# Influence of Resonances on the Noise Performance of SQUID Susceptometers

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## Abstract

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## 1. Introduction

## 2. Modeling

#### 2.1. IR$\Phi $ Characteristics

#### 2.2. Noise

#### 2.3. Summary of Noise Calculations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Clarke, J.; Braginski, A.I. The SQUID Handbook; Wiley Online Library: Hoboken, NJ, USA, 2004. [Google Scholar]
- Vinante, A.; Mezzena, R.; Prodi, G.A.; Vitale, S.; Cerdonio, M.; Falferi, P.; Bonaldi, M. Dc superconducting quantum interference device amplifier for gravitational wave detectors with a true noise temperature of 16 μK. Appl. Phys. Lett.
**2001**, 79, 2597–2599. [Google Scholar] [CrossRef][Green Version] - Harry, G.M.; Jin, I.; Paik, H.J.; Stevenson, T.R.; Wellstood, F.C. Two-stage superconducting-quantum-interference-device amplifier in a high-Q gravitational wave transducer. Appl. Phys. Lett.
**2000**, 76, 1446–1448. [Google Scholar] [CrossRef][Green Version] - Hari, R.; Salmelin, R. Magnetoencephalography: From SQUIDs to neuroscience: Neuroimage 20th anniversary special edition. Neuroimage
**2012**, 61, 386–396. [Google Scholar] [CrossRef] - Devoret, M.H.; Wallraff, A.; Martinis, J.M. Superconducting qubits: A short review. arXiv
**2004**, arXiv:cond-mat/0411174. [Google Scholar] - Kirtley, J.R.; Wikswo, J.P., Jr. Scanning SQUID microscopy. Annu. Rev. Mater. Sci.
**1999**, 29, 117–148. [Google Scholar] [CrossRef][Green Version] - Black, R.; Mathai, A.; Wellstood, F.; Dantsker, E.; Miklich, A.; Nemeth, D.; Kingston, J.; Clarke, J. Magnetic microscopy using a liquid nitrogen cooled YBa
_{2}Cu_{3}O_{7}superconducting quantum interference device. Appl. Phys. Lett.**1993**, 62, 2128–2130. [Google Scholar] [CrossRef] - Veauvy, C.; Hasselbach, K.; Mailly, D. Scanning μ-superconduction quantum interference device force microscope. Rev. Sci. Instrum.
**2002**, 73, 3825–3830. [Google Scholar] [CrossRef] - Finkler, A.; Segev, Y.; Myasoedov, Y.; Rappaport, M.L.; Ne’eman, L.; Vasyukov, D.; Zeldov, E.; Huber, M.E.; Martin, J.; Yacoby, A. Self-aligned nanoscale SQUID on a tip. Nano Lett.
**2010**, 10, 1046–1049. [Google Scholar] [CrossRef][Green Version] - Vu, L.; Wistrom, M.; Van Harlingen, D.J. Imaging of magnetic vortices in superconducting networks and clusters by scanning SQUID microscopy. Appl. Phys. Lett.
**1993**, 63, 1693–1695. [Google Scholar] [CrossRef][Green Version] - Kirtley, J.; Ketchen, M.; Stawiasz, K.; Sun, J.; Gallagher, W.; Blanton, S.; Wind, S. High-resolution scanning SQUID microscope. Appl. Phys. Lett.
**1995**, 66, 1138–1140. [Google Scholar] [CrossRef][Green Version] - Gardner, B.W.; Wynn, J.C.; Björnsson, P.G.; Straver, E.W.; Moler, K.A.; Kirtley, J.R.; Ketchen, M.B. Scanning superconducting quantum interference device susceptometry. Rev. Sci. Instrum.
**2001**, 72, 2361–2364. [Google Scholar] [CrossRef][Green Version] - Kirtley, J. Fundamental studies of superconductors using scanning magnetic imaging. Rep. Prog. Phys.
**2010**, 73, 126501. [Google Scholar] [CrossRef][Green Version] - Huber, M.E.; Koshnick, N.C.; Bluhm, H.; Archuleta, L.J.; Azua, T.; Björnsson, P.G.; Gardner, B.W.; Halloran, S.T.; Lucero, E.A.; Moler, K.A. Gradiometric micro-SQUID susceptometer for scanning measurements of mesoscopic samples. Rev. Sci. Instrum.
**2008**, 79, 053704. [Google Scholar] [CrossRef] [PubMed] - Kirtley, J.R.; Paulius, L.; Rosenberg, A.J.; Palmstrom, J.C.; Holland, C.M.; Spanton, E.M.; Schiessl, D.; Jermain, C.L.; Gibbons, J.; Fung, Y.K.K.; et al. Scanning SQUID susceptometers with sub-micron spatial resolution. Rev. Sci. Instrum.
**2016**, 87, 093702. [Google Scholar] [CrossRef] [PubMed] - Tesche, C.D.; Clarke, J. dc SQUID: Noise and Optimization. J. Low Temp. Phys.
**1977**, 29, 301–331. [Google Scholar] [CrossRef] - Bruines, J.; de Waal, V.; Mooij, J. Comment on: “Dc SQUID: Noise and optimization” by Tesche and Clarke. J. Low Temp. Phys.
**1982**, 46, 383–386. [Google Scholar] [CrossRef] - Hilbert, C.; Clarke, J. Measurements of the dynamic input impedance of a dc SQUID. J. Low Temp. Phys.
**1985**, 61, 237–262. [Google Scholar] [CrossRef] - Enpuku, K.; Cantor, R.; Koch, H. Modeling the dc superconducting quantum interference device coupled to the multiturn input coil. III. J. Appl. Phys.
**1992**, 72, 1000–1006. [Google Scholar] [CrossRef] - Huber, M.E.; Neil, P.A.; Benson, R.G.; Burns, D.A.; Corey, A.; Flynn, C.S.; Kitaygorodskaya, Y.; Massihzadeh, O.; Martinis, J.M.; Hilton, G. DC SQUID series array amplifiers with 120 MHz bandwidth (corrected). IEEE Trans. Appl. Supercond.
**2001**, 11, 4048–4053. [Google Scholar] [CrossRef] - Knuutila, J.; Ahonen, A.; Tesche, C. Effects on dc SQUID characteristics of damping of input coil resonances. J. Low Temp. Phys.
**1987**, 68, 269–284. [Google Scholar] [CrossRef] - Josephson, B.D. Possible new effects in superconductive tunnelling. Phys. Lett.
**1962**, 1, 251–253. [Google Scholar] [CrossRef] - IC Design Software for Linux, OS X and Windows. Available online: http://www.wrcad.com/ (accessed on 13 September 2019).
- JSPICE. Available online: https://ptolemy.berkeley.edu/projects/embedded/pubs/downloads/spice/jspice.html (accessed on 11 March 2019).

