Superconducting properties of Fibonacci chains with enhanced superconducting pairing at the boundaries

TL;DR

This work addresses end superconductivity in Fibonacci quasicrystals with $s$-wave pairing by solving the self-consistent Bogoliubov-de Gennes equations on finite off-diagonal Fibonacci chains with a Hubbard attraction. The analysis reveals three distinct critical temperatures, $T_{cb}$ for the bulk, $T_{cL}$ for the left end, and $T_{cR}$ for the right end, arising from the interplay between topological bound states and critical, multifractal states; end enhancements can be substantial, especially at the right end for odd sequence parity and $t_A/t_B oughly 1.6$. The left-end and bulk temperatures show weak dependence on the sequence index $n$, while the right-end temperature exhibits strong parity dependence and can exceed the bulk value by up to about $65 ext{--}68 ext%}$ under favorable conditions. These findings highlight a mechanism to engineer higher effective critical temperatures in quasicrystals via boundary interference effects and complex state structure, with potential implications for boundary-controlled superconductivity in quasiperiodic materials.

Abstract

Recently, the superconducting properties of Fibonacci quasicrystals have attracted considerable attention. By numerically solving the self-consistent Bogoliubov-de Gennes equations for an $s-$wave superconducting Fibonacci chain, we find that the system exhibits universal end superconductivity, where the pair potential at the chain ends can persist at higher temperatures compared to the bulk critical temperature ($T_{cb}$) of the condensate in the chain center. Furthermore, our study reveals two distinct critical temperatures at the left ($T_{cL}$) and right ($T_{cR}$) ends of the chain. This complex behavior arises from the competition between topological bound states and critical states, a characteristic of quasicrystals. With the chosen parameters, the maximal enhancement of $T_{cR}$ reaches up to $66\%$ relative to $T_{cb}$, while $T_{cL}$ can increase by up to $31\%$. Our study sheds light on the phenomenon of end superconductivity in Fibonacci quasicrystals, pointing to alternative pathways for increasing the superconducting critical temperature.

Figures (10)

• Figure 1: The pair potential $\Delta(i)$ and the corresponding spatial electron distribution $n_e(i)$ as functions of the site number $i$ for $T=0$, $0.2$, $0.258$, and $0.274$. The calculations are performed at $t_A=0.8$ for the Fibonacci chain $S_{n=13}$ with the total number of sites $N=F_{13}+1 = 234$.
• Figure 2: (a-c, e-g) Single-species quasiparticle contribution $\Delta_\alpha(i)$ as a function of the quantum number $\alpha$ at $i=1$ (red), $i={\color{black}N}$ (blue) and in the chain center (green) for $T=0.258$ and $0.274$. (d, h) The quasiparticle energy spectrum $\varepsilon_\alpha$ versus $\alpha$ at $T=0.258$ and $0.274$, respectively; the insets show a zoomed-in view of the zero-energy region. The calculations are performed for $S_{n=13}$ with $t_A=0.8$.
• Figure 3: (a-e) Typical examples of the spatial profile of the single-species contribution $\Delta_\alpha(i)$ for topological end quasiparticles with $\alpha=56$ and $166$ (in red), $81$ and $124$ (in blue), and for the critical quasiparticle state with $\alpha=35$ (in black). (f-j) The corresponding examples of the probability density distribution $|u_\alpha(i)|^2 + |v_\alpha(i)|^2$. The results are calculated for $S_{n=13}$ at $t_A=0.8$ and $T=0.258$.
• Figure 4: The same as in Fig. \ref{['fig3']} but at $T=0.274$.
• Figure 5: (a-d) Temperature-dependent pair potentials $\Delta(1)$ (blue circles), $\Delta(N)$ (red triangles) and $\Delta({\rm bulk})$ (green squares) for the chain $S_{n=13}$ with $t_A=0.8$, $1$, $1.6$, and $3$. In each panel, the two solid green curves represent the maximal and minimal values of the multifractal order parameter, calculated over the site domain $[0.4N,\,0.6N]$. The critical temperatures $T_{cL}$, $T_{cR}$, and $T_{cb}$ are defined in panel (a): each corresponds to the temperature at which the respective pair potential drops to zero. (e-h) $\Delta_{\alpha}(i=1,\,{\color{black}N})$ versus $\alpha$ at $T=0$ for the same chain and $t_A$ values. For reference, the end pair potentials $\Delta(1)$ and $\Delta(N)$ are also included.