Impatient Superconductivity - LK99 A tale of unconfirmed superconductivity

Superconductors, materials which do not consume any energy to move electric currents, are a holy grail to be discovered in a usable form at room temperature and ambient pressure. In other words, they do not drop any voltage (which is a measure of the energy spent per charge) but they let current to pass. An ambient room temperature, ambient pressure superconductor will change the world in ways imaginable and will leave many problems unchanged in ways unimaginable. From energy transport, generation, medical, computing to and interconnects, a RT, ambient superconductor will provide new and powerful tools to solve a plethora of problems. Starting with direct (unimaginative) changes to existing use of superconductors in MRI to hitherto uninvented computing and sensing, this will indeed be a breakthrough that will seed a generation of developments. The economic implications and scientific treasure is so large that we may see a new industrial revolution similar to the invention of the transistor.

Recently, there have been extraordinary claims of a new RT superconductor called LK-99 [1], developed by a bold team of scientists who have been working on this material since at least 1999. However, extraordinary claims need extraordinary proof, so the scientific community, especially materials and condensed matter physics will test and debate the results. For news coverage of LK-99 please see, see. What's next for LK-99 (jump to end of the article). While, we will let science run its course. I am going to try to convert the science behind the new generation of superconductors to “interested engineer” terms

Figure 1A. This particle picture is not fully accurate since the electrons are delocalised in a material as wavefunctions. Electrons don’t pass each other like objects, but in reality form a standing wave (!, welcome to the quantum world) with a fixed width.?

Intuitive minimum that you may skip: At the core of all types of superconductors are electrons which form pairs [4] allowing them to masquerade as bosons. (Reminder electrons are fermions and are not allowed to be in the same quantum state, but when they form composite bosons they are allowed to be in the same quantum state). So essentially,? electrons form pairs allowing them to masquerade as bosons. One up and one down spin electron form what is known as a Cooper pair. A cooper pair has no net spin and is a boson much like photons (another popular boson is the Higgs Boson). OK, so how do I visualize a Cooper pair in space: A Cooper pair has an up spin electron and a down spin electron paired together, but these electrons are moving in opposite directions ! Here I drew a picture of how they might look if the electrons were to be localized. The natural issue with choosing a particle model is to ask what happens when the electrons move away from each other or cross each other. So a more accurate picture is to consider wavefunctions.

Since a positive traveling wave function and negative traveling wave function (ei(kx-wt)+ei(kx+wt)) add to produce a standing wave [5, figure taken from blog 'this condensed life'], the net wavefunction looks as follows, forming a localized interference pattern [6], which in-turn can flex the lattice to reduce the total energy . Hence, the atoms and their arrangements, their ability to deform are a key consideration whether the copper pairs form. At high temperatures, the atoms move around so much that these subtle deformation of the atoms don’t really form stable cooper pairs. This is one of the reasons why it is so hard to get a room temperature superconductor ! The width of a cooper pair can be 10s to 100s of nm (that is they can easily span 100s-1000s of atoms), this makes the crystal quality very important and any defects in these lattice structures kills the cooper pairs.

Figure 1B, the localized wave picture of a cooper pair. The particle picture is not fully accurate since the electrons are wave functions in a material. Note the red balls, which are positively charged atoms, they move ever so slightly to reduce the energy of the system which keeps the Cooper pair stable. The motion of the atoms requires phonon physics to be supportive of cooper pair formation.

In plain terms, many of these Cooper pairs can condense into a single/similar quantum state hence producing the superconductivity phenomenon. Analogies for this exist for super fluidity and super solidity.? Resistance of a material is the result of the scattering of the electrons, but when the entire sea of electrons may behave as a wavefunction, they will transfer charge without scattering or with minimal scattering.

Unconventional superconductors [7] : superconductors that deviate from the ‘old’ wisdom

Here I will put in simple language what this specific style of superconductors are about. They belong to a class of superconductors called “Unconventional superconductors”, discovered a in 1979 by Frank Steglich [8]. The first material was CeCu2Si2 which was followed by a number of materials. What makes them unconventional is that they do not obey either the conventional BCS theory or Nikolay Bogolyubov's theory or its extensions.

