Scanning Quantum Interference Device (SQUID)

Johannes Ungar, winter semester 2019/20

A SQUID (Superconducting Quantum Interference Device) is a type of magnetometer^[1] which represents the most sensitive detector of magnetic flux known. Due to its high sensitivity, wide bandwidth and broad dynamic range, it is commonly used to measure brain activity and is a sophisticated tool in Non-Destructive Testing (NDT) for detecting material flaws at unprecedented resolutions and depths. (Jenks, 1997)

Fundamentals

SQUIDs combine two physical phenomena: flux quantization, the fact that the flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}(\boldsymbol{B},\boldsymbol{S})\end{array} //]]>$ in a closed superconducting loop is quantized, and Josephson tunneling (Clarke, 1996). It is, therefore, necessary to get the basic knowledge about these two fundamental physical effects in order to understand the working principle of a SQUID.

Magnetic Flux Quantization

The magnetic flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}(\boldsymbol{B},\boldsymbol{S})\end{array} //]]>$ is the surface integral of the normal component of the magnetic flux density $//<![CDATA[ \begin{array}{l}\boldsymbol{B}\end{array} //]]>$ over the surface $//<![CDATA[ \begin{array}{l}\boldsymbol{S}\end{array} //]]>$ the magnetic flux is passing through. In SI units $//<![CDATA[ \begin{array}{l}\boldsymbol{B}\end{array} //]]>$ is measured in teslas $//<![CDATA[ \begin{array}{l}\mathrm{[T]}\end{array} //]]>$ or $//<![CDATA[ \begin{array}{l}\mathrm{[Gauss]}\end{array} //]]>$ and the magnetic flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}(\boldsymbol{B},\boldsymbol{S})\end{array} //]]>$ in weber $//<![CDATA[ \begin{array}{l}\mathrm{[Wb]}\end{array} //]]>$. If the magnetic field is constant over and orthogonal to the surface the formula for magnetic flux can be simplified to $//<![CDATA[ \begin{array}{l}\mathit{\Phi}(\boldsymbol{B},\boldsymbol{S})=\boldsymbol{B}\cdot\boldsymbol{S}\end{array} //]]>$. Generally, the magnetic flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}(\boldsymbol{B},\boldsymbol{S})\end{array} //]]>$ can take any arbitrary value. However, if one deals with the super-conducting loop or a hole in a bulk superconductor, the magnetic flux passing through the hole or loop is no longer arbitrary but instead can only take certain values in discrete steps (quantization). More precisely, the value of the magnetic flux can only take integer multiples of the magnetic flux quantum. The superconducting magnetic flux quantum $//<![CDATA[ \begin{array}{l}\Phi_0 = \mathrm h/2\mathrm e=2.0678 \cdot 10^{-15}\end{array} //]]>$ Wb is a combination of fundamental physical constants (Planck constant^[2] $//<![CDATA[ \begin{array}{l}\mathrm h\end{array} //]]>$ and the electron charge^[3] $//<![CDATA[ \begin{array}{l}\mathrm e\end{array} //]]>$). Its value is, therefore, the same for any superconductor.

