic atom. Although a magnetic atom in a superconductor has been considered as a pair breaker since the 1960s, we propose an alternative physical picture to explain these reasons. The analytical expression of the Green's function indicates that a magnetic atom converts a spin-singlet s-wave Cooper pair into odd-frequency Cooper pairs rather than breaking it and that the odd-frequency pairing correlations coexist with the YSR states below the gap. The relationships among the free-energy density, the amplitudes of pairing correlation functions, and the sign change of the pair potential at a magnetic impurity are discussed utilizing the self-consistent solution of the Eilenberger equation. We conclude that the sign change of the pair potential happens only when the amplitudes of odd-frequency pairing correlations are dominant at the magnetic impurity. In the presence of the local pi-phase shift in the pair potential, odd-frequency pairs can decrease the free-energy density there because their response to a magnetic field is paramagnetic.