# Hybrid Superconducting/Superconducting Mesoscopic Heterostructure Studied by Modified Ginzburg–Landau Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Theoretical Approach

#### Appendix A.1. Dimensionless Formulas in the Ginzburg–Landau Theory

#### Dimensionless Form of the Free Energy Functional in a Hybrid System

#### Appendix A.2. Derivation of GL Equations in a Hybrid System

#### Appendix A.2.1. First GL Equation

#### Appendix A.2.2. Second GL Equation

## References

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**Figure 1.**(Color online) Schematic representation of a hybrid superconducting system made of two types of superconductors: ${S}_{1}$ and ${S}_{2}$. The lateral size of the superconducting sample is $a=$ 400 nm, and the width of the sample is $d=$ 20 nm. Two cases are studied: 1. ${S}_{1}$ and ${S}_{2}$ are both type II superconductors with the following parameters: a coherence length of ${\xi}_{1}\left(0\right)=$ 39 nm and ${\xi}_{2}\left(0\right)=$ 20 nm and a penetration depth of ${\lambda}_{1}\left(0\right)=$ 52 nm and ${\lambda}_{2}\left(0\right)=$ 200 nm; 2. ${S}_{1}$ is a type II superconductor with the parameters ${\xi}_{1}\left(0\right)=$ 39 nm and ${\lambda}_{1}\left(0\right)=$ 52 nm, but ${S}_{2}$ is a type I superconductor with the parameters ${\xi}_{2}\left(0\right)=$ 120 nm and ${\lambda}_{1}\left(0\right)=$ 72 nm.

**Figure 2.**(Color online) (

**Left**) The curve represents the magnetization as a function of an applied magnetic field H for a type II/type II superconducting square sample with $a=$ 400 nm and a width of $d=$ 20 nm. (

**Right**) Contour plots of the density of the superconducting order parameter for different selected vortex states which correspond to the selected points on the magnetization curve named with letters from (

**a**–

**g**); the upper section of the simulated sample is ${S}_{1}$, and the characteristic lengths are ${\xi}_{1}\left(0\right)=$ 39 nm and ${\lambda}_{1}\left(0\right)=$ 52 nm, whereas the lower section is ${S}_{2}$ with ${\xi}_{2}\left(0\right)=$ 20 nm, ${\lambda}_{2}\left(0\right)=$ 200 nm (first column), and the corresponding phase of the order parameter (second column).

**Figure 3.**(

**g**–

**i**) (Color online) Contour plots of the superconducting order parameter for selected vortex states that show that the vortices move from ${S}_{1}$ to ${S}_{2}$ with larger values of the applied magnetic field, when considering the case analyzed in Figure 2 to follow the evolution of the behavior of the vortex state. Snapshots (

**l**–

**o**) show contour plots of the superconducting order parameter for selected vortex states; these show the vortices that correspond to the lower jumps in the magnetization curve. The simulation corresponds to a type II/type II superconducting square sample with $a=$ 400 nm and a width of $d=$ 20 nm. The characteristic lengths for ${S}_{1}$ are ${\xi}_{1}\left(0\right)=$ 39 nm and ${\lambda}_{1}\left(0\right)=$ 52 nm, and for ${S}_{2}$, they are ${\xi}_{2}\left(0\right)=$ 20 nm and ${\lambda}_{2}\left(0\right)=$ 200 nm.

**Figure 4.**(Color online) The curve represents the vorticity as a function of the applied magnetic field H for a type II/type II superconducting square sample with $a=$ 400 nm and a width of $d=$ 20 nm. The characteristic lengths for ${S}_{1}$ are ${\xi}_{1}\left(0\right)=$ 39 nm and ${\lambda}_{1}\left(0\right)=$ 52 nm, and those for ${S}_{2}$ are ${\xi}_{2}\left(0\right)=$ 20 nm and ${\lambda}_{2}\left(0\right)=$ 200 nm.

**Figure 5.**(

**a**–

**g**) (Color online) The corresponding simulation of the behavior of the supercurrent for different selected vortex states in a type II/type II superconducting square sample with $a=$ 400 nm and a width of $d=$ 20 nm. The characteristic lengths for ${S}_{1}$ were ${\xi}_{1}\left(0\right)=$ 39 nm and ${\lambda}_{1}\left(0\right)=$ 52 nm, and those for ${S}_{2}$ are ${\xi}_{2}\left(0\right)=$ 20 nm and ${\lambda}_{2}\left(0\right)=$ 200 nm.

**Figure 6.**The curve represents the magnetization as a function of the applied magnetic field H for a type II/type I superconducting sample. The size of the superconducting sample is $a=$ 400 nm and $d=$ 20 nm. The following parameters are chosen for ${S}_{1}$ and ${S}_{2}$: a coherence length of ${\xi}_{1}\left(0\right)=$ 39 nm and ${\xi}_{2}\left(0\right)=$ 120 nm and a penetration depth of ${\lambda}_{1}\left(0\right)=$ 52 nm and ${\lambda}_{2}\left(0\right)=$ 72 nm.

