Signature of the long range triplet proximity effect in the density of states

We study the impact of the long range spin-triplet proximity effect on the density of states (DOS) in planar SF${}_1$F${}_2$S Josephson junctions that consist of conventional superconductors (S) connected by two metallic monodomain ferromagnets (F${}_1$ and F${}_2$) with transparent interfaces. We determine the electronic DOS in F layers for arbitrary orientation of the magnetizations using the solutions of Eilenberger equations in the clean limit and for a moderate concentration of impurities in ferromagnets. We find that a fully developed long range proximity effect occurs in Josephson junctions with a highly asymmetric ferromagnetic bilayer. For orthogonal magnetizations, the effect manifests itself as an enhancement in DOS and as a dominant second harmonic in the Josephson current-phase relation. Distinctive variation of DOS in ferromagnets with the angle between magnetizations is experimentally observable by tunneling spectroscopy. This can provide an unambiguous signature of the long range spin-triplet proximity effect.

Figure 1

Schematics of an SF${}_1$F${}_2$S heterojunction. The magnetization vectors lie in the $y$-$z$ plane at angles ${\alpha }_1$ and ${\alpha }_2$ with respect to the $z$ axis.

Figure 2

Normalized DOS, $N\left(\varepsilon \right)/N_0\left(0\right)$, in ferromagnets for an asymmetric SF${}_1$F${}_2$S junction with $d_1=10k_F^{-1}$ and $d_2=990k_F^{-1}$, $h_1=h_2=0.1E_F$. Superconductor is characterized by ${\Delta }_0\left(0\right)/E_{\mathrm{F}}={10}^{-3}$; the phase difference $\phi =0$. The results of non-self-consistent calculations in the clean limit $l\to \infty$ are shown for three values of the relative angle between magnetizations: $\alpha =0$ (dashed curve), $\pi /2$ (solid curve), and $\pi$ (dashed-dotted curve).

Figure 3

The Josephson current-phase relation $I\left(\phi \right)$ for an asymmetric SF${}_1$F${}_2$S junction with $d_1=10k_F^{-1}$ and $d_2=990k_F^{-1}$, $h_1=h_2=0.1E_F$, $l\to \infty$, $T/T_c=0.1$, and for three values of the relative angle between magnetizations: $\alpha =0$ (dashed curve), $\pi /2$ (solid curve), and $\pi$ (dashed-dotted curve). Inset: Magnified curves for $\alpha =0$ and $\pi$.

Figure 4

Normalized DOS, $N\left(\varepsilon \right)/N_0\left(0\right)$, in ferromagnets for an asymmetric SF${}_1$F${}_2$S junction with $d_1=10k_F^{-1}$ and $d_2=990k_F^{-1}$, $h_1=h_2=0.1E_F$, for $\alpha =\pi /2$. Superconductor is characterized by ${\Delta }_0\left(0\right)/E_{\mathrm{F}}={10}^{-3}$; the phase difference $\phi =0$. Results of self-consistent numerical solutions in the clean limit $l\to \infty$ (thin solid curve) and for moderately diffusive ferromagnets, $l=200k_F^{-1}$, $x=d_2/2$ (thick solid curve) are shown. For comparison, the non-self-consistent solution in the clean limit (dotted curve) is also shown.

Figure 5

Normalized DOS, $N\left(\varepsilon \right)/N_0\left(0\right)$, in ferromagnets for a symmetric SF${}_1$F${}_2$S junction with $d_1=d_2=500k_F^{-1}$, $h_1=h_2=0.1E_F$. Superconductor is characterized by ${\Delta }_0\left(0\right)/E_{\mathrm{F}}={10}^{-3}$; the phase difference $\phi =0$. Results of non-self-consistent calculations in the clean limit $l\to \infty$ are shown for three values of the relative angle between magnetizations: $\alpha =0$ (dashed curve), $\pi /2$ (solid curve), and $\pi$ (dashed-dotted curve).

Figure 6

Normalized DOS, $N\left(\varepsilon \right)/N_0\left(0\right)$, in ferromagnets for a symmetric SF${}_1$F${}_2$S junction with $d_1=d_2=500k_F^{-1}$, $h_1=h_2=0.1E_F$, for $\alpha =\pi /2$. Superconductor is characterized by ${\Delta }_0\left(0\right)/E_{\mathrm{F}}={10}^{-3}$; the phase difference $\phi =0$. Results of self-consistent numerical solutions in the clean limit $l\to \infty$ (thin solid curve) and for moderately diffusive ferromagnets, $l=200k_F^{-1}$, $x=d_2/2$ (thick solid curve) are shown. For comparison, the non-self-consistent solution in the clean limit (dotted curve) is also shown.