Superconducting proximity effect and order parameter fluctuations in disordered and quasiperiodic systems

We study the superconducting proximity effect in inhomogeneous systems in which a disordered or quasicrystalline normal-state wire is connected to a BCS superconductor. We self-consistently compute the local superconducting order parameters in the real-space Bogoliubov–de Gennes framework for three cases, namely, when states are (i) extended, (ii) localized, or (iii) critical. The results show that the spatial decay of the superconducting order parameter as one moves away from the normal-superconductor interface is power law in cases (i) and (iii), stretched exponential in case (ii). In the quasicrystalline case, we observe self-similarity in the spatial modulation of the proximity-induced superconducting order parameter. To characterize fluctuations, which are large in these systems, we study the distribution functions of the order parameter at the center of the normal region. These are Gaussian functions of the variable [case (i)] or of its logarithm [cases (ii) and (iii)]. We give arguments to explain the characteristics of the distributions and their scaling with system size for each of the three cases.

Figure 1

Superconducting proximity effect in a normal one-dimensional metal: (a) Sketch of a normal-superconductor (N-SC) hybrid ring: on the left side is the nonsuperconducting region, whereas on the right is an intrinsic superconductor. (b) Spatial variation of the superconducting order parameter ${\mathrm{\Delta }}_i$ along the chain for a clean N-SC hybrid ring—both parts are bulk-periodic. (c) Spatial decay of the superconducting order parameter in the normal part of a clean N-SC ring for zero and finite temperature ($\beta =1/kT$), extracted by finite-size scaling of ${\mathrm{\Delta }}_{\mathrm{mid}}$ for chains with varying $L$. The best-fit parameters are $x_o=10.0$ and $\zeta =6.4$. $\mathrm{\Delta }$ decays inversely with distance at zero temperature and exponentially with distance at finite temperature.

Figure 2

Superconducting proximity effect in a disordered one-dimensional metal: spatial profile of the superconducting order parameter $\mathrm{\Delta }$ along the chain in a hybrid N-SC system with off-diagonal disorder in the normal region. Disorder strength: $W/t=0.08$.

Figure 3

Superconducting proximity effect in the weak-disorder regime: (a) Finite-size scaling of $\langle {\mathrm{\Delta }}_{\mathrm{mid}}\rangle$ with the length of the normal region $L$ shows inverse-law decay of $\mathrm{\Delta }$ away from the interface in N-SC hybrid rings where N is weakly disordered. The variance of ${\mathrm{\Delta }}_{\mathrm{mid}}$ is also inversely proportional to the distance. Disorder strength: $W/t=0.005$. Fit offset $x_0=10.4$. (b) Normal distribution of ${\mathrm{\Delta }}_{\mathrm{mid}}$ in N-SC rings in the weakly disordered regime. 10 000 realizations, $W/t=0.005$, $L=90$, $L_{\mathrm{SC}}=200$. (c) The dependence of the mean and standard deviation of ${\mathrm{\Delta }}_{\mathrm{mid}}$ on the disorder strength $W/t$ in the weakly disordered regime.

Figure 4

Superconducting proximity effect in the strong-disorder regime: (a) Finite-size scaling of $\langle {\mathrm{\Delta }}_{\mathrm{mid}}\rangle$ with the length of the normal region $L$ shows stretched exponential decay of $\mathrm{\Delta }$ away from the interface in N-SC hybrid rings where N is strongly disordered. Disorder strength: $W/t=0.7$. $\zeta =8.4$. (b) Log-normal distribution of ${\mathrm{\Delta }}_{\mathrm{mid}}$ in N-SC rings in the strongly disordered regime. 9000 realizations, $W/t=0.2$, $L=90$, $L_{\mathrm{SC}}=200$. (c) The dependence of the mean and standard deviation of $ln\left({\mathrm{\Delta }}_{\mathrm{mid}}\right)$ on the disorder strength $W/t$ in the strongly disordered regime.

Figure 5

Superconducting proximity effect in a Fibonacci chain: (a) Real-space profile of the superconducting order parameter when the normal segment is a Fibonacci chain. Modulation strength: $W/t=0.1$. (b) Superconducting order parameter profile in the central 987 (top), 233 (middle), and 55 (bottom) segments of the Fibonacci chain in a hybrid Fibonacci-SC ring. The order-parameter profiles are self-similar upon scaling by the renormalization parameter ${\tau }^3$. The data in panel (b) are taken from a hybrid ring with 2585 and 200 sites in the normal and superconducting regions, respectively. (c) The hopping sequence corresponding to the sites shown in the last panel in (b).

Figure 6

The local order parameter and its correlation with the symmetry of the local environment: the peaks in ${\mathrm{\Delta }}_i$ occur at sites with high reflection symmetry represented by ${\lambda }_i$.

Figure 7

Superconducting proximity effect in a Fibonacci chain: (a) Finite-size scaling of $\langle {\mathrm{\Delta }}_{\mathrm{mid}}\rangle$ with the length of the normal region $L$ shows power-law decay of $\mathrm{\Delta }$ away from the interface in N-SC hybrid rings where N is a Fibonacci chain. Modulation strength: $W/t=0.005$. Fit offset $x_0=10.85$. (b) The distribution of $ln{\mathrm{\Delta }}_{\mathrm{mid}}$ in N-SC rings is symmetric when N is a Fibonacci chain. $L=988$, $L_{\mathrm{SC}}=200$. (c) The dependence of the mean and standard deviation of ${\mathrm{\Delta }}_{\mathrm{mid}}$ on the modulation strength $W/t$. Inset: The functional dependence of the width, $\sigma (ln{\mathrm{\Delta }}_{\mathrm{mid}})$, on the ratio of the hopping parameters.