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Silverious
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I'd like to know more about cooper pairing. If anyone has any good resources I'd appreciate it if they shared them. Thanks in advance.
Do Cooper pairs interact with Fermi surfaces?
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Discussion Overview
The discussion revolves around the concept of Cooper pairs and their interactions with Fermi surfaces, particularly in the context of superconductivity. Participants explore various aspects of Cooper pairing, including its mechanisms, the role of band gaps, and the relationship between superconductivity and entropy. The conversation touches on both conventional and high-temperature superconductors, as well as the implications of atomic mass on superconducting properties.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants express a desire for resources on Cooper pairing, indicating a need for clearer explanations.
- One participant critiques the Hyperphysics website for its vague treatment of Cooper pairing and emphasizes the importance of understanding the energy gap related to Cooper pairs.
- Another participant notes that high-Tc superconductors exhibit complex behavior, where "pre-formed" pairs exist above the critical temperature, complicating the understanding of superconductivity.
- There is a discussion about the nature of band gaps, with some participants suggesting they arise from periodic potentials, while others argue they are due to electron-phonon interactions.
- One participant points out that not all electron-phonon interactions lead to band gaps, highlighting the complexity of the mechanisms involved.
- Entropy's role in superconductivity is discussed, with references to specific heat capacity and the behavior of Cooper pairs as temperature decreases.
- Participants debate the relationship between atomic mass and superconductivity, with some suggesting that heavier elements might enhance conductivity, while others argue that lighter elements typically lead to higher transition temperatures in conventional superconductors.
- A question is raised about the potential for superconductivity at temperatures higher than 40K if sound waves could influence atomic vibrations.
Areas of Agreement / Disagreement
Participants express a range of views on the mechanisms of Cooper pairing and the factors influencing superconductivity. There is no consensus on the exact nature of band gaps or the implications of atomic mass on superconducting properties, indicating ongoing debate and exploration of these topics.
Contextual Notes
Participants acknowledge the complexity of superconductivity, particularly in high-Tc materials, and the limitations of current understanding regarding the interactions that lead to Cooper pairing and the formation of band gaps.
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Originally posted by suyver
This, and the links therein, is a good place to start.
I am typically a fan of the Hyperphysics website. It has a wealth of solid info on just about everything and I always recommend people to go there first if they have a question. This, however, is the one exception. The info on Cooper paring is rather vague, and I don't understand this overemphasis on the "band gap"! The existence of the band gap does not automatically imply that these cooper pairs can conduct without resistance. If this were true, than insulators would have the same properties since there's a huge band gap in a typical insulator! In a superconductor, this band gap (more accurately the energy gap in the single-particle band structure) is the energy needed to break apart a cooper pair.
Instead of the hyperphysics site, I would recommend this one. It is low on math (in fact, non-existent), but it has a nice "cartoon" description of one mechanism for the formation of these cooper pairs.
http://www.chemsoc.org/exemplarchem/entries/igrant/bcstheory_noflash.html
Zz.
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suyver
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Originally posted by ZapperZ
http://www.chemsoc.org/exemplarchem/entries/igrant/bcstheory_noflash.html
That is indeed a nice site!
I quite like the cartoon for high Tc superconductivity!
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Originally posted by suyver
That is indeed a nice site!
I quite like the cartoon for high Tc superconductivity!
:)
I'm glad you like it. But please make sure you take descriptions like these with a grain of salt. There's nothing in the actual description of this phenomena that says that this is what it exactly looks like.
And the matter is even more complicated for high-Tc superconductors. An example would be that there are strong evidence that the electrons can assembled themselves into "pre-formed" pairs but do not yet form a condensate. This means that you can detect a gap in the energy band, but yet, it is still above the critical temperature for the superconductor. This means that there are "pairs" already, but the material isn't superconducting yet! It needs to go to an even lower temperature before such thing occurs. In conventional superconductor, both pairing and condensing occur at roughly the same temperature, Tc.
It is why this family of material has been keeping condensed matter physicists on their toes ever since they were discovered. The more layers you peel, the more complex and fascinating it becomes!
Zz.
Silverious
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Fascinating indeed. Thank you for the links zapperz and suyver.
garytse86
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what exactly is a band-gap? Is this linked with periodic potential?
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garytse86 said:what exactly is a band-gap? Is this linked with periodic potential?
