Physiology of a DC SQUID (Superconducting Quantum Interference Device)

DC-Squid, a wonderful man-made creature, can typically be found sitting in the bottom of a liquid helium tank, whispering secrets of the unexplored minuscule, or probably just lying on the bench of a researcher who is desperately trying to keep it alive with the periodical pouring of Liquid Nitrogen.

In frigid conditions, these are the most sensitive detectors of magnetic flux known – they can detect a change in applied magnetic flux as small as 10e-6 Φ0  Hz (0.5) (Φ0 = h/(2e) = 2.067833831(13)X1015 Wb). A number of other physical quantities such as voltage, magnetic field, current, mechanical displacements which can be converted to magnetic flux can also be measured. Furthermore, they have enormous operation bandwidth extending over from the minuscule to over 1GHz. As a consequence of this outstanding versatility and precision, their population has grown exponentially in recent years- from applications in magnetoencephalography (to measure neural electric currents in brain- finally creating an image of brain activity or so-called brain scan) and quantum computation hardware (The current buzz- computers exponentially faster than yours which can solve problems yours cannot imagine) to the search for dark matter axion– evolving and attracting attention every so often. Here, we shall discuss (very briefly) the integral elements that execute life in the very primitive form of this organism.

Cooper pairs- The Blood Corpuscles

For any tool capable of sensitive measurements, it is essential that variations in parameters due to circuit design are minimized. Attainability of such sensitivity as demonstrated by squids was imaginable only after the discovery of superconductors. Quite as blood corpuscles are responsible for the exchange of respiratory gases in a living macro-organism allowing the sustenance of essential processes, ‘Cooper pairs’ are responsible for the processes that allow the existence of superconductivity.

In ordinary conductors, electron flow receives resistance from collisions with other electrons and the crystal lattice vibrations (phonons). However, if a current is set up in a superconducting loop, it will continue to flow. In superconductors (some special materials), electrons move as ‘bound pairs’ called cooper pairs. When an electron passes through the lattice, positive ions are attracted towards it and distortion happens in the nominal positions of the positive ions. This disturbance in lattice attracts another electron. The net effect is a weak attractive force between the two electrons which now constitute a cooper pair. Each electron in the cooper pair forms another cooper pair, which makes another one, finally making a large network of interactions. Electrons have wave-like behavior and their wave function extends throughout the solid, overlapping with wave function of other electrons. As a result, they can be represented as a single wave function. This collective behavior creates substantial energy gap and unless energy from the environment is available (which is not available at low temperatures (<100 Kelvin)) none of the electrons collide with the lattice and thus encounter no resistivity.

Electrons in Cooper pairs have zero momentum (as constituent electrons move towards each other with same speeds). But when a potential is applied across a superconductor, the momentum of cooper pairs (initially zero) shifts to one electron slightly, allowing the flow of current. This momentum stays intact in absence of resistivity even after the current source or potential difference is removed and currently continues to flow if the superconductor circuit is closed.

Josephson Junction- The Vessels and The Flow of blood

The blood vessels, permeable, allow the exchanges of gasses and several other molecules. It is this boundary that allows a number of parameters like blood density, count of blood corpuscles, etc to vary. Many a time, measuring this variation is how a doctor can say that you have a state of diabetes.

Josephson Junction which is a thin slice of insulator (of the order of nanometers) sandwiched between two superconductors follows this analogy quite explicitly. The barrier(insulator) as must be clear doesn’t allow normal conduction, instead, the electrons or cooper pairs have to ‘tunnel’ through the barrier to complete the circuit. Subatomic particles can cross a barrier (if substantially thin by quantum tunneling) owing to the wave-like property as established by quantum mechanics. When in the barrier, Electrons can pick-up or lose energy from or to the environment. This leads to a change in current flowing through the circuit which when measured can imply something about the environment.

In quantum mechanics, exponential functions and their linear combinations form wave functions in most of the simple cases. We can define ψ1 = √n1 e 1 and ψ2 = √n2 e2 as wavefunctions on either side of the insulating junction where n1, n2 are the cooper pair densities and θ1, θ2 are the phases. Substituting these in the coupled Schrödinger equations (where µ1 and µ2 are lowest energy states on either side and K is coupling parameter):

We get:

The time derivative of the density of Cooper pairs describes a charge transport and thus, I = (n1 ) t   (Partial derivative wrt time).  If a voltage V is applied between the junctions the energy levels will shift according to µ2 1= 2eV .

