Ballistic Josephson junctions in the presence of generic spin dependent fields

Ballistic Josephson junctions are studied in the presence of a spin-splitting field and spin-orbit coupling. A generic expression for the quasiclassical Green's function is obtained and with its help we analyze several aspects of the proximity effect between a spin-textured normal metal (N) and singlet superconductors (S). In particular, we show that the density of states may show a zero-energy peak which is a generic consequence of the spin dependent couplings in heterostructures. In addition, we also obtain the spin current and the induced magnetic moment in a SNS structure and discuss possible coherent manipulation of the magnetization which results from the coupling between the superconducting phase and the spin degree of freedom. Our theory predicts a spin accumulation at the S/N interfaces, and transverse spin currents flowing perpendicular to the junction interfaces. Some of these findings can be understood in the light of a non-Abelian electrostatics.

Figure 1

Sketch of the Josephson junction, with $\phi$ the phase difference between the two superconducting electrodes (in gray). The white part corresponds to the normal region, with a spin texture giving rise to the Andreev-Wilson loop $W\left(x\right)$ at point $x$ [see Eq. (3.18)]. The electron spin precesses at a constant latitude around the local vector $\mathfrak{n}$ when traveling along the junction $[\mathfrak{n}$ evolves according to Eq. (3.31)]. Blue and red colors refer to a precession axis for electrons and holes, respectively. After completing the loop, the spin rotates an angle $\mathrm{\Phi }$ between the initial and final states.

Figure 2

Density of state with respect to the energy $\omega /\mathrm{\Delta }$ from the expression (4.3) for $\mathrm{\Phi }=0$. In each panel, different lengths of the junction are represented: $L/{\xi }_0=\left\{0,1,2,3,4\right\}$ (${\xi }_0=\hslash v_F/\mathrm{\Delta }$ is the coherence length), with a constant offset of each curve from the previous one for commodity, and an alternance of plain and dashed curves. Top-left panel: $\phi =0$; top-right panel: $\phi =\pi /4$; bottom-left panel: $\phi =\pi /2$; bottom-right panel: $\phi =\pi$. One sees that increasing the length of the junction allows the opening of new Andreev modes, which eventually goes to zero energy when $\phi$ approach $\pi$. Longer and longer junctions allow for more and more Andreev modes to appear.

Figure 3

Density of state at zero energy from the expression (4.3) with respect to $\mathrm{\Phi }$. Different phase differences of the junction are represented: $\phi =\left\{0,1,2,3,4\right\}\times \pi /4$, with a constant offset of each curve from the previous one for commodity, and an alternance of plain and dashed curves. The plot is for $L/{\xi }_0=4$, but any length displays the same zero-energy peaks, namely, when $\mathrm{\Phi }\pm \phi =\pi$, generally broader for longer junctions. Note that the possible length dependency of the parameter $\mathrm{\Phi }$ has not been taken into account for this check.

Figure 4

Energy $E\left(\phi \right)/\mathrm{\Delta }$ of the Andreev bound states with respect to the phase difference $\phi$ for a short junction $L/{\xi }_0\to 0$, from the expression (4.5). $\mathrm{\Phi }=\pi /4$ (left panel), and $\mathrm{\Phi }=\pi /2$ (right panel). At the values $\phi =\pi$ and $\phi =\pi \pm \mathrm{\Phi }$, the curves are spin degenerate in the upper or lower half-planes.

Figure 5

Spin polarization $S^z\left(\phi \right)/S^z\left(0\right)$ in 1D monodomain S/F/S Josephson junction from Eq. (5.3) with respect to the phase difference $\phi$ and for different values of the temperature $t=T/T_c=0.99$ (plain/red), $0.5$ (blue/dotted), $0.1$ (green/dashed), and $0.01$ (blue/dotted-dashed) showing steeper and steeper curves as the temperature is decreased. Left panel: $\mathrm{\Phi }=2hL/v_F=\pi /4$. Middle panel: $\mathrm{\Phi }=\pi /2$. Right panel: $\mathrm{\Phi }=3\pi /4$.

