Physical properties of weak-coupling quasiperiodic superconductors

We numerically study the physical properties of quasiperiodic superconductors with the aim of understanding superconductivity in quasicrystals. Considering the attractive Hubbard model on the Penrose tiling as a simple theoretical model, we calculate various basic superconducting properties and find deviations from the universal values of the Bardeen-Cooper-Schrieffer theory with a constant density-of-states approximation. In particular, we find that the jump of the specific heat at the superconducting transition is about 10%–20% smaller than that universal value, consistent with the experimental results obtained for the superconducting Al-Mg-Zn quasicrystalline alloy. Furthermore, we calculate current-voltage characteristics and find that the current gradually increases with the voltage on the Penrose tiling in contrast to a rapid increase in the periodic system. These distinctions originate from the nontrivial Cooper pairing characteristic of the quasiperiodic system.

Figure 1

(a) Spatially averaged superconducting order parameter $\overline{\mathrm{OP}}$ and superconducting gap $E_g$ at quarter filling for $U=-3$ as a function of the temperature. The inset shows the square of $\overline{\mathrm{OP}}$ and $E_g$ around $T_c$ with a linear fitting function (black line). (b) Spatial pattern of the site-dependent superconducting order parameter $\mathrm{O}{\mathrm{P}}_i$ at zero temperature in the Penrose tiling of 4181 sites.

Figure 2

Perpendicular space profile of the site-dependent superconducting order parameter $\mathrm{O}{\mathrm{P}}_i$ for (a) $z_{\mathrm{perp}}=0$, (d) $z_{\mathrm{perp}}=1$, (e) $z_{\mathrm{perp}}=2$, and (b) $z_{\mathrm{perp}}=3$ at zero temperature in the Penrose tiling of 11 006 sites. Each perpendicular space for (c) $z_{\mathrm{perp}}=0,3$ and (f) $z_{\mathrm{perp}}=1,2$ is divided into star-shaped sections which correspond to the vertex pattern. (g) Vertex patterns D, Q, K, J, S, S5, S4, and S3 defined for the Penrose tiling [35, 36].

Figure 3

Local density of states for geometrically inequivalent sites at zero temperature obtained for the Penrose tiling of 4181 sites at quarter filling for $U=-3$. The inset shows an enlarged view of Fig. 1 around the center of the Penrose-tiling cluster. To plot the density of states, we have added an imaginary part $i\eta$ (with $\eta =0.003$) to the energy $\omega$.

Figure 4

Temperature dependence of (a) entropy $S$ and (b) specific heat $C_e/T$ obtained in the Penrose tiling of 11 006 sites at quarter filling for $U=-3$. We note that the specific heat is shown in units of $C_{\mathrm{en}}/T_c$, where $C_{\mathrm{en}}$ denotes the specific heat in the normal state at $T_c$. The specific heats obtained for 4181 sites and in the experiment for the Al-Mg-Zn quasicrystalline alloy [4] are plotted for comparison. The dashed curve represents the results calculated for 11 006 sites while artificially restricting a solution to the normal state.

Figure 5

Temperature dependence of specific heat $C_e/T$ obtained in the Penrose tiling of 11 006 sites at different values of the average filling $n$ for $U=-3$. We note that the specific heat is shown in units of $C_{\mathrm{en}}/T_c$, where $C_{\mathrm{en}}$ denotes the specific heat in the normal state at $T_c$.

Figure 6

(a) Distribution of $E_{\alpha }^{\mathrm{diff}}$ in the superconducting state at $T=0.325$ (0.33) for the Penrose (square) lattice. (b) Distribution of $E_{\alpha }$ in the normal state at $T=0.34$ (0.36) for the Penrose (square) lattice. The calculations were done at quarter filling for $U=-3$ on the Penrose (square) lattice of 11 006 (10 000) sites, where $T_c$ is 0.328 (0.333).

Figure 7

(a) Current as a function of $eV/E_g^0$ and (b) the site-averaged density of states at zero temperature obtained for the Penrose tiling of 11 006 sites and for a square lattice of 10 000 sites at quarter filling for $U=-3$ and $\eta =0.001$. The inset shows an enlarged view around $\omega =E_g^0$.

Figure 8

$I-V$ characteristics at various temperatures for the (a) Penrose ($N=11006$) and (b) square ($N=10000$) lattices, calculated for quarter filling and $U=-3$. Insets show the enlarged views around the threshold voltages, $eV=E_g^0=0.555$ and 0.576 (denoted by solid vertical lines), respectively. The dashed vertical lines denote the voltages $eV=E_g^0+\mathrm{\Delta }E$ with $\mathrm{\Delta }E=0.002$ and 0.01, which are used to calculate the slope. (c) The temperature dependence of the slope at $eV=E_g^0$ for the Penrose and square lattices.