A band gap is a gap in the material's energy level. In solids, the energy levels are continuous but only over a certain range. In metals, if you estimate the conduction electrons as "free electrons", then you have a dispersion relation looking like
[tex]E = \hbar^2 k^2/2m[/tex]
This means that E and k are continuous over all range. However, if you turn on the periodic potential, then you no longer have completely free electrons, but rather a "nearly-free" electrons, and your electron wavefunction is now the Bloch wavefunction having the same periodicity. This is where you will start seeing a gap at the zone boundary because only a certain range of k's will be allowed, and that in turn will limit E.
Zz.
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nbo10
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In a superconductor the band gap is not caused by a periodic potential but is caused by the electron-phonon interaction.
JMD
JMD
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nbo10 said:In a superconductor the band gap is not caused by a periodic potential but is caused by the electron-phonon interaction.
JMD
Er.. not quite. The gap in the energy spectrum of a superconductor is caused by the energy required to break apart a Cooper Pair.
The mechanism for such pairing can be anything. In conventional superconductor, it is the electron-phonon interaction, but in high-Tc superconductors, it can easily be electron-magnon or electron-[insert favorite bosonic excitation here]. Also take note that electron-phonon interaction doesn't always produce a band gap. There are tons of electron-phonon interactions in a normal metal - this is the predominant origin of normal resistivity in metals. Yet, we don't see band gaps arising out of these interactions.
Zz.
kurious
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Is superconductivity associated exclusively with materials which have low entropy.
clint
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entropy
Entropy (or the sum of -p.lnp. over all probabilities p at a given temperature) is determined experimentally by first measuring the specific heat capacity (C), that is, the heat required to raise the temperature of a material a Delta amount (s.i. units of C are of the order 10^-23 joules/kelvin) under given experimental conditions, and then by partially integrating C/T over temperature (starting at near 0K).
As far as the conduction electron contribution to C, as cooper pairs progressively condense below the critical temperature Tc, there is an observed decrease in C/T with decreasing temperature, and thus a drop in the electronic entropy S. These observations are clear in all superconducting materials, conventional or not.
In relation to Zz's statement also, for non conventional high Tc superconductors, there is already an observed loss in entropy in the normal state, ie at temperatures above Tc, due to the "pre-formed" pairs. The onset temperature of this effect is known as the psuedo-gap temperature, T*.
Entropy (or the sum of -p.lnp. over all probabilities p at a given temperature) is determined experimentally by first measuring the specific heat capacity (C), that is, the heat required to raise the temperature of a material a Delta amount (s.i. units of C are of the order 10^-23 joules/kelvin) under given experimental conditions, and then by partially integrating C/T over temperature (starting at near 0K).
As far as the conduction electron contribution to C, as cooper pairs progressively condense below the critical temperature Tc, there is an observed decrease in C/T with decreasing temperature, and thus a drop in the electronic entropy S. These observations are clear in all superconducting materials, conventional or not.
In relation to Zz's statement also, for non conventional high Tc superconductors, there is already an observed loss in entropy in the normal state, ie at temperatures above Tc, due to the "pre-formed" pairs. The onset temperature of this effect is known as the psuedo-gap temperature, T*.
kurious
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If superconductors were made from the heaviest atoms known to science, this would
limit the amplitude of atomic vibration and should increase electronic conductivity at
high temperatures.Is this why most superconductors are heavy materials?
limit the amplitude of atomic vibration and should increase electronic conductivity at
high temperatures.Is this why most superconductors are heavy materials?
clint
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Not quite.
In conventional superconductors, the electron-electron attraction is a result of the electron-phonon interaction (where a phonon is a quanta of a lattice/atomic vibration), right?
According to conventional superconductor theory then, the onset temperature of superconductivity, Tc, is proportional to the characteristic phonon temperature, so called the Debye temperature Td. Td is in turn proportional to M^-1/2 , where M is the atomic mass of the metal. So in fact, for all other things constant, the lighter the element, or the lower the M, the larger the value of Td and therefore Tc (although Tc has a maximum theoretical value, around 40K or so) for conventional materials.
As for the high temperature normal state, well above Td, the conductivity is dominated by the number of phonons available to limit the mean free path of the conduction electrons, which doesn't depend on the M. Going down the alkali metals column, the conductivity at a fixed temperature, say room temperature, decreases with increasing atomic mass mostly due to the increasing separation between atoms.