**Figure 1.**Two types of susceptometer layouts: (

**a**) That of Huber et al. [14,15], without a damping resistor, and (

**b**) that of Gardner et al., [12] with a damping resistor. I labels the current leads, M the modulation coil leads, and $F.C.$ the field coil leads. The Josephson junctions are indicated by Xs. The semi-transparent regions indicate superconducting shields. Superconducting coaxial leads connect the central regions with junctions and modulation coils to the pickup loop/field coil pairs to the left and right. (

**c**) Current–voltage (IV) characteristic for an undamped susceptometer at various magnetic fluxes, and (

**d**) IVs for a damped susceptometer.

**Figure 2.**Adding a parasitic capacitance to an ideal Superconducting Quantum Interference Device (SQUID) produces a resonance. (

**a**) Ideal SQUID schematic, and (

**b**) calculated $dV/dI$ characteristic for an ideal SQUID with no parasitic capacitance at T = 4.2 K. In this instance, the upper inductances ${L}_{p}$ = 30 pH, the lower inductances ${L}_{p}$ = 1 pH, the Josephson critical currents ${I}_{0}$ = 22 $\mathsf{\mu}$A, the shunt resistors ${R}_{J}$ = 2 $\mathsf{\Omega}$, and the junction capacitances ${C}_{J}$ = 10 fF. (

**c**) Schematic with a parasitic capacitance, and (

**d**) calculated $dV/dI$ characteristic at T = 4.2 K. Here ${I}_{0}$ = 22 $\mathsf{\mu}$A, ${R}_{s}$ = 2 $\mathsf{\Omega}$, ${C}_{j}$ = 10 fF, upper ${L}_{p}$ = 30 pH, lower ${L}_{p}$ = 1 pH, upper ${C}_{p}$ = 10 pF, and lower ${C}_{p}$ = 1 pF.