The phenomenon of flux quantization was discovered experimentally at the Walther-Meißner Institute by Robert Doll and Martin Näbauer in 1961 (Gross, 2018). Fig.1(b) is showing this experimental evidence. In Fig.1(a) a small superconducting tube is aligned with a magnetic field $//<![CDATA[ \begin{array}{l}\boldsymbol{B}(\boldsymbol{x})\end{array} //]]>$ and steadily cooled below the critical temperature $//<![CDATA[ \begin{array}{l}T_c\end{array} //]]>$. Below $//<![CDATA[ \begin{array}{l}T_c\end{array} //]]>$ the magnetic field $//<![CDATA[ \begin{array}{l}\boldsymbol{B}(\boldsymbol{x})\end{array} //]]>$ is turned off, inducing a small current in the superconducting tube which is again causing a magnetic field. Since the current keeps flowing because of the superconducting state of the tube, the magnetic field will also persist or in other words, gets “frozen”. Fig.1(b) The frozen magnetic flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}(\boldsymbol{B},\boldsymbol{S})\end{array} //]]>$ in the tube vs. the magnetic flux density $//<![CDATA[ \begin{array}{l}\boldsymbol{B}\end{array} //]]>$ shows a quantized step-function. The flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}(\boldsymbol{B},\boldsymbol{S})\end{array} //]]>$ is always an integer multiple of the magnetic flux quantum $//<![CDATA[ \begin{array}{l}\Phi_0\end{array} //]]>$. The picture shows n = 0,1 and 2 times the magnetic flux quantum. Without magnetic flux quantization, the function should look like the dashed line. (Huebener, 2017)

Josephson Junction

The essential element of every SQUID is the Josephson Junction which is a connection of two superconductors separated by a very thin insulating layer or narrow constriction, called weak link (Jenks, 1997).

The occurring effect in such type of connection was theoretically explained by Brian David Josephson in 1962 (Nobel Prize 1982). The equations which describe the electrical properties are known as Josephson equations:

The DC-relation (1) deals with the fact that a current of Cooper Pairs (Bardeen, Cooper, Schriefer, 1957) can flow through the junction without showing any electrical resistance. In other words, a superconducting tunneling current can flow through such a junction without any voltage drop appearing across the junction (Weinstock, 1989). The supercurrent^[4] tunneling through the junction is dependent on the phase difference $//<![CDATA[ \begin{array}{l}\varphi(\boldsymbol{r},t)\end{array} //]]>$ on both sides of the junction (Huebener, 2017). The equations above are based on the concept that superconductivity can be described by a quantum mechanical wave function (Ginzberg and Landau, 1950) with a single amplitude $//<![CDATA[ \begin{array}{l}\psi_0 = |\psi(\boldsymbol{r},t)|\end{array} //]]>$ and phase $//<![CDATA[ \begin{array}{l}\theta(\boldsymbol{r},t)\end{array} //]]>$ according to the relation

The AC-relation (2) describes the behavior of the Josephson Junction when the applied current is greater than the critical current $//<![CDATA[ \begin{array}{l}I_0\end{array} //]]>$. When $//<![CDATA[ \begin{array}{l}I > I_0\end{array} //]]>$ a voltage drop $//<![CDATA[ \begin{array}{l}\Delta V\end{array} //]]>$ appears, and $//<![CDATA[ \begin{array}{l}\varphi\end{array} //]]>$ evolves with time according to the voltage-frequency relation (Clarke, 1996).

Note that the reduced Planck constant $//<![CDATA[ \begin{array}{l}\hbar = \mathrm h/2\pi\end{array} //]]>$ is used in Eq.(2) and (5). An applied electric voltage $//<![CDATA[ \begin{array}{l}\Delta V\end{array} //]]>$ on a Josephson Junction is always accompanied by an alternating supercurrent between the two superconductors of high frequency. The frequency of this Josephson oscillation is proportional to the applied voltage (Huebener, 2017). A derivation of those equations could be done with the time-dependent Schrödinger equation, but this is out of the scope of this article. The interested reader is referred to Gross (2018).

Dependence of the Josephson Junction on an external magnetic field

We are assuming a Josephson Junction with a rectangular cross-section (length  $//<![CDATA[ \begin{array}{l}L\end{array} //]]>$ and width $//<![CDATA[ \begin{array}{l}W\end{array} //]]>$) and the Josephson current is running orthogonally to this cross-section. Furthermore, a variable external magnetic field in $//<![CDATA[ \begin{array}{l}y\end{array} //]]>$-direction is applied (see Fig.3).