Simplifying we get:

Where I0 = 2K√n1n2 /ђ, δ = θ1 – θ2.

 

In presence of magnetic field, the phase difference which was considered constant previously varies along the insulating junction. However, when we consider one loop through the insulating barrier, the phase change must be an integer multiple of 2π since ψ = √ne = √nei(θ+2πn).

Summing up the phase contributions along the loop we get:

The integral in the first term on the right-hand side is recognized as the total magnetic flux φ in the loop. The integral in the second term on the right-hand side can be neglected since it is only non-zero in the very small barrier region. One can reach at the conclusion that:

where teff is the effective barrier thickness. Deploying this result in equation 1 and integrating, we get

This result is remarkably similar to the intensity pattern obtained on the screen due to diffraction of light by a single slit (Fraunhofer diffraction).

The Primitive Organism

A dc squid consists of two Josephson junctions connected in parallel. If the two links have If the two links have the phase difference δ1 and δ2 respectively the total current through the SQUID will be:

I = I0 sin δ1 + I0 sin δ2        (2)

The phase differences δ1 and δ2 are dependent on total flux φ threading

the ring. By summing up phase contributions from the flux and the junctions we get:

By choosing an integration path well inside the two superconductors the integral in the first term on the right-hand side is recognized as the total magnetic flux φ. Since the current inside the superconductor is zero, there will be no contribution from the second term and the expression can be simplified to

      

Combining equation (2) and equation (3) gives us the expression for the total current:

    

 

  (4)

 

This variation of current with respect to magnetic flux is similar to the behavior of light when passed through two slits in Young’s double slit experiment i.e., an interference pattern modulated by a diffraction pattern. Such behavior is far more sensitive than a single Josephson junction as the current intensity cycles several times for same variation in magnetic field.

An Essential Appendage- the Damping Circuit

A Josephson junction alone shows hysteresis in its I-V characteristics i.e., the voltage rises as current is increased, finds a saturation and remains there until the current is reduced to some relatively low value after which the voltage drops to zero rather sharply. Such behavior is undesirable and over time, it evolved into a resistively- and capacitively-shunted junction (RCSJ). These have resistance and capacitance connected in parallel and like any other RC circuit, they provide damping.

Using Kirchoff’s law or conservation of charge, we can say

Using equation (1), this becomes

Using variables:

This can be written in a neat fashion as:

As one may notice, the form of this equation is quite similar to that of a damped harmonic oscillator except that it is non-linear in δ.

β is the inertial term here. For overdamping, β will be negligible. A negligible inertial term would mean that by reducing the bias current from above Io, the particle gets trapped instantly in one of the minima of the potential at I = Io, which results in non-hysteretic I–V characteristics. For very small values, β can be ignored (or considered zero) and the resultant equation then gives the following solution:

The dependence is given in figure:

An almost linear dependence.

In the case of squid, at zero flux, the I-V characteristic is similar to that of a single Josephson junction with 2Io critical current and R/2 resistance.

When the DC SQUID is biased at an appropriate constant current Ib [Figure1.3(b)], changes in applied magnetic flux cause the voltage to swing between two extrema, producing the oscillations with period φo shown in Figure 1.3(c).

For an increasing flux, the total current will decrease down to a minimal value Itot min when the flux is φo/2. When the flux is increased even more, the total current is increased back to its maximum value at the flux φo.

If the current is biased above 2Io = Itotmax ,the modulation depth is equal to change in voltage which we can measure.  Using this measured voltage, we can calculate the flux. Knowing flux and the set-up parameters, one can calculate magnetic fields, orientations, etc.

Conclusion

Squids as we began, whispered secrets to us. Secrets of minuscule, secrets of the immense world made from minuscules. As one can see, these creatures show sensitive responses to the environment. We took advantage of their functional behavior and their responses are all that we have deciphered using the mathematical models and physical laws.

References

  1. The SQUID Handbook: Fundamentals and Technology of SQUIDs and SQUID Systems, I by Prof. Dr. John Clarke Prof. Dr. Alex I. Braginski
  2. Superconductivity by Joe Khachan and Stephen Bosi
  3. The Josephson Effect by  Jakob Blomgren 1998, Revised: Per Magnelind 2005
  4. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/4%3A_Diffraction/4.3%3A_Double-Slit_Diffraction
  5. Image Source: https://www.researchgate.net/figure/Superconducting-Quantum-Interference-Devices-SQUIDS_fig1_282760178

 

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6 thoughts on “Physiology of a DC SQUID (Superconducting Quantum Interference Device)”

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