**Figure 3.**Modeling of $dV/dI$ vs. I and $\Phi $ ($IR\Phi $) for two types of SQUID susceptometers: (

**a**) Undamped schematic, (

**b**) experimental $dV/dI$ characteristic, and (

**c**) calculated $dV/dI$ characteristic at T = 4.2 K for a SQUID with the layout of Figure 1a [14,15]. In this model, ${I}_{0}$ = 25 $\mathsf{\mu}$A, ${R}_{J}$ = 2 $\mathsf{\Omega}$, ${C}_{J}$ = 10 fF, ${L}_{m}$ = 30 pH, ${L}_{p}$ = 4 pH, and ${C}_{p}$ = 8 pF. There are a total of five ${L}_{p},{C}_{p}$ pairs in each arm to the left and right of the schematic, representing the coaxial leads to the pickup loops. (

**d**) Damped schematic, (

**e**) experimental $dV/dI$ characteristic, and (

**f**) calculated $dV/dI$ characteristic at T = 4.2 K for a SQUID with the layout of Figure 1b [12]. In this model, ${I}_{0}$ = 12 $\mathsf{\mu}$A, ${R}_{J}$ = 2 $\mathsf{\Omega}$, ${R}_{D}$ = 4 $\mathsf{\Omega}$, ${C}_{j}$ = 10 fF, ${L}_{m}$ = 30 pH, ${L}_{p}$ = 1 pH, and ${C}_{p}$ = 8 pF. There are five ${L}_{p},{C}_{p}$ pairs in each arm of the center of the schematic, representing the coaxial leads to the pickup loops.

**Figure 4.**Comparison with previous work: (

**a**) Schematic of the model used. In this case, the junction critical current ${I}_{0}$ = 17.2 $\mathsf{\mu}$A, junction capacitance ${C}_{j}$ = 0 pF, modulation inductance ${L}_{m}$ = 30 pH, shunt resistance ${R}_{J}$ = 2 $\mathsf{\Omega}$, $\Phi =0.25\phantom{\rule{3.33333pt}{0ex}}{\Phi}_{0}$, and T = 20.56 K. This choice of parameters leads to $\beta =1.0$, $\mathsf{\Gamma}=0.05$, for direct comparison with Figures 13a, 14a, and 15a of Tesche and Clarke [16], as well as Figures 1a and 2a of Bruines et al. [17]. The curve labelled Bruines in (

**b**) is inferred from the curves labelled Bruines in (

**c**,

**d**).

**Figure 5.**Noise calculations for an undamped SQUID along the critical curve: (

**a**) Plot of $dV/dI$ vs. current (I) and flux ($\Phi $). The crosses correspond to the values of I and $\Phi $ for which noise was calculated. There are two sets of crosses, separated by 0.02 ${\Phi}_{0}$, to enable the calculation of the derivative $dv/d\varphi $ at each flux value. (

**b**) Plots of the dimensionless low-frequency voltage noise power ${S}_{v}^{0}/2\mathsf{\Gamma}$ vs. current I for four different flux values. (

**c**) Plots of the dimensionless transfer junction $|d\nu /d\varphi |$. (

**d**) Plots of the dimensionless flux noise ${\zeta}_{\varphi}^{1/2}$. The schematic used for these calculations was that of Figure 3a, with ${I}_{0}$ = 25 $\mathsf{\mu}$A, ${L}_{m}$ = 30 pH, ${R}_{J}$ = 2 $\mathsf{\Omega}$, ${L}_{p}$ = 4 pH, ${C}_{J}$ = 10 fF, ${C}_{p}$ = 8 pF, and $T=4.2$ K.

