Confused with the formation of Cooper pairs

[paulzhen]

Confusion over the formation of cooper pairs

Hi all,

There are 2 points I do not get:

1. I read something saying that phonon is an energy packet of "heat and sound", and propagates through lattice vibration...so, I thought it is saying that phonon is emitted by one electron, then absorbed by the lattice (and excited one of its vibration mode), then (the phonon) re-emitted via vibration, finally absorbed by another electron.
But when I read on, I found that the electron-phonon interaction is a direct process between 2 electrons (one emitted a phonon, another one immediately absorbed it, nothing to do with lattice). So what is the role of "lattice vibration" playing in this phonon exchanging process? Or what exactly is "phonon exchange between electrons"?

2. How can it possible that "a cooper pair travel in one direction, but 2 electrons have opposite momenta (means traveling in opposite direction)"? Just desperately asking...how can this happened?

Thanks so much for helping!

 

[K^2]

It's field theory stuff. You have electron interacting with particles in the lattice. The total of these interactions can be written as if a quaziparticle, a phonon, is emitted or absorbed by an electron. So yes, first electron interacts with the lattice, and the lattice interacts with the second electron. But since we are only interested in interactions due to excitation of lattice, we can write it as particle emitted by first electron and absorbed by second.

As for the propagation of pair, in simplest way, think of it as center of mass motion. Center of mass can be moving in a particular direction, while the two particles move in opposite directions with slightly different momenta. Reality is a bit more complicated, since electrons themselves are delocalized, but it's the same principle.

 

[paulzhen]

It's field theory stuff. You have electron interacting with particles in the lattice. The total of these interactions can be written as if a quaziparticle, a phonon, is emitted or absorbed by an electron. So yes, first electron interacts with the lattice, and the lattice interacts with the second electron. But since we are only interested in interactions due to excitation of lattice, we can write it as particle emitted by first electron and absorbed by second.

As for the propagation of pair, in simplest way, think of it as center of mass motion. Center of mass can be moving in a particular direction, while the two particles move in opposite directions with slightly different momenta. Reality is a bit more complicated, since electrons themselves are delocalized, but it's the same principle.

Thanks for your answer!
For the second part, since 2 electrons have opposite momenta, does it imply that they just "pair" with each other for a short term (say within coherence length?), and will change their "partner" consistently?

 

[DrDu]

The two electrons forming a Cooper pair have only opposite momenta if there is no current flowing. If there is a superconducting current, the momentum of the two electrons adds up to the average momentum of the Cooper pairs.

 

[Davephaelon]

Due to the probabilistic nature of Quantum Mechanics, is it possible that a small number of Cooper pairs might come into existence in a type 1 superconductor, such as niobium, even at liquid nitrogen temperatures (niobium requires liquid helium temperature to attain superconductivity)? What I had in mind was maybe one out of every million possible Cooper pairings momentarily forming in a piece of niobium before rapidly dissipating, even though the temperature of liquid nitrogen is far above that of liquid helium.

Also are Roman Numerals available? I couldn't find them. I used the Arabic Numeral 1 in place of the Roman Numeral for 1 in "Type 1".

 

[ZapperZ]

Due to the probabilistic nature of Quantum Mechanics,

This is actually an extremely vague, and frankly, a "cop-out" answer that is used to explain everything. One might as well say that "due to the probabilistic nature of QM, the Cooper pair can also tunnel out of the superconductor and appear in air"! What exactly does that mean?

...is it possible that a small number of Cooper pairs might come into existence in a type 1 superconductor, such as niobium, even at liquid nitrogen temperatures (niobium requires liquid helium temperature to attain superconductivity)? What I had in mind was maybe one out of every million possible Cooper pairings momentarily forming in a piece of niobium before rapidly dissipating, even though the temperature of liquid nitrogen is far above that of liquid helium.

Now, separate from the use of the all-encompassing QM as the "excuse", we know already that there are dips (pseudogap) in the single-particle spectral function and tunneling density of states measurement of various superconductors, especially the underdoped cuprate superconductors, that can be interpreted as having some sort of pairing, but without long-range coherence. This means that there are signs of formation of pairs (are they really "Cooper pairs"?) but no superconductivity. This has been reinforced recently from AFM data on strontium titanate that showed clearly the formation of pairs without any superconductivity.

Please note that one does not use the vague "probabilistic nature of Quantum Mechanics" as an explanation for these phenomena.

Zz.

 

[DrDu]

Yes, in principle this is correct. Cooper pairs can exist as short lived resonances above the critical temperature.
In fact, when one calculates the critical temperature one calulates for which temperature these resonance states become stable. However, at the temperature of liquid nitrogen the livetime of these resonances will be so short that one cannot speak of resonances anymore. I.e., the picture breaks down.

 

[Davephaelon]

Yes, in principle this is correct. Cooper pairs can exist as short lived resonances above the critical temperature.
In fact, when one calculates the critical temperature one calulates for which temperature these resonance states become stable. However, at the temperature of liquid nitrogen the livetime of these resonances will be so short that one cannot speak of resonances anymore. I.e., the picture breaks down.

For niobium metal at liquid nitrogen temperature, how brief would these Cooper pairings be? Would it be like micro-seconds, nano-seconds? Is there a way to roughly calculate this?

 

[Davephaelon]

The two electrons forming a Cooper pair have only opposite momenta if there is no current flowing. If there is a superconducting current, the momentum of the two electrons adds up to the average momentum of the Cooper pairs.

Would it be correct to say that the net momentum of any given Cooper pair is zero, for the case where no supercurrent is actually flowing?

