Why modeling superconductors is so hard

Author:Jukka Knuutinen
Expert contributor:Dr. Valtteri Lahtinen
SIMULATIONS -️ April 5, 2022

Superconductors are awesome.

This article will take you through their amazing properties and showcase some of their innovative applications.

Also, we’re going to explain why modeling them is so difficult - and why neglecting to do so can turn out to be costly.

Whether you own a particle accelerator or not, this article will provide you with some interesting pieces of information along with a more straightforward understanding of the challenges regarding the modeling of superconductors.

Superconductivity in a nutshell

Superconductors are materials that lose all of their resistivity under a certain temperature, being able to conduct electricity with zero losses.

The temperature that allows for this superconductive state is usually really low, near absolute zero.

The temperature, however, is not the only parameter dictating the superconductivity. The strength of the current and any magnetic fields have to also be under a certain threshold.

Once these so-called critical conditions are met, a material can enter and maintain the superconductive state.

Once in this state, another effect can be observed, as superconductive materials have the tendency to expel any magnetic flux from their interior.

This expulsion of the magnetic field is called the Meissner effect and can be neatly observed as superconductive materials levitating over magnets, as seen in the picture below.

A superconducting material levitating over a magnet.
A superconducting material levitating over a magnet.

Applications of superconductors

These characteristics - the perfect absence of both resistivity and the magnetic field - are exploited in the applications of superconductors.

The loss of resistivity allows for the transportation of high electric currents without any heat losses. Since these currents form a magnetic field, we can create strong magnets that, likewise, have zero dissipation or heat losses.

 Maglev train at speed on test track.
Maglev trains employ superconducting magnets and can reach speeds up to 600 km/h.

One exciting application of these superconducting magnets is maglev. Maglev (magnetic levitation) is a train transportation system that employs superconducting magnets to elevate the train off the track and move it forward.

Maglev trains can reach record-breaking speeds due to the absence of friction; however, their true potential in the future of rail transportation is still under debate.

Another - somewhat more down-to-earth - application is magnetic resonance imaging (MRI) machines, found in most major hospitals. The functionality of these machines is based on the strong magnetic field powered by superconducting magnets.

The fact that superconductors can conduct electricity with zero losses has incredible potential in power transmission, and sure enough, promising research is being undertaken. However, the demanding requirements to enter and maintain the superconductive state pose a significant challenge.

Different types

Superconductors are grouped into type 1 and type 2 based on slight differences in properties and behavior.

Type 1 superconductors expel the magnetic field from a material entirely. Type 2, on the other hand, only partly obeys the Meissner effect. This means some of the magnetic fields can, in fact, penetrate the material in the form of quantized flux vortices.

This by no means is an unfavorable occurrence since it allows the superconducting material to sustain stronger currents, therefore increasing the strength of the magnetic field.

Some impurities can also be added to the material to prevent the movement of the flux vortices, allowing even more current to be sustained without additive heat losses.

However, heat losses do occur in superconducting materials whenever there is a change in the magnetic field. Alternating current (AC), for example, inflicts these changes and therefore causes the superconducting material to exhibit some heat loss.

An interesting history

In 1957 a quantum theory was proposed that could describe the behavior of superconductors in great detail.

This theory - called the BCS theory - also stated that materials could only enter a superconductive state in temperatures below 30 kelvin (-243,15° Celsius).

The researchers (Bardeen, Cooper, and Schrieffer) received a Nobel Prize in Physics for their celebrated theory in 1972.

In 1986, however, so-called high-temperature superconductivity was discovered.

These high-temperature superconductors operate in temperatures up to 130 kelvin (-143,15° Celsius), although still fairly cool for any everyday practices, it’s certainly higher than the previously thought limit of 30 kelvin.

Interestingly, much of the behavior of high-temperature superconductors is still unsolved, as BCS theory alone cannot explain this phenomenon.

A theory covering the quantum level details of both high- and low-temperature superconductors is still needed.

Luckily, when modeling the large-scale applications of superconductors, whether it be a maglev train or an MRI machine, the quantum level theories can be neglected.

Unfortunately - as we will soon see - this is a small consolation.

Why is electromagnetic modeling so hard?

As stated, when modeling the applications of superconductors, quantum level theories are not often necessary. More often than not, the classical equations of electromagnetics - the Maxwell equations - along with some slight modifications suffice.

Let’s examine an example from an AC standpoint: we want to predict heat losses in a superconductive material experiencing a magnetic field varying in time.

We can start with the Maxwell equations, which already by themselves become tricky for superconductors due to the nonlinear nature of the materials. In particular, the resistivity of the superconductor is modeled using a so-called power-law relation.

This means that there is a non-linear relationship between the current density and the strength of the electric field.

Furthermore, the material's resistivity has a nonlinear dependence on not only the strength of the magnetic field but also the direction, making the material anisotropic. This is true in particular for high-temperature superconductors.

These multiple nonlinear dependencies make solving electromagnetism problems tricky, not even considering the multiphysics aspect of combining, for example, classical mechanics into the equation.

Moreover, we have to keep in mind that the heat losses that we are modeling also affect the temperature and, therefore, the material properties of the superconductor.

This is yet another nonlinear dependence to account for.

Accidents can happen - and they can be expensive

Although electromagnetics modeling can be challenging, it is essential.

A fascinating thing can happen in a superconducting magnet when for some reason - whether it be heat losses due to AC or a beam of electrons hitting the magnet - there is a sudden increase in temperature.

The sudden increase in temperature results in a loss of the superconductive state locally, which means that the electricity is now met with resistance.

This resistance results in more heat being generated and, therefore, the loss of more extensive areas of the superconducting magnet.

In a blink of an eye, the magnet will lose its superconductive abilities and, in the worst case, burn, rendering an expensive piece of equipment useless.

This chain-reaction-like spreading of the normal zone caused by an increased temperature is called a magnet quench.

CERN got a first-hand experience of this phenomenon in 2008 when their Large Hadron Collider quenched, requiring multiple of their - rather expensive - magnets to be replaced.

Failsafes are in place to prevent massive damage, but better and more detailed modeling is still needed to better help us protect equipment against this quenching.

In conclusion

  • The development of superconductors has had a massive impact on many fields with lots of potential for further use.
  • Although modeling the electromagnetic, thermal and mechanical behavior of superconductors is complex due to the multitude of nonlinear dependencies and multiphysics involved, it is vital for R&D in many fields.
  • At Quanscient, we research better models for superconductors, while also developing quantum algorithms that help us transfer these problems into the realm of quantum computation.