The confusion may arise from non-conventional materials, high Tc, where the chemical unit cell is more complicated than in conventional superconductors, and contains heavy elements like Hg, La, etc. amongst the lighter ones like Cu, O. However, the phonon coupling to the conduction electrons is not believed to be the dominant interaction behind superconductivity in this case, and has no theoretical limit to Tc (that we know of).
In conventional superconductors, the electron-electron attraction is a result of the electron-phonon interaction (where a phonon is a quanta of a lattice/atomic vibration), right?
According to conventional superconductor theory then, the onset temperature of superconductivity, Tc, is proportional to the characteristic phonon temperature, so called the Debye temperature Td. Td is in turn proportional to M^-1/2 , where M is the atomic mass of the metal. So in fact, for all other things constant, the lighter the element, or the lower the M, the larger the value of Td and therefore Tc (although Tc has a maximum theoretical value, around 40K or so) for conventional materials.
As for the high temperature normal state, well above Td, the conductivity is dominated by the number of phonons available to limit the mean free path of the conduction electrons, which doesn't depend on the M. Going down the alkali metals column, the conductivity at a fixed temperature, say room temperature, decreases with increasing atomic mass mostly due to the increasing separation between atoms.
The confusion may arise from non-conventional materials, high Tc, where the chemical unit cell is more complicated than in conventional superconductors, and contains heavy elements like Hg, La, etc. amongst the lighter ones like Cu, O. However, the phonon coupling to the conduction electrons is not believed to be the dominant interaction behind superconductivity in this case, and has no theoretical limit to Tc (that we know of).
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kurious
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superconductivity
for all other things constant, the lighter the element, or the lower the M, the larger the value of Td and therefore Tc (although Tc has a maximum theoretical value, around 40K or so) for conventional materials
Would superconductivity be present at higher temperatures than 40K if sound waves were fed into a crystal lattice.Can sound waves increase the phonon quanta by reducing the inertia of atoms?
for all other things constant, the lighter the element, or the lower the M, the larger the value of Td and therefore Tc (although Tc has a maximum theoretical value, around 40K or so) for conventional materials
Would superconductivity be present at higher temperatures than 40K if sound waves were fed into a crystal lattice.Can sound waves increase the phonon quanta by reducing the inertia of atoms?
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Silverious
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Well, haven't superconductors been made at 150K by adding an Oxygen? Seems like I heard that somewhere.
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Silverious said:Well, haven't superconductors been made at 150K by adding an Oxygen? Seems like I heard that somewhere.
Most of the high-Tc superconductors (the cuprates) are doped with oxygen to add "holes" to the insulating parent compounds. These are the ones with the highest Tc at optimal doping. There is a sister compound of the cuprates that are electron doped. They tend to not have the same high Tc as the hole doped. There are also indications that their physics may be different than the hole doped.
The 150K superconductor is a Hg-family of hole-doped cuprates, but it gets to 150K Tc only under pressure.
Zz.
kurious
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Why does high pressure cause superconductivity at higher temperatures?
Does the pressure force electron pairs to stay together?
Does the pressure force electron pairs to stay together?
kurious
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The copper-oxygen planes that
are believed to be responsible for superconductivity are separated by
many insulating layers layers. So there might not be much overlap between
atomic orbitals between separated Cu-O planes. However, electrons can
still tunnel through the insulating layers. This tunneling is believed
to affect superconductivity, but it is not well understood how.
In type 1 superconductors there are lots of cooper pairs and there is a rapid change in conductivity.Type 2 superconductors show a gradual change from normal to super conductivity .Perhaps type 2 superconductors conduct increasingly better because electron movement through a lattice causes the formation of "groups" of atoms in the lattice which then cause further electron movements in such a way as to increase the formation of yet more groups and so on, as the temperature decreases
are believed to be responsible for superconductivity are separated by
many insulating layers layers. So there might not be much overlap between
atomic orbitals between separated Cu-O planes. However, electrons can
still tunnel through the insulating layers. This tunneling is believed
to affect superconductivity, but it is not well understood how.