 

[Davephaelon]

"Now, separate from the use of the all-encompassing QM as the "excuse", we know already that there are dips (pseudogap) in the single-particle spectral function and tunneling density of states measurement of various superconductors, especially the underdoped cuprate superconductors, that can be interpreted as having some sort of pairing, but without long-range coherence. This means that there are signs of formation of pairs (are they really "Cooper pairs"?) but no superconductivity. This has been reinforced recently from AFM data on strontium titanate that showed clearly the formation of pairs without any superconductivity."

"Please note that one does not use the vague "probabilistic nature of Quantum Mechanics" as an explanation for these phenomena."

OK, I used a broad brush treatment, being a novice in these sort of things. I can see I have a lot of learning to do before I can carry on an intelligent conversation in these topics. But I have a few questions. It sounds like physicists are not quite certain that cooper pairing is the mechanism for cuprate superconductivity, though the evidence is suggestive. Might there be some hybrid mechanism that combines Cooper pairing with something else to induce superconductivity in cuprates?

Also, from reading on the internet I learned that Cooper pairs are spaced on the order of 1000 times the lattice spacing: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/coop.html That really surprised me. I visualized that the Cooper pairs were separated by one, or several, lattice spacing. From reading elsewhere on the net I knew that quantized packets of sound, phonons, were exchanged between the Cooper pairs to bind them together, and overcome the Coulomb repulsion. Sound waves move much faster in solids than in air. So, has there ever been an estimate, or measurement of the velocity of these phonons in type 1 superconductors?

 

[DrDu]

Also, from reading on the internet I learned that Cooper pairs are spaced on the order of 1000 times the lattice spacing: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/coop.html

Just to clear this up: This is not the spacing between Cooper pairs but between the two electrons making up a Cooper pair.

 

[DrDu]

I think the answer to your original question, i.e. the scattering in the Cooper channel abouve TC, can be found in Mattuck, "A guide to Feynman diagrams in the many body problem".

 

[Davephaelon]

DrDu and ZapperZ, thank you very, very much for your responses! I"ll have to see if Mattuck's book is available on Amazon. I was just going over other threads on the Cooper pair/superconductor issues, and it is immensely helpful. I'm starting to catch on to the terminology, which is almost like learning a new language. But ultimately I know that I have to learn the math to fully comprehend the deep concepts here.

One question that I have for ZapperZ, who wrote the following in this thread (https://www.physicsforums.com/threads/superconductivity-cooper-pair.39913/), which is closed for further replies, so hopefully I'm not violating forum rules, by posing a question here from that thread.

Here's what ZapperZ wrote "...In a superconductor, the quasiparticle is a single-particle excitation in the NORMAL state, i.e. not in a condensed Cooper pair. In fact, to be exact, the quasiparticles are the ones that form the Cooper pairs, not the bare electrons or holes."

I don't understand what the distinction is between a quasiparticle electron and "bare electron", it seems like they should be the same thing; e.g. just individual electrons not bonded with another electron as in a Cooper pair.

 

[ZapperZ]

DrDu and ZapperZ, thank you very, very much for your responses! I"ll have to see if Mattuck's book is available on Amazon. I was just going over other threads on the Cooper pair/superconductor issues, and it is immensely helpful. I'm starting to catch on to the terminology, which is almost like learning a new language. But ultimately I know that I have to learn the math to fully comprehend the deep concepts here.

One question that I have for ZapperZ, who wrote the following in this thread (https://www.physicsforums.com/threads/superconductivity-cooper-pair.39913/), which is closed for further replies, so hopefully I'm not violating forum rules, by posing a question here from that thread.

Here's what ZapperZ wrote "...In a superconductor, the quasiparticle is a single-particle excitation in the NORMAL state, i.e. not in a condensed Cooper pair. In fact, to be exact, the quasiparticles are the ones that form the Cooper pairs, not the bare electrons or holes."

I don't understand what the distinction is between a quasiparticle electron and "bare electron", it seems like they should be the same thing; e.g. just individual electrons not bonded with another electron as in a Cooper pair.

In a metal, if you carefully measure the "mass" of the conduction electron, via the electron dispersion curve, you'll find that it is different than the bare mass of an electron. That Mattuck book that you have been referred to will explain this further. What happens here is that the electrons in the metal are undergoing many-body interactions. It is interacting not only with the background potential of the lattice ions, but also the potential of OTHER electrons, and there are many, many of these other electrons around it. This is why this is a "many-body" problem. This is a difficult problem to solve, because essentially, you have a single, many-body interaction, and the equation to solve this is practically impossible.

Landau came up with his Fermi Liquid theory. In it, he said that if the interaction, or couple, is under some weak limit, one can actually simplify this from one, many-body problem into many, one-body problem by lumping all the many-body effects into what is called the "self-energy" of the interaction. We know how to solve this one-body problem. This one-body is no longer the bare electron, but rather a quasiparticle, or quasielectron in this case. (or the quasi-horse in Mattuck's text). A consequence of this is that you have quasiparticles with "effective mass" that can be different than the bare mass of an electron. In the Ruthenates, for example, the mass of the charge carrier than be as high as 200 times the bare mass. So essentially, all the many-body interactions have been "renormalized", and lumped into the effective mass, but we got back our "one-body" scenario that we know and love.

Of course, there are limitations to this, especially the weak-coupling limit. There are many scenarios where the Fermi Liquid model doesn't work very well to describe the more complicated systems.

Zz.

 

[Davephaelon]

Zz

Thank you for this very detailed explanation! I now understand what is meant by a quasiparticle; a term I've come across repeatedly in condensed matter literature. Aside from Mattuck's book, might there be some other good primers for the field of superconductivity? I do intend to purchase the Mattuck book, though.