Spin-Orbital Coupling in a Triplet Superconductor-Ferromagnet Junction

We study the interplay of spin and orbital degrees of freedom in a triplet superconductor-ferromagnet junction. Using a self-consistent spatially dependent mean-field theory, we show that increasing the angle between the ferromagnetic moment and the triplet vector order parameter enhances or suppresses the $p$-wave gap close to the interface, according to whether the gap antinodes are parallel or perpendicular to the boundary, respectively. The associated change in condensation energy establishes an orbitally dependent preferred orientation for the magnetization. When both gap components are present, as in a chiral superconductor, first-order transitions between different moment orientations are observed as a function of the exchange field strength.

Figure 1

(a) Schematic diagram of the two-dimensional TSC-FM junction. The FM region is located at $x<0$, while the TSC is realized for $x>0$. The magnetization $\mathbf{M}$ of the FM is collinear to the exchange field $\mathbf{h}$ and forms an angle $\varphi$ with the $\mathbf{d}$-vector of the TSC, which defines the $z$ axis. We study TSC states with $p_x$, $p_y$, and $p_x+i{p}_y$ symmetry. (b) Evolution of the bulk FM magnetization and the majority and minority spin concentrations as a function of the exchange field $\mathbf{h}$.

Figure 2

Zero temperature pairing amplitude scaled to its bulk value as a function of the distance $i_x$ from the interface for $h=1.5$, $t_{\mathrm{int}}=1$, and several different angles $\varphi$ of the $\mathbf{d}\mathrm{\text{−}}\mathbf{M}$ misalignment. The spin-triplet orbital symmetry is of (a) $p_x$, (b) $p_y$, and chiral type with (c) a real $p_x$ and (d) an imaginary $p_y$ component, respectively.

Figure 3

(a) Dependence of the condensation energy $E_{\Delta }$ on the angle $\varphi$. (b)–(d) Dependence of the Gibbs energy $F$ on $\varphi$ for various fixed $h$ and for $p_x$, $p_y$, and $p_x+i{p}_y$ orbital symmetries of the TSC. $E_{\Delta ,\min }$ and $F_{\min }$ are the minimum amplitudes of the related energies. All panels are for $t_{\mathrm{int}}=1$ and temperature $k_{B}T=0.05$.

Figure 4

(a) Gibbs and (b) condensation energy difference energy difference between the parallel ($\varphi =0$) and perpendicular ($\varphi =\pi /2$) configurations of the moment as a function of the exchange field strength $h$ and $t_{\mathrm{int}}$, at temperature $k_{B}T=0.03$ for the $p_x+i{p}_y$ TSC. Bottom panel: Sketch of the most favorable magnetic (small red arrow) configurations with respect to the orbital symmetry and the $\mathbf{d}$-vector (large blue arrow) of the TSC as well as the character of interface transparency.