In type 1 superconductors there are lots of cooper pairs and there is a rapid change in conductivity.Type 2 superconductors show a gradual change from normal to super conductivity .Perhaps type 2 superconductors conduct increasingly better because electron movement through a lattice causes the formation of "groups" of atoms in the lattice which then cause further electron movements in such a way as to increase the formation of yet more groups and so on, as the temperature decreases
nbo10
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kurious said:In type 1 superconductors there are lots of cooper pairs and there is a rapid change in conductivity.Type 2 superconductors show a gradual change from normal to super conductivity .Perhaps type 2 superconductors conduct increasingly better because electron movement through a lattice causes the formation of "groups" of atoms in the lattice which then cause further electron movements in such a way as to increase the formation of yet more groups and so on, as the temperature decreases
Carefull... Only a small portion of electrons are invloved in the pairing.
JMD
clint
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Pressure effects on Tc, and Cooper pair tunnelling
With regards to external pressure effects on Tc.
Well ,the application of external pressure (usually in the 10,000's of atmospheres) on high Tc materials is primarily believed to increase the hole concentration N in the CuO2 planes.
Specifically, Tc is believed to have a dome like structure with respect to N, or
Tc ~ Tcmax( 1 - (N - Nopt)^2 ) ,
where Tcmax is the maximum Tc for a class of materials, and Nopt is optimal hole doping, and both parameters are independent of pressure. So the goal is to reach optimal doping N=Nopt where Tc is highest by increasing N by applying pressure. Typically, Tcmax = 160K and Nopt = 0.16. So for underdoped materials (N<Nopt), pressure will increase N towards Nopt, and Tc will increase. For overdoped materials (N>Nopt), pressure will decrease Tc.
And where do the added holes come from with increasing pressure? well, the insulating layers inbetween the CuO2 planes.
Regarding tunneling of Cooper pairs between CuO2 planes.
Cooper pairs have a finite spatial extent, called the coherence length L, i.e. their spatial size. In high Tc materials, L is anisotropic, i.e. L is larger in the CuO2 planes (the ab-axis) than perpendicular to the planes (the c-axis), Lab >> Lc. Typically at 0K, Lab = 3nm, and Lc = 0.3nm.
The consequence of this anisotropy is interesting (and is absent in conventional superconductors). First it is worth knowing that high Tc materials are composed of adjacent slices of CuO2 planes (1 to 3 slices), then an insulating layer, then another set of slices of CuO2 planes etc.. YBCO for instance has 2 slices of CuO2 planes between the insulating layer. Hg and Tl-based high Tc can have up to 3 slices between insulating layers.
Now the typical Lc is much less than the thickness of the insulating layers, which basically means that the Cooper pairs remain within one set of slices of CuO2 planes, and do not tunnel through the insulating layer to an adjacent set of slices. This is what is meant by low dimensional superconductivity, i.e. 2-dimensional.
What is REALLY interesting is that the larger the "slice number", the higher the Tc ! YBCO with 2 has a Tc=90K, while Hg/Tl-based materials with 3 have Tc=150K. So, the goal would seem to be to make 4 or 5 or higher CuO2 planes adjacent to each other, thereby increasing Tc. So far, 3 is the maximum found. And, as one adds more and more CuO2 slices per unit cell, the material becomes more and more 3-dimensional; this makes sense because afterall, strictly speaking, superconductivity is a bulk 3-dimensional phenomenon.
With regards to external pressure effects on Tc.
Well ,the application of external pressure (usually in the 10,000's of atmospheres) on high Tc materials is primarily believed to increase the hole concentration N in the CuO2 planes.
Specifically, Tc is believed to have a dome like structure with respect to N, or
Tc ~ Tcmax( 1 - (N - Nopt)^2 ) ,
where Tcmax is the maximum Tc for a class of materials, and Nopt is optimal hole doping, and both parameters are independent of pressure. So the goal is to reach optimal doping N=Nopt where Tc is highest by increasing N by applying pressure. Typically, Tcmax = 160K and Nopt = 0.16. So for underdoped materials (N<Nopt), pressure will increase N towards Nopt, and Tc will increase. For overdoped materials (N>Nopt), pressure will decrease Tc.
And where do the added holes come from with increasing pressure? well, the insulating layers inbetween the CuO2 planes.
Regarding tunneling of Cooper pairs between CuO2 planes.
Cooper pairs have a finite spatial extent, called the coherence length L, i.e. their spatial size. In high Tc materials, L is anisotropic, i.e. L is larger in the CuO2 planes (the ab-axis) than perpendicular to the planes (the c-axis), Lab >> Lc. Typically at 0K, Lab = 3nm, and Lc = 0.3nm.
