Inverse proximity effect in superconductor-ferromagnet structures: From the ballistic to the diffusive limit

The inverse proximity effect, i.e., the induction of a magnetic moment in the superconductor, in superconductor-ferromagnet (S/F) junctions is studied theoretically. We present a microscopic approach which combines a model Hamiltonian with elements of the well-established quasiclassical theory. With its help we study systems with arbitrary degree of disorder, interface transparency, and thickness of the layers. In the diffusive limit we recover the result of previous works: the direction of the induced magnetization $M$ is opposite to the one of the F layer. However, we show that in the ballistic case the sign of $M$ may be positive or negative depending on the quality of the interface and thickness of the layers. We show that, regardless of its sign, the penetration length of the magnetic moment into the superconductor is of the order of the superconductor coherence length, which demonstrates that the effect has a superconducting origin.

Figure 1

Geometry of the S/F structure.

Figure 2

Induced magnetization $M∕M_P$ as a function of the transmission $\tau$ for (a) $h=0.2$ and (b) $h=0.4$ and different values of $\alpha$. We have chosen $\Delta =0.1$ and $T={10}^{-4}$. All energies are given in units of $t_S=t_F$.

Figure 3

Sketch of the position of the Andreev states in the spectral density for the majority $(\uparrow )$ and minority $(\downarrow )$ spins as a function of the hopping parameter $t_{\mathrm{SF}}$. The three panels at the bottom show the DOS of the spin-up (solid line) and spin-down (dashed line) electrons for three different values of $t_{\mathrm{SF}}$. At low transparencies and below the Fermi level there is a peak in the spin-up channel (left panel). Increasing the value of $t_{\mathrm{SF}}$, this peak moves to the right while the peak in the spin-down channel moves to the left. At some $t_{\mathrm{SF}}$ of the order of $h$ both peaks coincide (middle panel). Further increase of $t_{\mathrm{SF}}$ leads to a spin-down peak below the Fermi level (right panel).

Figure 4

The induced magnetization $M∕M_P$ at the S side of the interface as a function of the transmission coefficient $\tau$ for different values of the F-layer thickness $d_F$. The latter is given in units of $\pi {\xi }_F$. The S layer is assumed to be semi-infinite. We have chosen $h=0.2$, $\Delta =0.1$, and $T=0$.

Figure 5

The spatial dependence of the magnetization induced in a semi-infinite S layer in contact with a F layer of thickness $\pi {\xi }_F$ and $2\pi {\xi }_F$. We have chosen $h=0.2$ and $\Delta =0.1$.

Figure 6

Magnetization in the superconductor as a function of $\tau$ for different types of interfaces: $\alpha =1$ corresponds to the ballistic while $\alpha =0$ to the diffusive one. The other values chosen for $\alpha$ are 0.5 and 0.75. Here $d_F=2\pi {\xi }_F$, $h=0.2$, $\Delta =0.1$, and $T=0$.

Figure 7

The self-energy within the self-consistent Born approximation. The thick lines in this figure correspond to the fully dressed Green function, the crosses represent the hopping processes, and the dashed semicircle indicates averaging over the parallel momentum $k$ .