Based on Eq.(1) the following relation for the supercurrent density $//<![CDATA[ \begin{array}{l}J_s(y,z,t)\end{array} //]]>$ in the cross-section can be derived (Gross, 2018):

In Eq.(6) the wavenumber $//<![CDATA[ \begin{array}{l}k:=2\pi/\lambda\end{array} //]]>$ with the wavelength $//<![CDATA[ \begin{array}{l}\lambda=\Phi_0/dB_yz\end{array} //]]>$. The maximum Josephson current is denoted as $//<![CDATA[ \begin{array}{l}J_c(y,z)\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}d\end{array} //]]>$ is the thickness of the magnetic conduction band. In this case $//<![CDATA[ \begin{array}{l}d\end{array} //]]>$ is the thickness of the isolation layer since we neglect any penetration of the magnetic field into superconducting material according to the Meissner-Ochsenfeld effect. $//<![CDATA[ \begin{array}{l}\boldsymbol{B}_\textbf{y}\end{array} //]]>$ is the magnetic flux density pointing in $//<![CDATA[ \begin{array}{l}y\end{array} //]]>$-direction. The magnetic flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}\end{array} //]]>$ is simply the product of $//<![CDATA[ \begin{array}{l}\boldsymbol{B}\end{array} //]]>$ with the cross-sectional area of the magnetic conduction band $//<![CDATA[ \begin{array}{l}S=L\cdot d\end{array} //]]>$.

If Eq.(6) is applied to the example above (Fig.3) the spatial distribution of the Josephson current density $//<![CDATA[ \begin{array}{l}J_s(y,z)\end{array} //]]>$ in a Josephson Junction can be obtained. In Figure 4 the distribution for four different magnetic flux values $//<![CDATA[ \begin{array}{l}\mathit{\Phi}\end{array} //]]>$ is shown. This visualizes the dependency of the Josephson current on the magnetic flux density $//<![CDATA[ \begin{array}{l}\boldsymbol{B}\end{array} //]]>$.

When we integrate the supercurrent density $//<![CDATA[ \begin{array}{l}J_s(y,z)\end{array} //]]>$ over the cross-sectional plane $//<![CDATA[ \begin{array}{l}A=L\cdot W\end{array} //]]>$ we obtain the overall Josephson current $//<![CDATA[ \begin{array}{l}I_s\end{array} //]]>$ running through the contact:

When we calculate this integral, we get the relation of the max Josephson current to the magnetic flux $//<![CDATA[ \begin{array}{l}\mathit{\Phi}\end{array} //]]>$ (Gross, 2018).

If we plot this equation (see Fig.5) the dependency of the Josephson current on the magnetic flux gets even more apparent. The pattern has a strong resemblance to the interference pattern of the double-slit experiment in optics (Gross, 2018). $//<![CDATA[ \begin{array}{l}\mathit{\Phi}=B\cdot d\cdot L\end{array} //]]>$ is the magnetic flux threading through the contact and $//<![CDATA[ \begin{array}{l}I_c=i_cL\end{array} //]]>$, assuming a constant current density along the $//<![CDATA[ \begin{array}{l}z\end{array} //]]>$-direction. The plot is zero at integer multiples of $//<![CDATA[ \begin{array}{l}\Phi_0\end{array} //]]>$. As shown in Figure 4(c) the Josephson current flows in opposite directions and is eliminating itself (destructive interference).

SQUID

Since the maximum Josephson current in a Josephson Junction is dependent on an external magnetic field $//<![CDATA[ \begin{array}{l}\boldsymbol{B}\end{array} //]]>$ (see Fig.4 and 5), they possess the general ability to detect magnetic fields. In a very basic way, we can describe the sensitivity $//<![CDATA[ \begin{array}{l}s\end{array} //]]>$ of a magnetometer as the change of Josephson current for a unit change of magnetic flux density (Gross, 2018).