Figure 2. A timeline of superconductors between 1979 and early 2000s. The breakneck speed of inventions slowed down as both the scientific community and funding agencies lost their optimism to bridge the last ~3X gap to room temperature. Complicating matters, unconventional superconductors come in as many as 11 varieties, namely 1. Heavy fermions (HF are further divided into high and low heat capacity), 2. Cuprates (Hold and electron doped), 3. Iron based superconductors, 4. Strontium Ruthenate, 5. Non-centrosymmetric superconductors, 6. Organic superconductors, 7. Layered Nitrides, 8. Superconductors with ferromagnetism, 9. Cobalt oxide Hydrates, 10. High pressure H2S class and 11. Topological superconductors & Interracial superconductors. Infact, the list of potential materials and mechanisms is ever expanding as both the synthesis and metrology continue to advance, there was a genuine reason to feel this entire field was searching a needle in the haystack of nearly infinite possible materials formed by subtle variations in the structure and electronic states. Note that the funding sources for condensed matter used to be major electrical and telecom companies such as GE/AT&T, as these labs failed the sources shifted to intense focuse on microelectronics away from superconductors.

Heavy Electrons - or Heavy fermions in Unconventional superconductors

One of the main themes with the latest class of superconductors (including LK-99, Magic Angle twisted Graphene and chalcogenides is the presence of heavy electrons. They depend on a somewhat intuitive explanation (which I will attempt now). Recall from high school physics that, the kinetic energy of a moving object is given by?

which is another way of saying the sharper the bands are the smaller is the effective mass Or the flatter the bands are the higher are the effective masses of the electrons in the band.

The electrons in material can show effective masses that are greater or smaller than their masses outside the materials. Thanks to the effects of bloch waves. This is no different than the magical way Mexican waves in a stadium can defy the mass, force relation of humans making the human wave.

Figure 3 (Bloch wave analogy to Mexican wave): Mexican wave and effective mass of a wave function- Much like mexican wave, electronic wavefunction can travel through an atomic lattice, except electrons are actually characterized by wave functions fully. Depending on the details of the crowd, these waves can have “effective inertia” i.e mass higher, lower or equal to mass of electrons outside the medium. For example, a mexican wave can move much faster than a normal person !

The Bloch waves, which are the mathematical description of electrons moving inside a period potential, allow electrons to defy their F=ma properties outside the crystals. Much of condensed matter class is spent learning this amazing result.?

Back to unconventional superconductors, in some materials the bands can get so flat (i.e the curvature is so low) that the energy of the electron does not change with its momentum. The effective mass of these electrons can be as high as 1000 times the normal mass !

Heavy Electrons in the “free electron sea” can support superconductivity:?

This is where the physics gets complex or even muddy [9], filled with complex equations but lacking the simple intuitive explanations from the old generation superconductors as to what happens to a free electron gas with sluggish electrons [10]. Yet, the experimental progress in this area as well as increasing theoretical confidence is remarkable.

Imagine a sea of electrons, but made with electrons that are 100-1000 times heavier than normal electrons. For a moment consider, what happens to this sea of electrons if the electrons were all spin polarized in the same direction. You get what is known as a ferro-magnet or a paramagnet depending on the temperature. In fact, all? magnets, including the ones on your fridge, are formed this way !?

A less intuitive picture is if the electrons don't all align in the same direction, but align in alternating directions (i.e every electron looks in the opposite direction the electron next to it). This is known as an anti-ferromagnetic material.?

Let me explain an intuitive picture here (which is based on experimental evidence by Shamus Davis at Cornell).

A bridge to superconducting state - An antiferro-magnet forming a superconductor?

Consider an antiferromagnet (paramagnetism, ferro-magnetism, anti-ferromagnetism are 3 of the most common forms magnetism takes in a material). They approximately look like the following picture, A magnet is a reservoir of spins, sometimes all aligned, sometimes misaligned in specific ways.?

Figure 5A. The common magnetic states of materials. A magnet’s magnetism is almost completely formed by its free electrons’ spin and is one of the most evident results of quantum mechanics in everyday life. Ferro-magnetic state (all spins aligned the same way), Anti-ferro-magnetic state (with spins aligned in opposite directions) and paramagnets which are typically ferro-magnets that are too warm (i.e above the critical temperature for becoming a ferro-magnet). ??

Figure 5B. Anti-ferro-magnetic state (with spins aligned in opposite directions) can be a precursor to forming cooper pairs, especially if the electrons also happen to be slow moving (with very low fermi-velocity or? high effective masses)

Notice how an antiferromagnet already has the spins oppositely aligned living close together ? under specific conditions these spins can form a cooper pair. In fact it is not uncommon for the super conductors to be right next to an AFM state when you tune the composition or pressure.?

Let us look at the phase diagram of a heavy electron/flat band/heavy fermion material.?

Figure 6. Magnetic materials with heavy electrons (or flat bands or heavy fermions or narrow bandwidth materials) can become superconductors. They typically show conversion from AFM to PM or AF-SC or PM to SC changes as temperature or structure get modified. This is one common theme in the new superconductors invented since the 1970s. The LK-99 work lacks this level of characterization and hence is open to interpretation.

