# Superconducting Properties and Electron Scattering Mechanisms in a Nb Film with a Single Weak-Link Excavated by Focused Ion Beam

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Sample Characterization

#### 3.2. Superconducting Properties

#### 3.3. Normal State Properties

#### 3.4. The Evolution of the Critical Temperature with Disorder

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Abbreviations

FIB | Focused ion beam |

MO | Magneto-optical |

AFM | Atomic force microscopy |

EDS | Energy Dispersive X-ray Spectrometry |

SEM | Scanning Electron Microscope |

GF | Grooved Film |

PF | Plain Film |

S | Plain region of Grooved Film |

S′ | Groove region of Grooved Film |

${T}_{c}$ | Superconducting critical temperature |

WL | Weak-link |

${T}_{c}^{WL}$ | Weak-link critical temperature |

${H}_{\mathrm{rem}}$ | Remnant DC magnetic field |

h | Applied excitation field amplitude |

T | Temperature |

$\rho \left(T\right)$ | Temperature-dependent resistivity |

n | Power-law exponent in $\rho \left(T\right)$ |

${A}_{S}$ (${A}_{{S}^{\prime}}$) | Cross-section area of the grain (groove) |

${I}_{S}$ (${I}_{{S}^{\prime}}$) | Length of the grain (groove) |

${\rho}_{0}$ | Residual resistivity |

${\rho}_{{S}^{\prime}}$ | Groove resistivity |

${\rho}_{G}$ | Grain resistivity |

${\rho}_{epi}$ | Electron-phonon-impurity resistivity contribution |

${\rho}_{sd}$ | $sd$ interband scattering resistivity contribution |

${\rho}_{ss}$ | $ss$ intraband scattering resistivity contribution |

${\rho}_{10}$ | Resistivity at 10 K |

${\rho}_{300}$ | Resistivity at 300 K |

${V}_{\mathrm{G}}$ | Voltage measured between electrodes 2 and 3 (plain region) |

${V}_{\mathrm{SS}\prime \mathrm{S}}$ | Voltage measured between electrodes 3 and 4 (SS′S contribution) |

B | Electron-phonon-impurity coefficient |

${\Theta}_{D}$ | Debye temperature |

${T}_{\mathrm{max}}$ | Upper limit of temperature to fit $\rho \left(T\right)$ |

${\chi}_{AC}\left(T\right)$ | Temperature-dependent AC susceptibility |

${\chi}^{\prime}$ | Real part of ${\chi}_{AC}$ |

${\chi}^{\u2033}$ | Imaginary part of ${\chi}_{AC}$ |

f | Frequency |

R | Resistance |

$RRR$ | Residual resistivity ratio |

l | Mean free path |

$\xi $ | Coherence length |

$\lambda $ | Penetration depth |

${H}_{c2}$ | Upper critical field |

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**Figure 1.**(Color online) (

**a**) Magneto-optical (MO) image of the grooved Nb film taken at 8 K and 50 Oe after a ZFC procedure. (

**b**) Schematic representation of the sample and details of the contact leads used in transport measurements. (

**c**) AFM image and (

**d**) profile for the SS′S region. (

**e**) SEM image and the identification of three different points at which the EDS spectra were taken. (

**f**) EDS spectra showing the ${K}_{\alpha}$ line for gallium, which appears only in the region inside the groove (2).

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

**a**) Temperature-dependent AC susceptibility for two different samples: a plain film (PF) and a structured film with a central groove (GF). The measurements were taken for $100\phantom{\rule{3.33333pt}{0ex}}Hz$ and remnant DC field (${H}_{\mathrm{rem}}$). Open symbols (above the line at $\chi /{\chi}_{0}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$) represent the ${\chi}^{\u2033}$ component, while closed symbols (below $\chi /{\chi}_{0}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0$) are ${\chi}^{\prime}$ values. The inset presents the critical temperatures versus AC field amplitude (h) obtained for the GF specimen using different criteria. (

**b**) Detail of magnetic moment versus DC applied field (H) curves taken at different temperatures (T) for the GF sample. The upper left inset presents the complete hysteresis loops for the GF sample, whereas the upper right inset shows the forth quadrant of the $M\left(H\right)$ curves for the PF sample at different temperatures. The axis units are consistent throughout the panels.

**Figure 3.**(Color online) (

**a**) Temperature-dependent resistance curves for the SS′S contribution (main panel) and for the grain region (inset) for different applied currents ranging from 0.1 to 5.0 mA at ${H}_{\mathrm{rem}}$. (

**b**) Temperature-dependent resistance normalized by the resistance at 9.5 K for the SS′S contribution (closed symbols) at i = 0.1 mA and different applied DC magnetic fields ranging from remnant field to 7 kOe, and for remnant field, 2 kOe, 4 kOe, 5 kOe, and 6 kOe for the grain region (open symbols).

