Theory of superconductor-ferromagnet point-contact spectra: The case of strong spin polarization

We study the impact of spin-active scattering on Andreev spectra of point contacts between superconductors (SC) and strongly spin-polarized ferromagnets (FM) using recently derived boundary conditions for the quasiclassical theory of superconductivity. We describe the interface region by a microscopic model for the interface scattering matrix. Our model includes both spin filtering and spin mixing and is nonperturbative in both transmission and spin polarization. We emphasize the importance of spin-mixing caused by interface scattering, which has been shown to be crucial for the creation of exotic pairing correlations in such structures. We provide estimates for the magnitude of this effect in different scenarios and discuss its dependence on various physical parameters. Our main finding is that the shape of the interface potential has a tremendous impact on the magnitude of the spin-mixing effect. Thus, all previous calculations, being based on delta-function or box-shaped interface potentials, underestimate this effect gravely. As a consequence, we find that with realistic interface potentials the spin-mixing effect can easily be large enough to cause spin-polarized subgap Andreev bound states in SC/FM point contacts. In addition, we show that our theory generalizes earlier models based on the Blonder-Tinkham-Klapwijk approach.

Figure 1

(Color online) SC/sFM interface, showing the Fermi surfaces on either side (thick lines). Assuming momentum conservation parallel to the interface $\left({\vec{k}}_{\parallel }\right)$, a quasiparticle incident from the SC can either scatter into two (a) or into only one (b) spin band of the FM.

Figure 2

(Color online) (a) The Andreev point contact with spin-active interface; (b) The spherical angles $\alpha$ and $\phi$ characterize the orientation of the interface exchange field ${\vec{J}}_{\text{I}}$ with respect to the FM exchange field ${\vec{J}}_{\text{FM}}$. The dashed arrow indicates the area where the misaligned interface magnetic moment resides.

Figure 3

(Color online) Sketch of the box-potential model that we consider in this section (right) and of the corresponding Fermi-surface geometry (left). The model parameters are indicated.

Figure 4

(Color online) The spin-mixing angle $\vartheta$ as function of the momentum component parallel to the interface, shown for various barrier thicknesses. (a) $d=0.1$, 0.5, and $1.0{\lambda }_{\text{F}}/2\pi$, (b) $d=2.0$, 3.0, and $5.0{\lambda }_{\text{F}}/2\pi$. The remaining parameters are $E_2=0.1E_{\text{F}}$, $E_3=0.9E_{\text{F}}$, $U_{+}=1.1E_{\text{F}}$, $U_{-}=1.9E_{\text{F}}$, and $\alpha =0.5\pi$ (see text and Fig. 3).

Figure 5

(Color online) (a) Spin-mixing angle $\vartheta$ as a function of impact angle for (a) $d=0.5{\lambda }_{\text{F}}/\pi$ and (b) $d=5.0{\lambda }_{\text{F}}/2\pi$. In both plots, the curves are for $U_{-}=1.2,\dots ,2.0E_{\text{F}}$, $E_3=U_{-}-1.0$. The corresponding value of the exchange field $J$ is indicated. The remaining parameters are $E_2=0.1E_{\text{F}}$, $U_{+}=1.1E_{\text{F}}$, and $\alpha =0.5\pi$.

Figure 6

(Color online) The spin-mixing angles ${\vartheta }_2$ ($k_{\parallel }$) [(a) and (b)], and ${\vartheta }_3$ ($k_{\parallel }$) [(c) and (d)] for thin [left column: $d=0.1$ (solid), 0.5 (dashed-dotted), and 1.0 (dashed) ${\lambda }_{\text{F}}/2\pi$] and thick [right column: $d=2$ (dashed), 3 (dashed-dotted), 5 (solid) ${\lambda }_{\text{F}}/2\pi$] interfaces. The remaining parameters are the same as in Fig. 4.

