# Superfluid Properties of Superconductors with Disorder at the Nanoscale: A Random Impedance Model

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## Abstract

**:**

## 1. Introduction

## 2. The RRN Model

## 3. The RIN Model

- ${w}_{1}$: the superconducting fraction percolating at ${T}_{c}$, with local critical temperatures following a Gaussian distribution with parameters (${\mu}_{c,1}$, ${\sigma}_{1}$);
- ${w}_{2}$: the superconducting fraction arising after proximization, with local critical temperatures following a Gaussian distribution with parameters (${\mu}_{c,2}\le {\mu}_{c,1}$, ${\sigma}_{2}$);
- ${w}_{3}=1-{w}_{1}-{w}_{2}$: the fraction of the residual metallic matrix, that will never undergo proximization.

## 4. EMT Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

EMT | Effective Medium Theory |

RIN | Random Impedance Network |

RRN | Random Resistor Network |

TMD | Transition Metal Dichalcogenides |

## Appendix A

## Appendix B

## References

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**Figure 1.**Temperature dependence of the effective resistance ${R}_{em}$ for ${\mu}_{c,s}=0.15$ K, ${\sigma}_{s}=0.02$ K and two values of ${w}_{s}$. The red (orange) curve corresponds to the pink (yellow) shaded area. As one can see, as soon as ${w}_{s}<0.5$, the zero-temperature resistance saturates to a finite value. Observe that the area underneath each gaussian distribution equals the corresponding value of ${w}_{s}$.

**Figure 2.**EMT calculations of $-{g}_{em}^{\u2033}$ (upper panels) and ${g}_{em}^{\prime}$ (lower panels), as a function of the temperature T, for different tuning parameters. The pink filled Gaussian corresponds to the probability distribution of the percolating superconducting component with total fraction w

_{1}= 0.5, while the other filled gaussians to the distribution of the proximized bonds. The parameters tuned in each figures are respectively: (

**a**) the total fraction w

_{2}of the proximized component; (

**b**) its variance σ

_{2}; (

**c**) its mean value u

_{c}

_{,2}; and (

**d**) the value of its inductance L

_{2}keeping in this case fixed L

_{1}= 2 nH. Where not specified in the labels, L

_{1}= L

_{2}= 1 nH, w

_{2}= 0.03. In all cases, we kept fixed R

_{N}= 800 Ω, L

_{3}= 10 nH and ω = 2 GHz. The areas underneath all Gaussian distributions equal the corresponding fractions w

_{j}.

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**MDPI and ACS Style**

Venditti, G.; Maccari, I.; Grilli, M.; Caprara, S.
Superfluid Properties of Superconductors with Disorder at the Nanoscale: A Random Impedance Model. *Condens. Matter* **2020**, *5*, 36.
https://doi.org/10.3390/condmat5020036

**AMA Style**

Venditti G, Maccari I, Grilli M, Caprara S.
Superfluid Properties of Superconductors with Disorder at the Nanoscale: A Random Impedance Model. *Condensed Matter*. 2020; 5(2):36.
https://doi.org/10.3390/condmat5020036

**Chicago/Turabian Style**

Venditti, Giulia, Ilaria Maccari, Marco Grilli, and Sergio Caprara.
2020. "Superfluid Properties of Superconductors with Disorder at the Nanoscale: A Random Impedance Model" *Condensed Matter* 5, no. 2: 36.
https://doi.org/10.3390/condmat5020036