Creation of Spin-Triplet Cooper Pairs in the Absence of Magnetic Ordering

In superconducting spintronics, it is essential to generate spin-triplet Cooper pairs on demand. Up to now, proposals to do so concentrate on hybrid structures in which a superconductor (SC) is combined with a magnetically ordered material (or an external magnetic field). We, instead, identify a novel way to create and isolate spin-triplet Cooper pairs in the absence of any magnetic ordering. This achievement is only possible because we drive a system with strong spin-orbit interaction—the Dirac surface states of a strong topological insulator (TI)–out of equilibrium. In particular, we consider a bipolar TI-SC-TI junction, where the electrochemical potentials in the outer leads differ in their overall sign. As a result, we find that nonlocal singlet pairing across the junction is completely suppressed for any excitation energy. Hence, this junction acts as a perfect spin-triplet filter across the SC, generating equal-spin Cooper pairs via crossed Andreev reflection.

Figure 1

Schematic of the TI-SC-TI junction. While the voltages applied to the outer leads should be tunable, the SC is assumed to be grounded. The dispersion relations are depicted for illustration: solid (dashed) lines represent electron (hole) Dirac cones. A possible crossed Andreev reflection process—resulting in the injection of equal-spin-triplet Cooper pairs in the SC due to spin-momentum locking—is sketched.

Figure 2

Electron (solid lines) and hole (dashed lines) dispersion relations in $L$ and $R$, at $k_y=0$. Colored arrows on the branches indicate the Cooper pair spin expectation values according to Eq. (16). The SC is depicted by the gray domain. At $ϵ=\mu$, the incident electron (1) may be reflected as an electron (2) at the interface (NR) or transmitted as a hole (3) through the SC (CAR). LAR and CO are, instead, not permitted since the states at the Dirac points have zero momentum.

Figure 3

Averaged moduli of the nonlocal pairing amplitudes from one interface to the other one. For the bipolar setup, nonlocal singlet pairing is completely suppressed, while triplet pairing remains finite. We set $\mu =0.5{\mathrm{\Delta }}_0$, ${\mu }_S=10{\mathrm{\Delta }}_0$, and $L_S=\xi$ ($\xi =v_F/{\mathrm{\Delta }}_0$) with $\sigma \in \{\uparrow ,\downarrow \}$; $f_0^r$ illustrates both $f_0^r(0,L)$ and $f_0^r(L,0)$.

Figure 4

Averaged moduli of the local and nonlocal conductance for $L_S=1.1\xi$ (blue lines) and $L_S=2.3\xi$ (red lines). At $e{V}_L=\mu$, the curves touch, which is a characteristic feature of the bipolar setup. We set $\mu =0.5{\mathrm{\Delta }}_0$ and ${\mu }_S=10{\mathrm{\Delta }}_0$.

Figure 5

Nonequilibrium net spin ${\mathcal{S}}_x$ pumped into the SC as a function of ${\mu }_R$ and $V_L$. We set $\mu =0.5{\mathrm{\Delta }}_0$, ${\mu }_S=10{\mathrm{\Delta }}_0$, $L_S=1.1\xi$.