Figure 7

(Color online) The transmission parameters $\left|t_2{t}_2^{\prime }\right|$ ($k_{\parallel }$) [(a) and (b)], and $\left|t_3{t}_3^{\prime }\right|$ ($k_{\parallel }$) [(c) and (d)], for thin [left column: $d=0.1$ (solid), 0.5 (dashed-dotted), 1.0 (dashed) ${\lambda }_{\text{F}}/2\pi$] and thick [right column: $d=2$ (dashed), 3 (dashed-dotted), 5 (solid) ${\lambda }_{\text{F}}/2\pi$] interfaces. The remaining parameters are the same as in Fig. 4.

Figure 8

(Color online) (a) Spin-mixing angle $\vartheta$ as a function of $V_{+}/V_{-}$ for the delta-function potential. $E_2=0.1E_{\text{F}}$ and $E_3=0.7E_{\text{F}}$. (b) The same as (a) for $E_2=-0.7E_{\text{F}}$ and $E_3=-0.1E_{\text{F}}$.

Figure 9

(Color online) Sketch of the scattering potential for the smooth potential model (right) and the here considered Fermi-surface geometry (left). The parameters introduced in Eq. (40) are indicated.

Figure 10

(Color online) (a) Shape function of the scattering potential (average between both spin directions) for $\sigma =0,\dots ,0.7{\lambda }_{\text{F}}$ and $\sigma +d=0.7{\lambda }_{\text{F}}$, and $E_2=-0.1E_{\text{F}}$, and $E_3=-0.8E_{\text{F}}$. (b) The spin-mixing angle $\vartheta$ as a function of impact angle for the different potentials plotted in (a). $\sigma$ increases from bottom to top.

Figure 11

(Color online) Transport processes contributing to the current through the point contact (a) Normal AR (b) Normal transmission, requires $\epsilon >\Delta$ (c) SAR, and (d) Spin-flip transmission, requires $\epsilon >\Delta$.

Figure 12

(Color online) The denominator of Eq. (46) as a function of the quasiparticle energy $\epsilon$. $r_{\uparrow }=0.9$, $r_{\downarrow }=0.95$, and ${\alpha }_Y=0.5\pi$. Plots for $\vartheta =0.1\pi ,\dots ,1.0\pi$ in steps of $0.1\pi$. The maximum moves closer to zero energy with increasing $\vartheta$.

Figure 13

(Color online) The conductance $G$ of an S/F point contact as function of contact voltage $V$ (first row), and the corresponding SAR (second row) and AR (third row) contributions to it, for $d=0.1$, 0.5, and $1.0{\lambda }_{\text{F}}/2\pi$ (left column) and $d=2.0$, 3.0, and $5.0{\lambda }_{\text{F}}/2\pi$ (right column). The remaining parameters are $E_2=0.1E_{\text{F}}$, $E_3=0.9E_{\text{F}}$, $U_{+}=1.1E_{\text{F}}$, $U_{-}=1.9E_{\text{F}}$, and $\alpha =0.5\pi$.

Figure 14

(Color online) The conductance $G$ of an SC/FM point contact as function of contact voltage $V$, for (a) $T=0$ and (b) $T=0.1T_c$. In both cases, the values of $E_3=0.2,\dots ,0.9$ and $U_{-}=E_{-}+E_{\text{F}}$ are increasing in steps of $0.1E_{\text{F}}$ from top to bottom. The remaining parameters are $E_2=0.1E_{\text{F}}$, $U_{+}=1.1E_{\text{F}}$, $\alpha =0.5\pi$, and $d=5.0{\lambda }_{\text{F}}/2\pi$.

Figure 15

(Color online) The conductance $G$ of a point contact as function of contact voltage $V$, for (a) $T=0$ and (b) $T=0.1T_c$, shown for two values of $\alpha$. The remaining parameters in all plots are $E_2=0.1E_{\text{F}}$, $E_3=0.99E_{\text{F}}$, $U_{+}=1.1E_{\text{F}}$, $U_{-}=1.99E_{\text{F}}$, and $d=1.0{\lambda }_{\text{F}}/2\pi$.

Figure 16

(Color online) The differential conductance for the smooth potential model. The parameters are the same as in Fig. 10. The interface smoothness parameter $\sigma$ increases from back to front by steps of $0.1{\lambda }_{\text{F}}$. Temperatures are $T=0$ (top) and $T=0.1T_c$ (bottom).