Well, how exactly do flat bands lead to superconductivity ?

Note : One of the supporting factors for LK-99 is the theoretical work on the bands and the formation of spin polarized flat bands. What is missing is the link from flat bands to superconducting. I go over a detailed example of a mechanism that related flatbands to Cooper pair formation. THIS is how the missing link that theoreticians/physicists need to produce looks like.

In fact, this intuitive model was visualized in 2013/2014 by a magnum opus work at Cornell using highly sophisticated and skilled measurements of a superconductor [11,12]. It turns out the heavy electrons (the electrons who respond with high effective mass) are almost bound to their respective atoms (typically f electrons) while conduction electrons (c-electrons) are weakly bound. The heavy electrons (bound tighter to the atoms) form an antiferromagnetic order (i.e they are alternating in their orientations) and push the conduction electrons into similar ordering. These alternating up and down electrons can in-turn form Cooper-pairs !?

Figure 7: This picture shows two types of electrons in the unconventional superconductor, red electrons which are tightly bound and having higher mass, and blue electrons which are conduction electrons, weakly bound with lower mass. The red electrons are strongly bound to the atoms (they are f-electron, typically from 4f or 5f) which can be imagined to have higher mass due to their tighter binding to the atoms, these electrons have a anti-ferromagnetic order which they transfer to the conduction electrons which can form cooper pairs. The right panels show the band structure in B, the red bands are formed by the tightly bound f electrons (hence the band is flatter with higher mass), the sharp black band is formed by the fast moving conduction electron-sea. However, these two bands interact to create the structure in C (left side in blue), these bands on the left panel of C are still normal metallic bands. On the right side of the panel in C, the bands further form cooper pairs causing the little split hair sections where the cooper pairs.?

This level of clarity of mechanism (even a prospective mechanism) is missing in the theoretical work on LK-99. Showing flat bands is a “not impossible” statement, but showing the path to cooper-pair formation is the real valuable goal additive to the invention of the RT SC.

Complexity of unconventional superconductors:

Decades after the original contributions of Cooper, BCS , experts still argue vehemently about the exact mechanisms of superconductivity in systems that are in mass usage. A norm in this area for burden of proof is to go with massive amounts of data that are all self consistent (this is in the absence of a sure-shot theoretical bridge). The data typically looks as follows (yes, its complex) for getting wider acceptance and is repeated by multiple groups in short order.

Figure 8: Exemplary phase diagram of a flat-band superconductor (Magic angle graphene). Note the quality of data and how they can pick a tunable parameter space. The richer the phase diagram, with various classic phase transitions, the more credible is the full experiment. The theory often lags the experiments by decades and the phase diagrams provide deeper insight into the mechanisms which often stay heuristic.

You can find many “flat band” materials off the internet, some of these may be superconductor candidates- No one honestly knows which ones are which ones are not !

Figure 9: There is an abundance of flat band material candidates collected in libraries across the world.

What next for LK-99:

The reason for optimism - the possibility of a sputnik moment in unconventional superconductors:

Early theoretical papers are pointing to an unconventional source of magnetism and superconductivity, inline with the breakthroughs from Steglich, the father of heavy electron superconductors. Infact, Steglich’s work led to heavy electron (flat band a.k.a heavy fermion) superconductors in a range of materials even including in Graphene and Chalcogenides.

For example, when the Pb, Cu, PO4 are positioned in a very specific (unlikely entropic) configuration, the bands form in a configuration that does not prohibit more complex physics to happen. (I know this sounds odd, why can't theory give accurate answers already, the problem is we don't know the exact physics that these systems are following, hence the computer models are only as good as the physics included in them, not accounting for computational artifacts)?

A flat band, similar to the one seen in? is formed in the seminal work of heavy fermions, can be readily seen in the Pb, Cu, PO4 complex structure. I note that this occurrence of flat bands has been posted on multiple online accessible data banks for years with dozens more listed flat band databases.

Wait, didn't we say DFT is bad at getting the absolute energy levels wrong ??

Well, there is a way to test this by changing a parameter in the simulation (known as Hubbard U parameter), the authors did just that. They varied the fudge factor known as Hubbard U parameter. And it still shows the flat bands.