**Figure 6.**Noise calculations for an undamped SQUID along resonance at the lowest magnitude bias current (“first resonance”): (

**a**) Plot of $dV/dI$ vs. current (I) and flux ($\Phi $). The crosses correspond to the values of I and $\Phi $ for which noise was calculated. (

**b**) Plots of the dimensionless low-frequency voltage noise power ${S}_{v}^{0}/2\mathsf{\Gamma}$ vs. current I for four different flux values. (

**c**) Plots of the dimensionless transfer junction $|d\nu /d\varphi |$. (

**d**) Plots of the dimensionless flux noise ${\zeta}_{\varphi}^{1/2}$. The schematic used for these calculations was that of Figure 3a, with ${I}_{0}$ = 25 $\mathsf{\mu}$A, ${L}_{m}$ = 30 pH, ${R}_{J}$= 2 $\mathsf{\Omega}$, ${L}_{p}$ = 4 pH, ${C}_{J}$ = 10 fF, ${C}_{p}$ = 8 pF, and T = 4.2 K.

**Figure 7.**Noise calculations for a damped SQUID along the critical curve: (

**a**) Plot of $dV/dI$ vs. current (I) and flux ($\Phi $). The crosses correspond to the values of I and $\Phi $ for which noise was calculated. (

**b**) Plots of the dimensionless low-frequency voltage noise power ${S}_{v}^{0}/2\mathsf{\Gamma}$ vs. current I for four different flux values. (

**c**) Plots of the dimensionless transfer junction $|d\nu /d\varphi |$. (

**d**) Plots of the dimensionless flux noise ${\zeta}_{\varphi}^{1/2}$. The schematic used for these calculations was that of Figure 3d, with ${I}_{0}$ = 22 $\mathsf{\mu}$A, ${L}_{m}$ = 30 pH, ${R}_{J}$ = 2 $\mathsf{\Omega}$, ${R}_{d}$ = 2 $\mathsf{\Omega}$, ${L}_{p}$ = 1 pH, ${C}_{J}$ = 10 fF, ${C}_{p}$ = 8 pF, and T = 4.2 K.

**Figure 8.**Minimum flux noise for damped vs. undamped SQUIDs. The square blue symbols correspond to the undamped SQUID along the critical curve, the diamond red symbols are for the undamped SQUID along the first resonance, and the triangular green symbols correspond to the damped SQUID along the critical curve. These data were generated by varying the current (I) at fixed flux.

Parameter | Symbol | Conversion Formula |
---|---|---|

Voltage | v | $V/{I}_{0}{R}_{J}$ |

Magnetic flux | $\varphi $ | $\Phi /{\Phi}_{0}$ |

Thermal noise parameter | $\mathsf{\Gamma}$ | $2\pi {k}_{b}T/{I}_{0}{\Phi}_{0}$ |

Voltage noise power | ${S}_{v}^{0}$ | $2\pi {S}_{V}^{0}/{I}_{0}{R}_{J}{\Phi}_{0}$ |

Flux noise | ${\zeta}_{\varphi}^{1/2}$ | ${S}_{\Phi}^{1/2}{(\pi {I}_{0}{R}_{J}/\mathsf{\Gamma})}^{1/2}/{\Phi}_{0}^{3/2}$ |

Hysteresis parameter | $\beta $ | $2L{I}_{0}/{\Phi}_{0}$ |

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**MDPI and ACS Style**

Davis, S.I.; Kirtley, J.R.; Moler, K.A. Influence of Resonances on the Noise Performance of SQUID Susceptometers. *Sensors* **2020**, *20*, 204.
https://doi.org/10.3390/s20010204

**AMA Style**

Davis SI, Kirtley JR, Moler KA. Influence of Resonances on the Noise Performance of SQUID Susceptometers. *Sensors*. 2020; 20(1):204.
https://doi.org/10.3390/s20010204

**Chicago/Turabian Style**

Davis, Samantha I., John R. Kirtley, and Kathryn A. Moler. 2020. "Influence of Resonances on the Noise Performance of SQUID Susceptometers" *Sensors* 20, no. 1: 204.
https://doi.org/10.3390/s20010204