Why does BCS theory fail to explain superconductivity at high temperatures?

Technically, the answer is BCS theory does not explain superconductivity at high temperatures because electron-phonon coupling isn’t sufficient strong and (more important) because the superconductors with high Tc are deficient in the Fermi surface that is just above Tc.

Disclaimer: I’m responding to this question only to the copper-oxide (cuprate) superconductors with high temperatures (many of them have Tc that exceeds 100K) but not the iron-pnictide mediumly-high temperature superconductors.

Disclaimer 2. BCS superconductivity is not impossible at higher temperatures. Metal atomic hydrogen has been thought to be a superconductor at room temperature and the most superconducting threshold ever (H2S at the pressure of high) is probably to be a BCS superconductor (see: What is the mechanism behind sulphur hydride’s Superconductivity at high temperatures?). The question was initially addressed before H2S was included on the scene, so I interpreted it as “why does BCS theory fail to explain the high Tc in high Tc material when it doesn’t explain the high Tc the same way?

Long answer: As as I am aware I have seen every superconductor explained in the BCS theories

The formation of Cooper pair The s phenomenon is caused by electrons that interact with quantized atomic lattice oscillations (phonons). The problem is that BCS theories do not require pair of Boson as a possible phonon. Another excitation that provides attractive interactions with electrons at the Fermi surface could substitute for. Furthermore, BCS theory does not restrict superconductivity at very high temperatures. In fact it has been demonstrated that superconductivity occurs at the temperature of 190K in the Hydride ( [1412.0460Conventional superconductivity at high pressures at 190 K

) This is believed to be mediated by electrons according to isotope effects studies.

In this regard, let’s take a look at the electron-phonon connection that causes Cooper pairing in a variety of (low-Tc) superconductors.

The animation for the phonon-mediated Cooper pairing follows. An electron that is moving through an ion lattice that is positively charged may move an ion away and cause an local surplus of positive charge, which draws the next electron. This slowed attraction can be attributed because atoms are heavier than electrons, and are moving slower. In a different context it is said that electrons emit the energy of a phonon that is q

Then, it is taken up by another electron. (skip ahead to skip equations if you’re not a fan or find it offensive to unscientific formulas) The interaction is described by
Veff(o)=|geff|21o2-o2D
Where is the place? O is the energy of electrons that is being studied, ODD is the frequency of Debye (the theoretical maximum phonon frequency of the solid) as well as geff It is a highly effective electronic-phonon coupling vertex (which defines the likelihood that one electron will emit a phonon , and an electron absorption). A negative Veff This is a beautiful interaction between two electrons that exchange the (virtual) phonen. Also, only electrons in +-kBT of the Fermi levels are taken into consideration, that sets a maximum of the Fermi level. This sets an upper |o| ( kB The Boltzmann constant that transforms temperature into units of energy). This is due to the fact that they are the primary electrons that give an element its physical properties (see the answer below: What is the most simple explanation of Fermi surfaces? ). In low temperatures, KBT is always a lot smaller than ODD So Veff is negative, favoring Cooper pairing. The argument presented thus this point, Cooper pairing won’t happen at temperatures that are higher than Debye threshold (which is the result in the conversion of the Debye frequency into an actual temperature). kB

). If the atoms within crystals are thought of as being a mass of springs that is, the Debye frequency (temperature) is high for lighter atoms on springs, and low for heavy molecules on floppy springs. In the case of the elements that make up the crystal, the Debye temperature can vary from 1860K in the case of diamond*, to around 100K for rare earth elements. it’s somewhere between for compounds made up of an assortment of elements, according to the arrangement of the bonds. This argument starts to explain the reason BCS theory is unable to explain the high Tc and high Tc, but I’m far from finished yet.

After establishing attractive interaction, BCS theory proves that this interaction is attractive and creates bound states which means it is more energy-efficient for electrons to be part of the form of a Cooper pair than in singletons. I will not go through an easy formula, but the basic result is that there’s bound state and the energy associated with it is expressed by the following formula
|E|=2oDe-1/geffN(EF)1.76kBTC

Here, geff is the only effective appealing (electron-phonon) pairing possibility (related to the power of electron-phonon coupling) and N(EF)

This is the electron density is the density of electron states at that Fermi level. The Fermi factor of 1.76 is precise in the limit of weak coupling. In the case of more powerful couplings, the factor might be different, and the exponent may appear slightly different, however the fundamental ideas, which I’ll be discussing below, remain identical.

In the light of that How can we achieve a high Tc? It is possible to enhance the energy of Debye and we can also increase the electron-phonon co-coupling, or we could increase the number of states that exist in that Fermi level. Let’s look at the two latter terms on the basis of the expenditure. It is apparent that the two terms aren’t separate, and, unfortunately they can counteract each other. If you increase the amount of electron states electrons, they will block out Phonons, and the coupling between electrons and phonons is weak (one reason why solid metals like copper and gold do not superconduct). If you boost the electron-phonon interaction, you’ll eventually experience some sort of instability (a Polaron, also known as a charging density wave) that causes the density in electrons at the Fermi level to decrease. This is the basic argument. If we look at the high temperature superconductors made of cuprate specifically, we are aware of that their frequency is Debye (I do not want to research it, however) So we can calculate the amount of electron-phonon coupling required to produce the Tc we see (say 100K). Based on the majority of calculations, this electron-phonon coupling is far too big to be feasible.

As I stated in the past, BCS theory does not depend on electron-phonon coupling (though all superconductors explained in BCS theory happen to be controlled by the electron-phonon connection). The principle insight that was used to develop BCS theory was the fact that the Fermi surface is not able to withstand any type of interaction that is attractive. This is why as in this formula the oD

It is possible to replace it with an energy that is related to another kind of excitation (say magnon) and geff It could be replaced with an expression that describes electrons that are coupled to other excitations. This was the way first used to explain superconductivity in cuprates. But there’s an issue: N(EF)

. BCS theory, and the extensions of BCS theory, which call for the presence of a distinct Boson rely on the existence of the existence of a Fermi surface in its normal state. To put it more simply and more simply, at temperatures of just Tc the material has to be a metallic material; not an insulator and certainly not a semiconductor. Above Tc, the cuprates do not belong to the metals. In the vicinity of Tc is the mysterious pseudogap that’s identity is being discussed (see: What are the most recent theories on superconductivity in high temperature?). What we understand about this pseudogap phase is it could be metallic for electrons traveling in one direction however it’s a semiconductor for electrons traveling in different directions (technical observation: metal indicates that there isn’t a gap in the density of state at Fermi levels, while semiconductor implies there’s gaps).

The main reason why BCS theory fails to explain superconductivity at high temperatures in cuprates is because what is normal (what is found at or above the Tc) of the cuprates is totally distinct from that of the regular state BCS theory begins with.

Fun trivia that is not connected to cuprates. If diamond is coated with sufficient Boron (p-type) It transforms into superconductor and a metal at a lower temperatures (10K or less). It is believed that the Tc is restricted due to the numerous defects in a real boron-doped diamond and if they were eliminated, boron doped diamond could become the highest temperature superconductor due to of the huge Debye frequency. In addition, hydrogen is thought to be an ambient temperature (!!) superconductor when it is pressured enough to form an element, and superconductivity is observed in H2S at pressures at 190K. This is the highest level of Tc ever observed ( [1412.0460] Conventional superconductivity at 190 k when pressure is high

). Hooray for large Debye frequencies.

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