Surface effects in magnetic superconductors with a spiral magnetic structure

We consider a magnetic superconductor (MS) with a spiral magnetic structure. On the basis of generalized Eilenberger and Usadel equations, we show that near the boundary of the MS with an insulator (I) or vacuum, the condensate (Gor’kov’s) Green’s functions are disturbed by boundary conditions and differ essentially from their values in the bulk. Corrections to the bulk quasiclassical Green’s functions oscillate with the period of the magnetic spiral, $2\pi ∕Q$, and decay inside the superconductor over a length of the order $v_F∕2\pi T$ (ballistic limit) or $\sqrt{D∕\pi T}$ (diffusive limit), where $Q$ is the wave vector of the magnetic spiral, $D=v_{F}l∕3$ is the diffusion coefficient, $v_F$ is the Fermi velocity, and $l$ is the mean free path. We calculate the dc Josephson current in a MS/I/MS tunnel junction and show that the critical Josephson current differs substantially from that obtained with the help of the tunnel Hamiltonian method and bulk Green’s functions.

Figure 1

Contributions of the singlet (solid line) and triplet $S_z=0$ (dotted line) components to the Josephson critical current as a function of the exchange field. This dependence has been obtained on the basis of bulk quasiclassical Green’s functions and corresponds to the tunnel Hamiltonian method. The critical current $I_c\left(h\right)$ and the exchange field $h$ are plotted in units of $I_c\left(0\right)$ and $D{Q}^2∕2$, respectively. The temperature is chosen to be equal to $T=0.1D{Q}^2∕2\pi$.

Figure 2

Contributions of the singlet (solid line) and triplet (dotted lines) components to the Josephson critical current as a function of the exchange field $h$. The upper and lower dotted curves correspond to the triplet $\mid S_z\mid =1$ and $S_z=0$ condensate components. These curves are plotted on the basis of Eq. (29). The normalization units are the same as in Fig. 1.

Figure 3

The total critical current versus the exchange field obtained on the basis of the bulk Green’s functions (solid line) and of the Green’s functions taken at the MS/I interface (dotted line). The solid line corresponds to the tunnel Hamiltonian method. The normalization units are the same as in Fig. 1.