**Figure 4.**(Color online) Temperature-dependent resistivity for the grain contribution. In panel (

**a**), the red line is the fitting for $n\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}5$ in Equation (4) up to 150 K, whereas (

**b**) shows the fitting for n as a free parameter for same equation. The upper and lower insets for both panels show the R-square coefficient versus the upper limit of temperature ${T}_{\mathrm{max}}$ and the relative fit error versus temperature, respectively.

**Figure 5.**(Color online) Resistivity versus temperature (up to 150 K) curves for the groove. The fitting depicted by the red curve in (

**a**) represents the $sd$ band scattering ($n\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}3$), as described by Equation (4). In (

**b**), the red curve represents the fitting considering an additional contribution given by Equation (5) in comparison to the fitting in (

**a**). The lower inset in both panels represents the relative fit error and the upper insets shows the quality of the fit given by the R-square coefficient, which changes when the temperature upper limit is varied.

**Figure 6.**Relative change of the critical temperature as a function of the relative beam dose used during groove milling. Red solid curve is the fit of Equation (6) using the FIB dose as control parameter. This analysis demonstrates that the data follows the expected trend for strong-coupled superconductors.

**Table 1.**Resistivity at 300 K and 10 K; RRR; Superconducting critical temperature, ${T}_{c}$; mean free path, l; coherence length, $\xi \left(0\right)$; penetration depth, $\lambda \left(0\right)$. All listed parameters were obtained for the grain and weak-link contributions.

Parameters | Grain | Weak-Link |
---|---|---|

${\rho}_{300}$ [$\mathsf{\mu}\mathsf{\Omega}$cm] | 22.48 | 302.49 |

${\rho}_{10}$ [$\mathsf{\mu}\mathsf{\Omega}$cm] | 3.11 | 114.62 |

RRR | 7.23 | 2.64 |

${T}_{c}^{onset}$[K] | $9.25\pm 0.05$ | $8.75\pm 0.05$ |

l [nm] | $12.02\pm 0.05$ | $0.33\pm 0.03$ |

$\xi \left(0\right)$ [nm] | $18.51\pm 0.04$ | $3.06\pm 0.14$ |

$\lambda \left(0\right)$ [nm] | $43.06\pm 0.09$ | $259\pm 12$ |

Contribution | $\mathit{\rho}\left(\mathit{T}\right)=$ | n | ${\mathit{\rho}}_{0}$ ($\mathsf{\mu}\mathsf{\Omega}$cm) | B (${10}^{-5}$ K${}^{-2}$) | ${\mathit{K}}_{0}$ ($\mathsf{\mu}\mathsf{\Omega}$cm) | Adj. R-Square |
---|---|---|---|---|---|---|

Grain | ${\rho}_{0}+{\rho}_{ss}$ | 5 | 3.426 ± 0.018 | - | 80.5 ± 0.3 | 0.99799 |

Grain | ${\rho}_{0}+{\rho}_{sd}$ | 3.000 ± 0.013 | 3.095 ± 0.004 | - | 39.21 ± 0.25 | 0.99998 |

Groove | ${\rho}_{0}+{\rho}_{sd}$ | 3 | 117.8 ± 0.3 | - | 437 ± 2 | 0.99601 |

Groove | ${\rho}_{0}+{\rho}_{sd}+{\rho}_{epi}$ | 3 | 113.30 ± 0.09 | 3.93 ± 0.06 | 307 ± 2 | 0.99988 |

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**MDPI and ACS Style**

Valerio-Cuadros, M.I.; Chaves, D.A.D.; Colauto, F.; Oliveira, A.A.M.d.; Andrade, A.M.H.d.; Johansen, T.H.; Ortiz, W.A.; Motta, M.
Superconducting Properties and Electron Scattering Mechanisms in a Nb Film with a Single Weak-Link Excavated by Focused Ion Beam. *Materials* **2021**, *14*, 7274.
https://doi.org/10.3390/ma14237274

**AMA Style**

Valerio-Cuadros MI, Chaves DAD, Colauto F, Oliveira AAMd, Andrade AMHd, Johansen TH, Ortiz WA, Motta M.
Superconducting Properties and Electron Scattering Mechanisms in a Nb Film with a Single Weak-Link Excavated by Focused Ion Beam. *Materials*. 2021; 14(23):7274.
https://doi.org/10.3390/ma14237274

**Chicago/Turabian Style**

Valerio-Cuadros, Marlon Ivan, Davi Araujo Dalbuquerque Chaves, Fabiano Colauto, Ana Augusta Mendonça de Oliveira, Antônio Marcos Helgueira de Andrade, Tom Henning Johansen, Wilson Aires Ortiz, and Maycon Motta.
2021. "Superconducting Properties and Electron Scattering Mechanisms in a Nb Film with a Single Weak-Link Excavated by Focused Ion Beam" *Materials* 14, no. 23: 7274.
https://doi.org/10.3390/ma14237274