Magnetic exchange interaction in a spin valve with a chiral spin-triplet superconductor

The coupling between two ferromagnets separated by a superconductor has been mostly investigated for the case of Cooper pairs with spin-singlet symmetry. Here, we consider a spin-triplet superconductor with chiral pairing. By full self-consistent analysis of the spatial dependent superconducting order parameter, we determine the magnetic ground state of the superconducting spin valve. The study is performed by investigating the role of the orientation and strength of the magnetization in the ferromagnets including spin-valve asymmetries in the magnetic configurations. Due to the nonvanishing angular momentum of the spin-triplet Cooper pairs, we demonstrate that the induced magnetic coupling has an anisotropic character and a structure that can favor collinear or noncollinear magnetic orientations, thus mimicking a magnetic interaction of the Heisenberg or Dzyaloshinskii-Moriya type, respectively. We investigate the role of the physical parameters controlling the character of the magnetic exchange: the amplitude of the magnetization in the ferromagnets, and the length of the superconducting spacer in the spin valve. Our study demonstrates that spin-triplet superconductors can be employed to devise anisotropic magnetic exchange and to allow for transitions in the spin-valve state from a collinear to noncollinear magnetic configuration.

Figure 1

Schematic of the investigated spin-valve heterostructure with a chiral spin-triplet superconductor having $p_x+i{p}_y$ structure sandwiched between two ferromagnets. Magenta arrows in the middle layer denote parallel-spin Cooper pairs lying in the $x\text{−}y$ plane, while green and red arrows along the interfaces stand for spin-up and spin-down charge currents due to the chiral symmetry of the superconductor. The magnetizations $M_L$ and $M_R$ on the left and right side of the junction, respectively, can have a relative orientation marked by an angle $\varphi$ with respect to the out-of-plane perpendicular direction as in panel (a). (b) for $\varphi =0$ ($\varphi =\pi$) the magnetic moments $M_L$ and $M_R$ are ferromagnetically (antiferromagnetically) aligned, respectively. The effective magnetic exchange for parallel or antiparallel alignment is of Heisenberg-type. For the case of perpendicularly oriented magnetization ($\varphi =\pi /2$), the exchange mimics the form of an effective Dzyaloshinskii-Moriya interaction.

Figure 2

Zero-temperature superconducting order parameter for several choices of the exchange fields in the left (L) and in the right (R) layer, respectively: equal magnitude and same direction, parallel to the $\mathbf{d}$-vector [panels (a) and (b)]; different magnitude and same direction, parallel to the $\mathbf{d}$-vector [panels (c) and (d)]; equal magnitude and directions parallel and perpendicular to the $\mathbf{d}$-vector, respectively [panels (e) and (f)]; different magnitude and directions parallel and perpendicular to the $\mathbf{d}$-vector, respectively [panels (g) and (h)]. The order parameters are normalized to the value that they assume in the middle of the superconducting layer, in the case of a normal-superconductor-normal (NSN) junction. $\xi =10a$ is the zero-temperature coherence length, $a$ being the lattice constant.

Figure 3

Temperature dependence of the magnitude of the triplet order parameter at the center of the superconducting layer for different thicknesses $d_S/\xi$ of the latter, $\xi$ being the zero-temperature superconductor coherence length: panels (a) and (b) show $p_x$-real and $p_y$-imaginary parts in the case of equal exchange field in the two ferromagnetic layers, respectively, whereas panels (c) and (d) show the same quantities in the case of different magnetic fields. $T_{c,\mathrm{NSN}}$ is the critical temperature in the NSN case for a thickness $d_S/\xi =4$, and the order parameters are normalized to their zero-temperature values in the NSN case. The exchange fields in the ferromagnetic layers are both aligned to the d-vector thus realizing a $z$-oriented parallel spin-valve configuration (e.g., $\uparrow ,\uparrow$).

Figure 4

Magnetization evolution in the right ferromagnetic layer for $T/T_c=0.16, d_S/\xi =4$, and different choices of the magnetic exchanges $h_L$ and $h_R$.

Figure 5

Free-energy evolution of the spin-valve magnetic states at $T/T_c=0.16$ and $d_S/\xi =3$ for symmetrical [panel (a)] and asymmetrical [panel (b)] amplitudes of the exchange fields in the two ferromagnetic layers. Energies of the different magnetic configurations are measured with respect to the case of a magnetic state with out-of-plane magnetizations, both parallel to the $\mathbf{d}$-vector. Panel (c) reports the difference between the energies of the parallel and antiparallel in-plane configurations.

Figure 6

Same as in Fig. 5 for $T/T_c=0.16, d_S/\xi =4$, and different choices of the exchange field in the two ferromagnetic layers. We observe that the main competition for the ground state of the spin valve is between the antiferromagnetic planar configuration ($\to ,\leftarrow$) and the perpendicularly oriented magnetic state ($\uparrow ,\leftarrow$).

Figure 7

Temperature evolution of the ground-state energy for different magnetic configurations and different thicknesses $d_S/\xi$ of the superconducting layer. Energies are measured with respect to that of the case of exchange fields both parallel to the $\mathbf{d}$-vector, and they are normalized to $V{\mathrm{\Delta }}_0^2, {\mathrm{\Delta }}_0$ being the zero-temperature order parameter in the NSN (normal-superconductor-normal) junction. Vertical (horizontal) arrows in the legend denote exchange fields in the two ferromagnetic layers perpendicular (parallel) to the planar junction.

Figure 8

Same as in Fig. 5 for different values $d_S/\xi$ of the thickness of the superconducting layer at a representative value of the temperature $T/T_c=0.16$.