• What is the critical temperature for LK-99 as predictable from the models ?
• Is there a theoretical possibility of how LK99 can beat the Tc of heavy fermion SC by such a large margin ??
• What is the mechanism of cooper pair formation? (This answer may not come for a long time)

For clarity, I am reposting how an exemplary work relating flat bands to cooper-pair formation looks like. In this work, Shamus Davis shows exactly how the f-electron flat bands, interact with conduction electrons to form an AFM order, which can lead to cooper-pair formation (now that there is energy reduction when the conduction electrons are interacting with the f-electrons to form the cooper pairs, what's even more impressive is the experimental work to show the formation of cooper pairing)

• What is the full phase diagram of LK99 with varying stoichiometry and varying strain???
• What are the limits to conductivity ? Is it bound to be low due to the inherently low density of states (DOS) available for a flat band conductor ? Note how the flat band SCs are all bunched to the lower left corner in the figure comparing the SCs.
• What does MD simulation predict for the structure ?

The reason for caution - the possibility of an USO (unidentified superconducting/superfluid/supersolid objects)

History of SCs is full of USOs. The best of the best have erred and came back to fix their errors. A very public fight on super solid Helium between a researcher and his once PhD advisor (Prof. Reppy) that lasted years is a great example of good science [14] done with conviction and adherence to scientific enquiry. The controversy simmered for years [15, 16] before the original discovery was found to be erroneous and gracefully, masterfully explained by the discovery team [17], who continue to make extraordinary progress in their fields [18].

A massive list of criticisms, all well placed on the validity of the results, are already articulated in arXiv and viral posts on X. These include

- The unusually high resistivity of the SC phase

- Lack of a complex phase diagram. A well done controlled SC study typically produces a complex and holistic phase diagram which often looks like the one in figures above

- Potential artifacts due to stochastic conductive network effects. Hersch shows that the usual 4-point measurements can yield erroneous results based on the quality of the samples.

- Imperfect meissner effect : The variety of viral videos posted leave many simple questions such as absence of flux locking and partial levitation

- Weak magnetic threshold

- Lack of direct TEM imaging of the samples?

-The units and dimensions in the LK-99 experimental research papers leave a big question mark on the experiments

-Role of contact formation and access to raw data without any subtraction and format conversions (yes these mistakes also happen in science)

In the end, we remember that “Science is the collective belief in the ignorance of the experts”. There can be no doubt that there will be discoveries that will shock the experts, especially when barriers to doing science lower as better tools become available. Even if the LK-99 discovery is confirmed there are dozens of complex problems to solve riding on top of this potential result. How does one enhance the SC to allow practical current densities, how to increase the current carrying capacity and allow high current injection, how to take advantage of a RTSC in computing and sensing. What LK-99 event shows is that there is an immense amount of passion for breakthroughs in the world. We are left wondering, what happens If many sets out to solve this problem with the same intensity as finding a pandemic vaccine ?

[1]?https://arxiv.org/abs/2307.12008

[2] https://www.economist.com/the-economist-explains/2023/08/03/have-scientists-really-found-a-room-temperature-superconductor

[3] https://www.bloomberg.com/news/articles/2023-08-03/room-temperature-superconductor-and-lk-99-what-to-know?in_source=embedded-checkout-banner

[4] Cooper, Leon N. (1956). "Bound electron pairs in a degenerate Fermi gas". Physical Review. 104 (4): 1189–1190.

[5] https://thiscondensedlife.wordpress.com/2015/09/12/draw-me-a-picture-of-a-cooper-pair/#:~:text=In%20a%20Cooper%20pair%2C%20two,carry%20electric%20current%20without%20dissipation.

[6] https://arxiv.org/ftp/cond-mat/papers/0510/0510279.pdf

[7] Stewart, G.R., 2017. Unconventional superconductivity. Advances in Physics, 66(2), pp.75-196.? Microsoft Word - AdvPhysReview5_29 (arxiv.org)

[8] ?https://en.wikipedia.org/wiki/Frank_Steglich#cite_note-1

[9] ?Phys. Rev. B 102, 201112(R) (2020) - Superconductivity, pseudogap, and phase separation in topological flat bands (aps.org)

[10] ?https://arxiv.org/ftp/arxiv/papers/1106/1106.1213.pdf

[11] https://www.nature.com/articles/nphys2671

[12] ?https://www.pnas.org/doi/full/10.1073/pnas.1409444111

[14] ?Kim, E. and Chan, M.H.W., 2004. Probable observation of a supersolid helium phase. Nature, 427(6971), pp.225-227

[15] Physics - How Solid is Supersolid? (aps.org)

[16] https://news.cornell.edu/stories/2010/07/john-reppy-challenges-supersolid-helium

[17] Rev. Mod. Phys. 92, 045002 (2020) - Mechanical behavior of solid helium: Elasticity, plasticity, and defects (aps.org)

[18] Is solid helium a supersolid? | Physics Today | AIP Publishing

[19] ?Electrical Resistance of Hydrides Under High Pressure: Evidence of Superconductivity or Confirmation Bias? | SpringerLink