Superconductor-Proximity Effect in Hybrid Structures: Fractality versus Chaos

We study the proximity effect of a superconductor to a normal system with a fractal spectrum. We find that there is no gap in the excitation spectrum, even in the case where the underlying classical dynamics of the normal system is chaotic. An analytical expression for the distribution of the smallest excitation eigenvalue $E_1$ of the hybrid structure is obtained. On small scales it decays algebraically as $\mathcal{P}(E_1)\sim E_1^{-D_0}$, where $D_0$ is the fractal dimension of the spectrum of the normal system. Our theoretical predictions are verified by numerical calculations performed for various models.

Figure 1

The distribution of the lowest excited energy, generated over an ensemble of different Fermi energies, for the KH model (4). Various symbols correspond to different system sizes and numbers of channels. All data overlap with one another indicating that $\mathcal{P}(E_1)$ is insensitive to the number of channels and the system size. The dashed line is the theoretical prediction of Eq. (2). In the insets we show the numerically evaluated $⟨E_1⟩$ versus the system size $L$ and the number of channels $M$.

Figure 2

The distribution $\mathcal{P}(E_1)$ of a hybrid structure consisting of a Fibonacci sample of size $L$ (normal part) attached with $M$ semi-infinite one-dimensional superconductors. The data for various $L,M$ fall one on top of the other, indicating that $\mathcal{P}(E_1)$ is independent of $L,M$. In the insets we report the mean value $⟨E_1⟩$ of the distribution as a function of system size $L$ and number of channels $M$.

Figure 3

(a) The distribution $\mathcal{P}(E_1)$ for hybrid structures where the normal sample is a Fibonacci lattice with $L=987$ and $M=1$ and various $V$’s. The dashed lines are the theoretical predictions (2); (b) the mean value $⟨{\tilde{E}}_1⟩$ versus the theoretical prediction (3) for various $V$ values. The straight line $y=-0.039+0.993x$ represents the best linear fit.

Figure 4

The distribution $\mathcal{P}({\tilde{E}}_1)$ for a hybrid structure where the normal part is a Fibonacci system with $L=987$ and $M=50$ and $V=1.4$ (○). Overplotted (★) is the corresponding distribution $\mathcal{P}({\tilde{E}}_1)$ for the same Fibonacci system disconnected from a superconductor.