Proximity effect in a superconductor-quasicrystal hybrid ring

We compute the real-space profile of the superconducting order parameter (OP) in a hybrid ring that consists of a one-dimensional superconductor connected to a Fibonacci chain using a self-consistent approach. In our paper, the strength of the penetration as measured by the order parameter at the center of the quasicrystal depends on the structural parameter $\varphi$ or phason angle that characterizes different realizations of the Fibonacci chains of a given length. We show that the penetration strength dependence on $\varphi$ reflects properties of the topological edge states of the Fibonacci chain. We show that the induced superconducting order parameter averaged over all chains has a power-law decay as a function of distance from the superconductor-normal interface. More interestingly, we show that there are large OP fluctuations for individual chains and that the penetration strength in a finite Fibonacci chain can be larger than in a normal periodic conductor for special values of $\varphi$.

Figure 1

(a) A sketch of a quasicrystal-superconductor hybrid chain. On the left is a Fibonacci chain with nearest-neighbor hopping terms that take one of two values $t_A,t_B$ modulated according to the Fibonacci chain; on the right is a superconductor with constant hopping $t_{\mathrm{SC}}$ and on site attraction $V$ between opposite spins. (b) Successive phason flips $\left(AB\to BA\right)$ in the six-site Fibonacci chain as $\varphi$ is varied. (a) The normalized integrated density of states (IDOS) for the Fibonacci chain. The four largest gaps are marked with their $q$ labels. The figure is taken from Ref. [14] with modifications.

Figure 2

Self-consistently computed order parameter in real space for various $t_A/t_B$'s. The 90-site quasicrystal is on the left, and the 90-site superconductor is on the right. The blue (green) curves have been shifted up by 0.025 (0.05) for clarity.

Figure 3

The order parameter profile averaged over $\varphi$ compared with a typical OP profile for $t_A/t_B=0.9$. The 90-site quasicrystal is on the left, and the 35-site superconductor is on the right. The red curve has been shifted up for clarity. The inset: The average penetration strength as a function of the modulation strength. We measure the average penetration strength as the OP amplitude at the center of the quasicrystal in the $\varphi$-averaged OP profile.

Figure 4

(a) and (b) The penetration strength ${\mathrm{\Delta }}_{\mathrm{mid}}$ of the order parameter into the Fibonacci chain as a function of the $\varphi$ used to generate the chain. In (a), the FC is 145-site long, and the different colors represent different modulation strengths. In (b), the modulation is constant $(t_A/t_B=0.9)$, but the length of the quasicrystal $N$ is varied. (c) Power spectrum of the penetration strength [blue curve in (a)] $|\mathcal{F}\left[{\mathrm{\Delta }}_{\mathrm{mid}}\left(\varphi \right)\right]{|}^2$ with the prominent peaks highlighted.

Figure 5

The lower half of the spectrum of the open Fibonacci chain of length 145 as a function of $\varphi$ with $t_A/t_B=0.5$. The gaps closest to the Fermi level are highlighted.