Unified Formalism of Andreev Reflection at a Ferromagnet/Superconductor Interface

We present a unified formalism of Andreev reflection of a partial polarized current at a ferromagnet/superconductor interface instead of assuming a linear combination of unpolarized and polarized currents. The Andreev reflection is limited by the states of minority spins and the extra majority spins become evanescent wave. We further study the effects of spin polarization, inelastic scattering, and interfacial scattering on the Andreev reflection, normal reflection, and transmitted probabilities in equilibrium as well as under a bias. Our model, which reduces to those of Blonder, Tinkham, and Klapwijk, Mazin, and Dynes in three limiting cases, provides a significantly better description of the experimental results.

Figure 1

Effect of polarization $\alpha$, interfacial scattering $Z$, and inelastic scattering $\Gamma$ on transmission and reflection probabilities $A$, $B$, $C$, and $D$ for (a)–(d) $Z=0$, 0.25, 0.5, 1, 3, (e)–(h) $\alpha =0$, 1.16, 2, 3.46, 100, and (i)–(l) $\Gamma =0$, 0.1, 0.2, 0.5 1.

Figure 2

Effect of spin polarization on conductance spectra from this model (solid lines) and the linear model (dashed lines) for various $\alpha =0$, 1.16, 2, 3.46, 100 and $P_l=0$, 0.25, 0.5, 0.75, 1 with (a) $Z=\Gamma =0$, (b) $Z=0$, $\Gamma =0.1$, (c) $Z=0.25$, $\Gamma =0$, and (d) $Z=0.25$, $\Gamma =0.1$.

Figure 3

Calculated spin polarization ($P_l=0.5$, 0.75, and 0.95 at $Z=0$) of the linear combination model as a function of $Z$ ($\Gamma =0$ and $V=\Delta /e$) at 1.5 K (dot line) and 4.2 K (dashed lines). Solid lines are spin polarization from this model.

Figure 4

Two Andreev spectra (open circles) of $\mathrm{Nb}/{\mathrm{Co}}_{40}{\mathrm{Fe}}_{40}{\mathrm{B}}_{20}$ contacts with $R_C=128\Omega$ (a) and $11\Omega$ (b) analyzed by this model (solid) and the linear model (dashed) with $T=4.25\mathrm{K}$, $\Delta =1.42\mathrm{meV}$ fixed as experimental values and $\Gamma =0$, (e) determined spin polarization value as a function of $Z$. Additional resistance $r_E$ is addressed in Ref. 19.