Spin-flip scattering and nonideal interfaces in dirty ferromagnet/superconductor junctions

We study the proximity-induced superconducting correlations as well as the local density of states of a ferromagnet, in a ferromagnet/$s$-wave superconductor heterostructure. We include the effects of spin-flip scattering, nonideal interfaces, and the presence of impurities in the sample. We employ the quasiclassical theory of superconductivity, solving the Usadel equation with emphasis on obtaining transparent analytical results. As our main result, we report that in a certain parameter regime the spatial oscillations of the anomalous (superconducting) part of the Green’s function induced in the ferromagnet by the proximity effect from the $s$-wave superconductor, are damped out due to the presence of spin-flip processes. As a consequence, spin-flip scattering may under certain conditions actually enhance the local density of states due to the oscillatory behavior of the latter in ferromagnet/superconductor structures. We also conjecture that the damping could be manifested in the behavior of the critical temperature $\left(T_c\right)$ of the $s$-wave superconductor in contact with the ferromagnet. More specifically, we argue that the nonmonotonic decrease of $T_c$ in ferromagnet/$s$-wave superconductor junctions without magnetic impurities is altered to a monotonic, nonoscillatory decrease when the condition $1>16{\tau }_{\mathrm{sf}}^2{h}^2$ is fulfilled, where ${\tau }_{\mathrm{sf}}$ is the spin-flip relaxation time and $h$ is the exchange field.

Figure 1

(Color online) The figure shows geometries that will be considered in this paper. In the top figure, we consider a $F∕F∕S$ junction consisting of a dirty ferromagnet sandwiched between a ferromagnetic and superconducting reservoir where the Green’s functions are described by their bulk values. In the bottom figure, we consider a $F∕S$ junction consisting of a dirty ferromagnet connected to a superconducting reservoir.

Figure 2

(Color online) Spatial variation of the deviation from the LDOS $\left(\delta N\right)$ for a $F∕S$ junction in the presence of spin-flip scattering. We have chosen ${\epsilon }_T∕\mid \Delta \mid =1$ and $\gamma =5$. The qualitative features are the same for the $F∕F∕S$ junctions for this particular set of parameters. It is seen that for a given energy, increasing spin-flip scattering $\Gamma =1∕{\tau }_{\mathrm{sf}}$ will increase the oscillation length and reduce the amplitude of the Green’s function.

Figure 3

(Color online) (a) Plot of the characteristic decay and oscillation lengths $\left({\xi }_x\right)$ of the anomalous Green’s function in the presence of spin-flip scattering with $h∕\mid \Delta \mid =5$. (b) Spatial variation of the deviation from the LDOS $\left(\delta N\right)$ for a $F∕S$ junction in the presence of spin-flip scattering. We have chosen ${\epsilon }_T∕\mid \Delta \mid =1$, $\gamma =5$, and $h∕\mid \Delta \mid =10$. The quasiparticle energy has been set to $\epsilon ∕\mid \Delta \mid =0.5$, but the qualitative behavior is identical for all $\epsilon <\mid \Delta \mid$. As shown in (b), there are no oscillations of the Green’s function in the ferromagnet in the parameter range $\Gamma ∕h>4$ $(\Gamma =1∕{\tau }_{\mathrm{sf}})$. Inclusion of the energy contribution $\epsilon$ brings small corrections to this result, but as seen from the figure the condition Eq. (37) is a very good approximation. This behavior is to be contrasted with the usual oscillations in $F∕S$ junctions as obtained without spin-flip scattering. Note that in (a), ${\xi }_{\mathrm{osc}}$ formally diverges near $\Gamma ={\Gamma }_c$, which separates the two parameter regimes where oscillations occur and where they do not occur.

Figure 4

(Color online) Correction to the LDOS $\left(\delta N\right)$ for $\Gamma ∕h=0.01$ $(\Gamma =1∕{\tau }_{\mathrm{sf}})$. Surface plot of the deviation from the LDOS in the $(x,\epsilon )$ plane for a junction of width $d=2{\xi }_S$ and with transparency parameter $\gamma =5$ and exchange field $h∕\mid \Delta \mid =30$. The most protruding feature is the peak emerging in the LDOS right at the $F∕S$ interface, followed by a dip structure at low energies.