Theory of tunneling spectroscopy of normal metal/ferromagnet/spin-triplet superconductor junctions

We study the tunneling conductance of a ballistic normal metal/ferromagnet/spin-triplet superconductor junction using the extended Blonder-Tinkham-Klapwijk formalism as a model for a $c$-axis-oriented Au/${\mathrm{SrRuO}}_3\text{/}{\mathrm{Sr}}_2{\mathrm{RuO}}_4$ junction. We compare chiral $p$-wave (CPW) and helical $p$-wave (HPW) pair potentials, combined with ferromagnet magnetization directions parallel and perpendicular to the interface. For fixed ${\theta }_M$, where ${\theta }_M$ is a direction of magnetization in the ferromagnet measured from the $c$ axis, the tunneling conductances of CPW and HPW clearly show different voltage dependencies. It is found that the cases where the $d$ vector is perpendicular to the magnetization direction (CPW with ${\theta }_M=\pi /2$ and HPW with ${\theta }_M=0$) are identical. The obtained results serve as a guide to determine the pairing symmetry of the spin-triplet superconductor ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

Schematic of the three-dimensional normal metal (N)/ ferromagnet (F)/superconductor (S) junction. We assume the structure extends to infinity in all directions.

Figure 2

The dimensionless tunneling conductance using pair potentials (a) SW, (b) CPW, and (c) HPW without barriers ($Z_1=0$, solid lines) and including a small barrier at the first interface ($Z_1=0.8$, dashed lines). The SW case is independent of the magnetization angle. For CPW and HPW, the magnetization angle varies from ${\theta }_M=0$ (blue lines) to ${\theta }_M=\pi /2$ (red lines). $X=0.6$, $k_{F}L=11$.

Figure 3

Dimensionless tunneling conductance using pair potentials (a) SW, (b) CPW with ${\theta }_M=0$, (c) HPW with ${\theta }_M=0$, which is identical to CPW with ${\theta }_M=\pi /2$, and (d) HPW with ${\theta }_M=\pi /2$. Magnetization strengths vary from $X=0$ (normal metal) to $X=0.99$ (fully polarized). $Z_1=0.8$, $k_{F}L=11$.

Figure 4

The dimensionless tunneling conductance using pair potentials (a) SW, (b) CPW with ${\theta }_M=0$, (c) HPW with ${\theta }_M=0$, which is identical to CPW with ${\theta }_M=\pi /2$, and (d) HPW with ${\theta }_M=\pi /2$ without a ferromagnet ($k_{F}L=0$) and for varying thicknesses $k_{F}L$ of the ferromagnet. $Z_1=0.8$, $X=0.6$.

Figure 5

Effects of an external magnetic field on the dimensionless tunneling conductance in the absence of the barrier potential. The pair potential is assumed to be (a) SW, (b) CPW with ${\theta }_M=0$, (c) HPW with ${\theta }_M=0$, and (d) HPW with ${\theta }_M=\pi /2$. The results for the CPW with ${\theta }_M=\pi /2$ are identical to the results in (c). The parameters are set to $Z_1=0$, $X=0.6$, and $k_{F}L=11$.

Figure 6

Effects of an external magnetic field on the dimensionless tunneling conductance in the presence of the barrier potential $Z_1=0.8$ in the same manner as in Fig. 5.

Figure 7

The dimensionless tunneling conductance for CPW with ${\theta }_M=\pi /2$ and HPW with ${\theta }_M=0$, including spin-orbit coupling $Z_{\mathrm{SO}}=1$. $Z_1=0.8$, $X=0.6$, $k_{F}L=11$.