**Figure 7.**(

**a**–

**c**) (Color online) Contour plots of the density of the superconducting order parameter for different selected vortex states (first column), the phase of the order parameter (second column), and the supercurrent corresponding to the first three vortex states in the sample (third column). The size of the considered superconducting sample is $a=$ 400 nm and $d=$ 20 nm. The following parameters are chosen for ${S}_{1}$ and ${S}_{2}$: a coherence length of ${\xi}_{1}\left(0\right)=$ 39 nm and ${\xi}_{2}\left(0\right)=$ 120 nm and a penetration depth of ${\lambda}_{1}\left(0\right)=$ 52 nm and ${\lambda}_{2}\left(0\right)=$ 72 nm.

**Table 1.**The vortex states in ${S}_{1}$ and ${S}_{2}$, the applied magnetic field $H/{H}_{c2,1}\left(0\right)$, the entry of magnetic flux $\Phi /{H}_{c2,1}\left(0\right){\xi}_{1}^{2}\times {10}^{-4}$ into the sample, and the variations in magnetization $\Delta M/{H}_{c2,1}\times {10}^{-5}$ that we obtained for the magnetization curve (Figure 2). The superconducting square sample had a length of $a=$ 400 nm and a width of $d=$ 20 nm. For ${S}_{1}$, the characteristic lengths were ${\xi}_{1}\left(0\right)=$ 39 nm and ${\lambda}_{1}\left(0\right)=$ 52 nm; for ${S}_{2}$, they were ${\xi}_{2}\left(0\right)=$ 20 nm and ${\lambda}_{2}\left(0\right)=$ 200 nm.

Point | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | $\mathbf{\Phi}$ | $\mathbf{\Delta}\mathit{M}$ |
---|---|---|---|---|

(a) | 0 | 2 | 2.32 | 0.455 |

(b) | 1 | 2 | 4.67 | 0.893 |

(c) | 1 | 4 | 1.91 | 0.364 |

(d) | 2 | 3 | 3.76 | 0.715 |

(e) | 3 | 4 | 2.574 | 0.522 |

(f) | 4 | 6 | 2.571 | 0.489 |

**Table 2.**The vortex states in ${S}_{1}$ and ${S}_{2}$, the applied magnetic field $H/{H}_{c2,1}\left(0\right)$, the entry of magnetic flux ${\Phi}_{0}/{H}_{c2,1}\left(0\right){\xi}_{1}^{2}\times {10}^{-4}$ into the sample, and the variations in magnetization $\Delta M/{H}_{c2,1}\times {10}^{-5}$ that we obtained for the magnetization curve shown in Figure 6 (points (a–g)). The size of the considered superconducting sample was $a=$ 400 nm and $d=$ 20 nm. The following parameters were chosen for ${S}_{1}$ and ${S}_{2}$: a coherence length of ${\xi}_{1}\left(0\right)=$ 39 nm and ${\xi}_{2}\left(0\right)=$ 120 nm and a penetration depth of ${\lambda}_{1}\left(0\right)=$ 52 nm and ${\lambda}_{2}\left(0\right)=$ 72 nm.

Point | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | H | $\mathbf{\Phi}$ | $\mathbf{\Delta}\mathit{M}$ |
---|---|---|---|---|---|

(a) | 0 | 1 | 0.038 | 4.328 | 0.824 |

(b) | 0 | 2 | 0.063 | 2.363 | 0.450 |

(c) | 0 | 3 | 0.088 | 0.493 | 0.094 |

(d) | 1 | 3 | 0.110 | 4.242 | 0.807 |

(e) | 2 | 5 | 0.161 | 4.164 | 0.792 |

(f) | 3 | 7 | 0.211 | 3.430 | 0.653 |

(g) | 4 | 9 | 0.260 | 2.689 | 0.512 |

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**MDPI and ACS Style**

González, J.; Melendez, A.; Camargo, L.
Hybrid Superconducting/Superconducting Mesoscopic Heterostructure Studied by Modified Ginzburg–Landau Equations. *Condens. Matter* **2023**, *8*, 104.
https://doi.org/10.3390/condmat8040104

**AMA Style**

González J, Melendez A, Camargo L.
Hybrid Superconducting/Superconducting Mesoscopic Heterostructure Studied by Modified Ginzburg–Landau Equations. *Condensed Matter*. 2023; 8(4):104.
https://doi.org/10.3390/condmat8040104

**Chicago/Turabian Style**

González, Jesús, Angélica Melendez, and Luis Camargo.
2023. "Hybrid Superconducting/Superconducting Mesoscopic Heterostructure Studied by Modified Ginzburg–Landau Equations" *Condensed Matter* 8, no. 4: 104.
https://doi.org/10.3390/condmat8040104