Figure 6

Spin polarization $S^z$ in 1D monodomain S/F/S Josephson junction from Eq. (5.3) with respect to the temperature $T/T_c$ for different values of $\mathrm{\Phi }=2hL/v_F=\pi /4$ (red/plain), $\pi /2$ (blue/dotted), $3\pi /4$ (green/dashed), $5\pi /6$ (blue/dotted-dashed), at zero-phase difference $\phi =0$ (upper left panel), $\pi /2$ (upper right panel), and $\pi /4$ (lower left panel). The diamagnetic magnetization saturates at low temperatures and vanishes for $T\to T_c$. A reversal of the spin polarization is predicted in the bottom panel. This is understood when plotting the density of state for $\phi =\pi /4$ and $\mathrm{\Phi }=5\pi /6$ (lower right panel), when one superimposes a Fermi distribution law, since the heights of the different peaks are similar.

Figure 7

The upper panel shows the spatial dependence of the spin polarizations $S^x$ (plain curve) and $S^z$ (dashed curve) for a Rashba SOC and an exchange field in $z$ direction. Lower panel shows the spatial dependence of the spin currents ${\mathfrak{J}}_x^y$ (plain curve) and ${\mathfrak{J}}_y^x$ (dotted curve) in the same situation with ${\mathfrak{J}}_i\propto {\mathfrak{D}}_0{F}_{0i}$ in analogy with a displacement current [see (6.3)]. The spin capacitor effect is illustrated by the $S^x$ curve, being odd in space and originating from the term $S\propto {\mathfrak{D}}_i{F}_{i0}$ in (6.2). We have chosen $\mathit{h}=\left(0,0,5\right)\hslash v_F/{\xi }_T,{\xi }_{T}r=3,L/{\xi }_T=1$, and the curves are normalized by the quantity $2N_0{\mathrm{\Delta }}^{2}cos\phi /\pi k_B{T}_c$. For a Dresselhaus SOC $\left(\gamma =\pi /2\right)$, one obtains similar curves by making the changes $S^x\to -S^y,S^z\to S^z,{\mathfrak{J}}_x^y\to {\mathfrak{J}}_x^x$, and ${\mathfrak{J}}_y^x\to {\mathfrak{J}}_y^y$.

Figure 8

Illustration of the transverse spin current ${\mathfrak{J}}_y^{z,x}$ (plain/dashed curves, respectively) along the axis perpendicular to the junction for a Rashba spin-orbit effect with respect to the junction length. We choose $L/{\xi }_T=1,{\xi }_{T}r=3,{\xi }_T{h}_x=5\hslash v_F$ (upper panel) and ${\xi }_T{h}_y=5\hslash v_F$ (lower panel) in Eqs. (3.33) and (3.34) after injection of $\mathfrak{n}$ from (6.6) and (6.8). The antisymmetric ${\mathfrak{J}}_y^z$ comes from the contribution $〈v_{i}sgn\left(v_x\right){\mathfrak{D}}_0{v}_j{\mathfrak{D}}_j{F}_{0k}{v}_k〉$ in (6.3). The vertical axis is in unit of $2N_0{\mathrm{\Delta }}^{2}cos\phi /\pi k_B{T}_c$. When $h_x=h_y=0$ but $h_z\ne 0$ this term is zero, and no asymmetric spin current is flowing (see Fig. 7). The current ${\mathfrak{J}}_y^x$ stems for higher-order terms in the expansion (6.3).

Figure 9

Magnetic interaction $cos\mathrm{\Phi }$ for a Bloch domain wall with respect to the parameter $hL/\left|v_x\right|$, from (7.3), and for different values of $\kappa =v_{x}q/2h$. Solid line: $\kappa =0$. Dotted line $\kappa =1.5$. Dashed line $\kappa =3$.