Tuning pairing amplitude and spin-triplet texture by curving superconducting nanostructures

We investigate the nature of the superconducting (SC) state in curved nanostructures with Rashba spin-orbit coupling (RSOC). In bent nanostructures with inhomogeneous curvature we find a local enhancement or suppression of the SC order parameter, with the effect that can be tailored by tuning either the RSOC strength or the carrier density. Apart from the local SC spin-singlet amplitude control, the geometric curvature generates nontrivial textures of the spin-triplet pairs through a spatial variation of the $\vec{d}$ vector. By employing the representative case of an elliptical quantum ring, we demonstrate that the $\vec{d}$ vector strongly depends on the local curvature and it generally exhibits a three-dimensional profile whose winding is tied to that of the single electron spin in the normal state. Our findings unveil paths to manipulate the quantum structure of the SC state in RSOC nanostructures through their geometry.

Figure 1

Schematics of the geometric profile of (a) straight, (b) planarly curved nanostructure, (c) ring, and (d) elliptically deformed ring with semiaxes $a$ and $b$. Red (gray) arrows indicate the perpendicular (tangential) direction of the spin orientation. $K$ is the curvature of the ring, e.g., the inverse of its radius $R$. Black (blue) dots indicate a position with minimum $K_{\min }$ (maximal $K_{\max }$) amplitude of the curvature in the elliptical ring, respectively.

Figure 2

Contour map of the averaged spin-singlet OP for a ring ($L=100$ sites) by varying the electron density via $\mu$, the RSOC ${\alpha }_{SO}$, and the SC coupling $g$. $a_0$ is the interatomic distance and $t$ the nearest-neighbor hopping amplitude. In (a) $\mu =0$ (half-filling) and $g/\left(2t\right)=1.0$ in (b).

Figure 3

Spin-singlet OP close to the maximal curvature ($K_{\max }$) (i.e., $s/L=0.25$) in the elliptical ring as a function of (a) $a/b$, (b) the RSOC, and (c) the chemical potential. (d) Difference between the local density of states at $K_{\max }$ and $K_{\min }$ in the elliptical ring (see Fig. 1). Microscopic parameters are (a) $\mu =0.0, {\alpha }_{SO}=1.0, g=1.2$; (b) $a/b=0.1, \mu =0.0, g=1.0,1.18,1.66$ for ${\alpha }_{SO}=0.5,1.0,0.5$, respectively. In (c) ${\alpha }_{SO}=1.0, a/b=0.1, g=1.2$. For (d) we have ${\alpha }_{SO}=1.0, a/b=0.1$. All the energies are in units of $2t$.

Figure 4

Electron spin orientation $\vec{\sigma }$ in the normal state (i.e., $g=0$) for the wire (a), ring (b), and elliptical ring in (c) and (d). (c) and (d) are for the electron spin with a winding around $\mathcal{N}$ and $\mathcal{N}-z$ directions. The $\vec{d}$ vector in (e), (f), (g), and (h) correspond to the spin patterns in (a), (b), (c), and (d), respectively. In the SC state we have $g=0.5$ and $\mu =0.5$ for all the geometrical configurations, i.e., (e)–(h). The RSOC is ${\alpha }_{SO}=0.053$ for the wire and the ring. For the elliptical ring the RSOC is ${\alpha }_{SO}=0.085$ in (c) and (g), while ${\alpha }_{SO}=0.12$ is chosen for (d) and (h). The configurations for the elliptical ring are for $a/b=0.3$. All the energy scales are in units of $2t$.