The product $//<![CDATA[ \begin{array}{l}L \cdot d\end{array} //]]>$ describes the cross-sectional area, the magnetic flux is passing through. Hence, by increasing this area the sensitivity can be increased as well. One possibility to increase the area is the use of a superconducting coil with two Josephson Junctions. It can be shown that in such a case the relevant cross-section for the magnetic flux is the enclosed area of the loop and not $//<![CDATA[ \begin{array}{l}L \cdot d\end{array} //]]>$ anymore (Gross, 2018). A setup like this where a superconducting loop is interrupted by one or multiple Joseph-son Junctions (Fig.6(a)) in general is called a Superconducting Quantum Interference Device (SQUID). SQUIDs are basically flux-to-voltage transducers which allow very precise measurements of all physical quantities involved.

There are two types of SQUIDs available today. The first, the DC SQUID, consists of two Josephson junctions connected in parallel in a superconducting loop. It can be operated with a direct current (DC). The second, the RF-SQUID, involves a single Josephson Junction in the superconducting loop and is operated with a radiofrequency flux bias. In both cases, the output from the SQUID is periodic with period $//<![CDATA[ \begin{array}{l}\Phi_0\end{array} //]]>$ in the magnetic flux applied to the loop (Clarke, 1996). In the following, only the DC-SQUID is explained in greater detail.

DC SQUID

The schematic of the DC-SQUID is shown in Fig.6(a). Two Josephson junctions (indicated by $//<![CDATA[ \begin{array}{l}\times\end{array} //]]>$) are connected in parallel on a superconducting loop. The $//<![CDATA[ \begin{array}{l}I-V\end{array} //]]>$ characteristic is shown in Fig.6(b) for $//<![CDATA[ \begin{array}{l}\mathit{\Phi} = n\Phi_0\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\mathit{\Phi} = (n+1/2)\Phi_0\end{array} //]]>$, where $//<![CDATA[ \begin{array}{l}\mathit{\Phi}\end{array} //]]>$ is the external flux applied to the loop and $//<![CDATA[ \begin{array}{l}n\end{array} //]]>$ is an integer. If we bias the SQUID with a constant current ($//<![CDATA[ \begin{array}{l}>2 I_0\end{array} //]]>$), the voltage across the SQUID oscillates with period $//<![CDATA[ \begin{array}{l}\Phi_0\end{array} //]]>$ as we steadily increase $//<![CDATA[ \begin{array}{l}\mathit{\Phi}\end{array} //]]>$, as indicated in Fig.6(c). The SQUID is generally operated on the steep part of the $//<![CDATA[ \begin{array}{l}V-\mathit{\Phi}\end{array} //]]>$ curve where the slope, $//<![CDATA[ \begin{array}{l}V_{\mathit{\Phi}}=|\partial V/\partial \mathit{\Phi}|\end{array} //]]>$, is a maximum (Point A or C). Thus, the SQUID produces an output voltage in response to a small input flux $//<![CDATA[ \begin{array}{l}\delta \mathit{\Phi} (\ll \Phi_0)\end{array} //]]>$ and is effectively a flux-to-voltage transducer (Clarke, 1996).

Practical System

In Fig.7 a cross-sectional schematic of a conventional Low-$//<![CDATA[ \begin{array}{l}T_c\end{array} //]]>$ SQUID system is shown. SQUID NDT systems are designed to measure the magnetic field very close to the surface of the samples under inspection. The resolution of the images is roughly determined by the greater of the size of the pick-up coil and its distance from the magnetic source. Hence, Dewar design (tail spacing) is very critical for non-destructive testing applications (Jenks, 1997).

Pick-up Coil

The SQUID is an excellent sensor for magnetic fluxes. To enhance its capabilities, most SQUID systems do not expose the bare SQUID to the magnetic field of interest. Rather, they typically employ a multi-turn pick-up coil inductively linked to the SQUID. Different designs, depending on the requirements, are used. One of the simplest instruments is the magnetometer (Fig.8(a)). A pick-up loop is connected across the input coil which results in a superconducting flux transformer.

