Spectral properties and quantum phase transitions in superconducting junctions with a ferromagnetic link

We study theoretically the spectral and transport properties of a superconducting wire with a magnetic defect. We start by modeling the system as a one-dimensional magnetic Josephson junction and derive the equation determining the full subgap spectrum in terms of the normal-state transfer matrix for arbitrary length and exchange field of the magnetic region. We demonstrate that the quantum phase transition predicted for a short-range magnetic impurity, and associated with a change of the total spin of the system, also occurs in junctions of finite length. Specifically, we find that the total spin changes discontinuously by integer jumps when bound states cross the Fermi level. The spin can be calculated by using a generalization of the Friedel sum rule for the superconducting state, which we also derive. With these tools, we analyze the subgap spectrum of a junction with the length of the magnetic region smaller than the superconducting coherence length and demonstrate how phase transitions also manifest as a change of sign of the supercurrent.

Figure 1

(a) The SFS junction under consideration. (b) Possible processes taking place in a Josephson junction: In the absence of normal reflection only Andreev reflections within the same propagation valley can occur (purple line). If normal backscattering is present valleys at $\pm k_F$ are coupled (green lines).

Figure 2

Energy of the bound states (solid lines) and the total spin $2S/\hslash$ (dashed line) of a SFS junction as a function of $L$ for three different values of $\mathrm{\Phi }$, and $\mu /\mathrm{\Delta }=100$. Red and blue colors correspond to spin projections of the electronic states.

Figure 3

(a) Phase diagram of the SFS Josephson junction in terms of the length of the junction $L$ and the magnetic phase $\mathrm{\Phi }$. The horizontal white dashed lines indicate the values of $\mathrm{\Phi }$ chosen in Fig. 2. (b) Total electronic spin of a SFS junction of length $k_{F}L=10$ [white dashed line in panel (a)] when one imposes 0 phase (dashed light) or $\pi$ phase (dashed dark). The red solid line shows the total spin when the junction stays in its ground state. Calculations have been done for $\mu /\mathrm{\Delta }=100$.