Proximity-induced triplet superconductivity in Rashba materials

We study a proximity junction between a conventional $s$-wave superconductor and a conductor with Rashba spin-orbit coupling, with a specific focus on the spin structure of the induced pairing amplitude. We find that spin-triplet pairing correlations are induced by spin-orbit coupling in both one- and two-dimensional systems due to the lifted spin degeneracy. Additionally, this induced triplet pairing has a component with an odd frequency dependence that is robust to disorder. Our predictions are based on the solutions of the exact Gor'kov equations and are beyond the quasiclassical approximation.

Figure 1

Bogoliubov-de Gennes excitation spectra of (a) a 1D ferromagnet and (b) a 1D Rashba metal with Rashba vector $\mathbf{g}=(0,0,-i{k}_x)$. Colors denote different spin states, while solid (dashed) lines represent electron (hole) bands. An incident electron ($e_{\uparrow }$ or $e_{\downarrow }$) with energy $\varepsilon <\mathrm{\Delta }$ is Andreev reflected as a hole of opposite spin ($h_{\downarrow }$ or $h_{\uparrow }$, respectively). The presence of a triplet pairing component with zero spin projection results from the lifted spin degeneracy. (c) Fermi surfaces of a 2D Rashba metal, with Rashba vector $\mathbf{g}=(i{k}_y,-i{k}_x,0)$, taking $E_F>0$. Because the two Rashba subbands are not associated with a definite spin, an incident electron ($e_1$ or $e_2$) with momentum $|k_y|<k_{F1}$ can be Andreev reflected into either subband ($h_1$ or $h_2$). The total pairing amplitude is a sum of oscillatory and nonoscillatory terms, resulting from interband and intraband Andreev reflection, respectively. Note: though they appear different, the 1D picture displayed in (b) and the $k_y=0$ limit of the 2D picture displayed in (c) are physically equivalent.

Figure 2

Geometry of the S/R proximity effect, together with the band structure of each material. System is taken to be infinite in the $y$ direction in 2D, while 1D corresponds to the single-channel limit $k_y=0$.

Figure 3

Spatial dependence of proximity-induced (a) singlet and (b) triplet pairing amplitudes, as defined in Eq. (43), in a 2D Rashba proximity junction for various strengths of SOC (${\xi }_S=v_F/\mathrm{\Delta }$ is the coherence length of the superconductor). Pairing amplitude is plotted in units of its value in the bulk of the superconductor. (c) Quantitative comparison between induced singlet and triplet amplitudes for $E_{so}/E_{\mathit{FR}}=2$ [dotted curves from (a) and (b)], which is a reasonable choice for BiTeI [38]. All pairing amplitudes are plotted with $\omega /\mathrm{\Delta }=0.8, m_R/m_S=0.1, E_{\mathit{FR}}/E_{FS}=0.05$, and $E_{FS}/\mathrm{\Delta }=200$. By the Pauli principle, the triplet component has an odd frequency dependence.

Figure 4

Quantitative comparison between induced triplet pairing amplitude in two-subband (solid line) and one-subband (dotted line) cases. Chosen parameters are the same as those in Fig. 3, with only a change in the sign of $E_{\mathit{FR}}/E_{FS}$ for the one-subband case.