The SQUID and input coil are generally enclosed in a superconducting shield (Clarke, 1996). Usually, a second-order gradiometer is employed, as this arrangement essentially suppresses the magnetic environmental noise. In Figure 8, we can see different pickup coil configurations used: (a) magnetometer, (b) first-order axial gradiometer, (c) second-order axial gradiometer.

SQUID for non-destructive testing

Current SQUID systems are used for the following materials/applications:

• Ferromagnetic materials (Fe, Ni, Co)

  • Flaw detection and characterization.
    In general, all methods applied for nonferrous metals (see below) are also applicable to ferrous metals with some added complication from the hysteretic nature of the material. One possibility is to apply a static magnetic field and scan it with a planar gradiometer which is connected to a SQUID (Bain, 1985).

  • Utilizing the magnetic properties of steel to evaluate stress and fatigue.
    SQUID magnetometry is also suitable for the study of stress-strain behavior in ferromagnetic materials (Weinstock and Niesenhoff, 1985). Magnetic flux is also changing as mechanical stress increases. Techniques were developed where it is possible to measure “fatigue” in steel by assessing the magneto-mechanical hysteresis (flux vs. stress curve) (Banchet, 1995).

• Nonferromagnetic materials (Al, Cu, etc.)

  • The current injection technique.
    An AC current is injected into the specimen or material of interest. The SQUID-system can measure the magnetic field distribution around the specimen even at very large distances to the material surface (≈ 20 cm) (Weinstock and Niesenhoff, 1985). This can be used to detect the presence of various holes or cracks.

  • The current induction technique.
    If the material or structure is isolated or painted with a non-conductive material the injection of electrical current is impossible. Therefore, it is necessary to induce eddy currents by an inductively coupled coil to the sample. Various methods are available and have been optimized for SQUIDs (Jenks, 1997). More information on the method and the physical background can be found in the article Fundamentals of eddy-current testing.

• Nonconducting materials (ceramics, composites)
  With a special SQUID-setup Wikswo (1993) showed that a high-resolution, susceptibility^[7]-sensitive imaging system can be used to measure the susceptibility distribution in a nonconducting material for detection and sizing of flaws. This technique is based on the fact that the susceptibility of a flaw in a material normally differs from the rest of the material. (Jenks, 1997)

• Localization of conducting/magnetic structures and current sources as for example buried conducting pipes, examining gas pipelines for stray currents (indicators for corrosion) and detection of defects in integrated circuits.

• Active Corrosion of metals
  SQUIDs offer a contactless method of detecting ongoing corrosion currents and mapping their distribution. (Jenks, 1997)

Comparison to conventional magnetic NDT-methods

The advent of SQUIDs as highly sensitive magnetic flux detectors has enabled measurements of small spatial variations of a weak magnetic field. For example, it is not necessary to apply a very large magnetizing field to the specimen in order to produce a detectable flux leakage, as is the case with conventional approaches. Also, it is possible to place the sensing probe at a relatively large distance from the specimen. (Jenks, 1997)

Advantages of SQUID systems

• High sensitivity

• Large penetration depths

• High spatial resolution

• Greater scanning distances (several cm distances between sample and detector possible)

• No requirement for large magnetic fields

• Examination of non-conducting materials possible

Disadvantages

• Increased system costs (cryocooler, Dewar, expensive liquid helium, etc.)
• Increased system size (heights range from 0.45 m to 2 m with weights from 4 kg to more than 1000 kg (Jenks, 1997)
• High maintenance costs
• Limited working time if coolant is not permanently cooled
• Academic personnel needed to operate the system and interpret the results
• Contamination of the signal by environmental noise because of ultra-high sensitivity of the magnetometer
• Time-consuming development process for new large-scale applications

In conventional eddy-current testing, the operating frequency typically ranges between 100kHz–10MHz. The signal decreases and instrumental noise increases as the operating frequency is decreased. As a result, since the current skin depth in metal is inversely proportional to the square root of the operating frequency, it follows that the eddy-current approach is inherently incapable of detecting deep flaws in good conductors. (Jenks, 1997)

Magnetoencephalography (MEG)

As mentioned in the beginning SQUIDs are commonly used to measure brain activity. This imaging technique is called Magnetoencephalography (MEG). It is a noninvasive method for investigating neuronal activity in the living human brain (Hämäläinen, 1993).

