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Evaluation of Self-Field Effects in Magnetometers Based on Meander-Shaped Arrays of Josephson Junctions or SQUIDs Connected in Series^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Ideal Devices

#### 2.1. Single SQUID

#### 2.2. Arrays of Josephson Junctions

- the input noise spectral density (NSD) $|{S}_{B}|$, assuming that the SQUID noise contributions are not correlated;
- the spur-free dynamic range (SFDR);
- and the dynamic range, as the output noise is independent of N.

## 3. Self-Flux

#### 3.1. Origin

#### 3.2. Layout Asymmetry

#### 3.3. Evaluation of Self-Flux in a Meander Arrangement

**i**and

**j**are unit vectors along the x and y-axes. The total magnetic field on segment i is:

#### 3.4. Josephson Asymmetry

## 4. Evaluation of Self-Flux Degradation on the Array Performance

#### 4.1. Impact of Layout

#### 4.2. Impact of Scattering

## 5. Compensation

- it is necessary to keep ${\beta}_{L}$ small to maintain the modulation amplitude of individual SQUIDs;
- smaller SQUIDs are less sensitive to “inter-SQUID” self-flux;
- and, as seen in Section 4.2, they are less sensitive to “intra-SQUID” self-flux.

- increasing the impedance of the readout electronics, which is possible only for low-frequency devices because it reduces the bandwidth of the system;
- associating devices in parallel to lower its output impedance;
- and using flux transformers and/or flux concentrators [39].

## 6. Josephson Junction Series Arrays

## 7. Discussion

## 8. Conclusions

## 9. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

JJ | Josephson junction |

JSA | Josephson junction series array |

LJJ | Long Josephson junction |

NSD | Noise spectral density |

SFDR | Spur-free dynamic range |

SNR | Signal-to-noise ratio |

SQUID | Superconducting quantum interference device |

SSA | SQUID series array |

WPD | Superconducting wave functions phase difference |

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**Figure 1.**Basic SQUID with perfect symmetry, with crosses representing JJs. The magnetic flux originating from the bias current has zero net value.

**Figure 2.**Transfer function made with series arrays of SQUIDs. Top curve: non-periodic with incommensurate SQUID areas; bottom curve: periodic with equal SQUID areas (${\Phi}_{M}$ refers to the flux coupled to the SQUIDs with area ${a}_{M}$). The curves are shifted vertically for clarity.

**Figure 3.**Illustration of the SQUID categories according to symmetry: (

**a**) Layout asymmetry in the SQUID loop. (

**b**) Layout asymmetry in a bias line. (

**c**) Josephson asymmetry (provided ${I}_{C1}\ne {I}_{C2}$).

**Figure 4.**Schematic representation (

**a**) of the meander geometry, with J = 28 including the edge segments represented in red; (

**b**) of the SQUID geometry (JJs are not represented).

**Figure 5.**Flux distribution over the SQUIDs located at the center of each segment of the meander geometry for a bias current $I=100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}A$ and different segment lengths ${W}_{y}=L\xb7{W}_{x}$. The symbols are located at the position of the center of each segment. The flux per unit length is given in ${\Phi}_{0}/m$.

**Figure 6.**Flux distribution over the SQUIDs located at the center of each segment of the meander geometry for a bias current $I=100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}A$. The flux per unit length is normalized by the flux quantum ${\Phi}_{0}$. The symbols refer to different configurations of the edge segments: (+) containing SQUIDs as all the other segments (same as Figure 5); (×) no SQUID on edge segments, and ${I}_{e}=0$; (⊕) no SQUID on edge segments, and ${I}_{e}$ as indicated in the table as inset; (⋄) idem, but edge segments are shifted by $d{X}_{0}$ as indicated in the table as inset.

**Table 1.**Scaling of arrays of perfectly identical SQUIDs, delivering the output signal to a matched, i.e., suitably scaled, load. M (resp. N) is the number of JJs in parallel (resp. in series). The frequency bandwidth to integrate the NSD for power evaluation is assumed independent of N and M.

Arrangement | Series | Parallel | 2D |
---|---|---|---|

Modulation $\Delta {V}_{N}$ | N | 1 | N |

Transfer Factor $\partial {V}_{N}/\partial B$ | N | 1 | N |

Impedance | N | ${M}^{-1}$ | $N{M}^{-1}$ |

Output Signal Power | N | M | $NM$ |

Output NSD ${S}_{V}$ | N | ${M}^{-1}$ | $N{M}^{-1}$ |

Output Noise Power | 1 | 1 | 1 |

Input NSD ${S}_{B}$ | ${N}^{-1}$ | ${M}^{-1}$ | ${\left(NM\right)}^{-1}$ |

SFDR | ${N}^{2/3}$ | ${M}^{2/3}$ | ${\left(NM\right)}^{2/3}$ |

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**MDPI and ACS Style**

Crété, D.; Kermorvant, J.; Lemaître, Y.; Marcilhac, B.; Mesoraca, S.; Trastoy, J.; Ulysse, C.
Evaluation of Self-Field Effects in Magnetometers Based on Meander-Shaped Arrays of Josephson Junctions or SQUIDs Connected in Series. *Micromachines* **2021**, *12*, 1588.
https://doi.org/10.3390/mi12121588

**AMA Style**

Crété D, Kermorvant J, Lemaître Y, Marcilhac B, Mesoraca S, Trastoy J, Ulysse C.
Evaluation of Self-Field Effects in Magnetometers Based on Meander-Shaped Arrays of Josephson Junctions or SQUIDs Connected in Series. *Micromachines*. 2021; 12(12):1588.
https://doi.org/10.3390/mi12121588

**Chicago/Turabian Style**

Crété, Denis, Julien Kermorvant, Yves Lemaître, Bruno Marcilhac, Salvatore Mesoraca, Juan Trastoy, and Christian Ulysse.
2021. "Evaluation of Self-Field Effects in Magnetometers Based on Meander-Shaped Arrays of Josephson Junctions or SQUIDs Connected in Series" *Micromachines* 12, no. 12: 1588.
https://doi.org/10.3390/mi12121588