Spin-active interfaces and unconventional pairing in half-metal/superconductor junctions

We study the physical properties of a half-metallic $\text{ferromagnet}\mid \text{superconductor}$ $\left(\text{HM}\mid \text{S}\right)$ bilayer, allowing for an arbitrary bulk pairing symmetry of the superconductor and spin-dependent processes at the interface. In particular, we study how the possibility of unconventional pairing such as $p$- and $d$-wave and a spin-active interface influence the (i) conductance spectra, (ii) proximity effect, and (iii) local density of states of such a bilayer. Our calculation is done both analytically and numerically in the ballistic limit, using both a continuum and lattice model. It is found that the spin-dependent phase shifts occurring at the $\text{HM}\mid \text{S}$ interface seriously influence all of the aforementioned phenomena. We explain our results in terms of Andreev reflection in the presence of a spin-active interface, allowing for both spin-filtering and spin-mixing processes. We demonstrate how the surface bound states induced by the anisotropy of the superconducting order parameter at the $\text{HM}\mid \text{S}$ interface are highly sensitive to these spin-dependent processes. Our results can be directly tested experimentally using scanning tunnel microscope measurements and/or point-contact spectroscopy.

Figure 1

(Color online) The superconductor/half-metallic ferromagnet bilayer studied in this paper. The barrier magnetic moment may in general be misaligned to the bulk magnetization in the ferromagnet, which is assumed to be directed along $\widehat{\mathbf{z}}$. The presence of a barrier magnetic moment may lead to both spin-split and spin-flip processes at the interface.

Figure 2

(Color online) Plot of the normalized conductance $G\left(eV\right)$ for a $s$-wave, chiral $p$-wave, $d_{x^2-y^2}$-wave, and $d_{xy}$-wave symmetry in the rows ranging from top to bottom. In the left panels, we consider the case of pure spin mixing for several values of $\rho$ with $\varphi =0$. In the right panels, we consider additionally spin-flip processes induced by a misaligned barrier moment at the interface for several values of $\varphi$ with $\rho =0.5$.

Figure 3

(Color online) Comparison of the conductance spectra for $\text{N}\mid \text{S}$ and $\text{HM}\mid \text{S}$ junctions with a $d_{xy}$-wave superconductor in the top row and a chiral $p$-wave superconductor in the bottom row. In the half-metallic case, we consider a spin-active interface with a misorientation angle $\varphi =0.5$. The ratio between the magnetic and nonmagnetic part of the barrier potential is denoted $\rho$. In all cases, the strong tunneling limit $Z=10$ is considered. The Andreev reflection probability in the right panels is calculated at the peak energy in the $d_{xy}$-wave case while it is calculated at $\epsilon =0$ in the chiral $p$-wave case.

Figure 4

(Color online) Andreev reflection scattering to first order for three cases: (a) $\text{N}\mid \text{S}$ junction without spin-active interface, (b) $\text{HM}\mid \text{S}$ junction without spin-active interface, and (c) $\text{HM}\mid \text{S}$ junction with spin-active interface. The incoming electronlike quasiparticle penetrates the superconducting region a distance ${\xi }_S$ before being backscattered as a hole by the superconducting gap.

Figure 5

(Color online) Schematic description of the lattice geometry for the $\text{HM}\mid \text{S}$ bilayer junction indicating the reference axis system, the notation for the size $\left(2L_x\times 2L_y\right)$, the position of the barrier $\left(x=0\right)$, the sketch of the pairing configurations ($d$ wave, $p$ wave, and local $s$ wave), respectively. On the left side, we show a sketch of the density of states for the half metal.

Figure 6

(Color online) Color map of the $s$-wave on-site pairing amplitude at the interface ${\Delta }_S\left[0\right]$ with respect to the bulk value ${\Delta }_{S,B}$ as a function of the misalignment angle $\varphi$ and the magnetic barrier strength $V_M$. The nonmagnetic scattering potential has an amplitude $V_0=2t$.

Figure 7

(Color online) Color map of the real part of the $p_x$ component of the pairing amplitude evaluated at the interface ${\Delta }_{p_x}\left[0\right]$ with respect to the bulk value ${\Delta }_{p_x,B}$ as a function of the scaled misalignment angle $2\varphi /\pi$ and the magnetic barrier strength $V_M/V_0$. The nonmagnetic scattering potential has a given amplitude of $V_0=2t$.

Figure 8

(Color online) Color map of the imaginary part of the $p_y$ component for the pairing amplitude evaluated at the interface ${\Delta }_{p_y}\left[0\right]$ with respect to the bulk value ${\Delta }_{p_y,B}$ as a function of the scaled misalignment angle $2\varphi /\pi$ and the ratio of the magnetic barrier strength with respect to the nonmagnetic one, $V_M/V_0$. The nonmagnetic scattering potential has an amplitude $V_0=2t$.

Figure 9

(Color online) Color map the $d_{x^2-y^2}$ pairing amplitude at the interface ${\Delta }_d\left[0\right]$ with respect to the bulk value ${\Delta }_{d,B}$ as a function of the scaled misalignment angle $2\varphi /\pi$ and the ratio of the magnetic barrier strength with respect to the nonmagnetic one, $V_M/V_0$. The nonmagnetic scattering potential has an amplitude $V_0=2t$.

Figure 10

(Color online) Density of states for the chiral $p$ wave evaluated at one representative site position $\left(i=2\right)$ within the range of one superconducting coherence length from the barrier $\left(i=0\right)$. The nonmagnetic potential has a value of $V_0=0.5t$.