SQUID-Technology is an essential part of it and leads to several significant improvements in the neuro-imaging sector. The time resolution of the method is better than 1 ms and the spatial discrimination is, under favorable circumstances, 2-3 mm (Hämäläinen, 1993). Further advantages are

• MEG-systems are independent of the anatomic head geometry
• Non-invasive
• No ionizing radiation
• Reference-free, because it´s a quantitative method

Spin Exchange Relaxation-Free (SERF) magnetometer

Recent developments have shown that optical magnetometers could achieve or even exceed the sensitivities of SQUIDs (Kominis, 2013). The so-called Spin Exchange Relaxation-Free (SERF) magnetometer is a new type of alkali metal atomic magnetometer, which operates in the SERF-regime and belongs to the optical magnetometer category. SERF refers to a property of the magnetic medium and to reach the required regime a certain atomic spin density for a given magnetic field is required, actually on the order of  $//<![CDATA[ \begin{array}{l}10^{14}\end{array} //]]>$$//<![CDATA[ \begin{array}{l}\mathrm{cm^{-3}}\end{array} //]]>$ (Savukov, 2017). Without explaining the physical concepts behind this technology, the advantage of such devices breaks down to the following points (J. Li, 2018)

• Equal or better sensitivity per volume (miniaturization)
• Cryogen-free
• High spatial resolution

Since the demand of more sensitive and miniaturized magnetometers is growing (analysis of the chemical composition, brain science, fundamental physical research), the SERF technology is gaining more importance (J.Li, 2018).

Summary

The SQUID has, except the SERF technology, no equal competitor for measuring magnetic fluxes regarding sensitivity. All other electromagnetic sensors e.g. Hall sensors, magnetostrictive sensors are fundamentally inferior to SQUIDs at frequencies below several kilohertz (see Fig.9). The SQUIDs can detect fields at frequencies from dc to gigahertz. The dynamic range is large, and the spatial resolution has improved more and more over the last years (Jenks, 1997).

Footnotes

[1] magnetometer: sensor for magnetic flux density

[2] Planck constant $//<![CDATA[ \begin{array}{l}\mathrm h =6.62607015\times 10^{-34}\end{array} //]]>$ $//<![CDATA[ \begin{array}{l}\mathrm{J/s}\end{array} //]]>$

[3] electron charge or elementary charge $//<![CDATA[ \begin{array}{l}\mathrm e =1.602176634\times 10^{-19}\end{array} //]]>$ $//<![CDATA[ \begin{array}{l}\mathrm{C}\end{array} //]]>$

[4] supercurrent: electric current which is flowing without dissipation (superconducting current)

[5] gradiometer: a device to measure the gradient of a physical quantity e.g. magnetic flux (see “Pick-up Coil")

[6] Dewar: well-insulated storage vessel, typically using vacuum between the inner and outer wall

[7] (magnetic) susceptibility $//<![CDATA[ \begin{array}{l}\chi\end{array} //]]>$: a measure of how much a material gets magnetized in a magnetic field; it is related to the relative magnetic permeability $//<![CDATA[ \begin{array}{l}\mu_r=1+\chi\end{array} //]]>$ and materials can be divided into three groups: $//<![CDATA[ \begin{array}{l}\chi< 0\end{array} //]]>$: diamagnetic material, $//<![CDATA[ \begin{array}{l}\chi>0\end{array} //]]>$: paramagnetic material, $//<![CDATA[ \begin{array}{l}\chi\gg0\end{array} //]]>$: ferromagnetic material

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