The consequence of this anisotropy is interesting (and is absent in conventional superconductors). First it is worth knowing that high Tc materials are composed of adjacent slices of CuO2 planes (1 to 3 slices), then an insulating layer, then another set of slices of CuO2 planes etc.. YBCO for instance has 2 slices of CuO2 planes between the insulating layer. Hg and Tl-based high Tc can have up to 3 slices between insulating layers.
Now the typical Lc is much less than the thickness of the insulating layers, which basically means that the Cooper pairs remain within one set of slices of CuO2 planes, and do not tunnel through the insulating layer to an adjacent set of slices. This is what is meant by low dimensional superconductivity, i.e. 2-dimensional.
What is REALLY interesting is that the larger the "slice number", the higher the Tc ! YBCO with 2 has a Tc=90K, while Hg/Tl-based materials with 3 have Tc=150K. So, the goal would seem to be to make 4 or 5 or higher CuO2 planes adjacent to each other, thereby increasing Tc. So far, 3 is the maximum found. And, as one adds more and more CuO2 slices per unit cell, the material becomes more and more 3-dimensional; this makes sense because afterall, strictly speaking, superconductivity is a bulk 3-dimensional phenomenon.
kurious
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CLINT:
Thanks for the answers - you've cleared up a lot of stuff the web doesn't explain clearly.
You said:
"And where do the added holes come from with increasing pressure? well, the insulating layers inbetween the CuO2 planes. "
What is the fundamental reason that pressure makes these holes appear?
Is oxygen moved into the CuO2 plane? How does electron doping help superconductivity?
Also why is it difficult to make thicker slices of CuO2?
And if a voltage is pulsed through a superconductor how does this affect the
number of cooper pairs?
Finally, could photons be injected into a superconductor to "squeeze" cooper pairs together?
Thanks for the answers - you've cleared up a lot of stuff the web doesn't explain clearly.
You said:
"And where do the added holes come from with increasing pressure? well, the insulating layers inbetween the CuO2 planes. "
What is the fundamental reason that pressure makes these holes appear?
Is oxygen moved into the CuO2 plane? How does electron doping help superconductivity?
Also why is it difficult to make thicker slices of CuO2?
And if a voltage is pulsed through a superconductor how does this affect the
number of cooper pairs?
Finally, could photons be injected into a superconductor to "squeeze" cooper pairs together?
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clint
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"What is the fundamental reason that pressure makes these holes appear?"
Actually, not surprisingly, the situation is not as straight forward as I suggested. What I was referring to was the case of CuO2 ladder systems which have recently been found to superconduct under application of high pressure. These systems are composed of CuO2 ladders (yes they actually look like ladders) and neighbouring insulating charge reservoirs. It has been suggested that applying pressure causes a transfer of holes onto the ladders from the reservoir, which then superconduct.
As for case of CuO2 planes; well pressure for one reduces the separation between existing holes by simply reducing the lattice parameter, thereby increasing the hole concentration N. It has also been suggested that there is an additional hole transfer, presumably the same mechanism as the ladders. Another possible effect of pressure is that the c-axis tunneling between sets of CuO2 planes increases. So there are at least 3 changes with pressure which generally always cause Tc to increase.
"Also why is it difficult to make thicker slices of CuO2?"
Actually 4 slice systems have been made, but their quality are somewhat dubious. It is believed that the holes don't distribute themselves evenly, or "homogeneously", amoung the 4 CuO2 planes. As a result Tc saturates, or stops increasing. This is more of a materials problem than a physics problem, and remains a challenge.
"And if a voltage is pulsed through a superconductor how does this affect the
number of cooper pairs? "
Well, if one flows a strong enough DC supercurrent in a superconductor, superconductivity can be lost, or Cooper pairs break. This is known as critical current density Jc. Typically, if one has a 1cm thick wire of say YBCO, at the lowest temperature one can flow 10^7 Amps without breakdown. This is an enormous current !
"Finally, could photons be injected into a superconductor to "squeeze" cooper pairs together?"
Superconductors have a frequency dependent conductivity. If one applies electromagnetic radiation at low temperatures, photons with frequencies less than the superconducting gap will go straight through the material. This is typically in the infrared range. So superconductors are transparent to microwaves for instance. If you apply optical waves, then they break the Cooper pairs and one just gets back the normal state properties.
Actually, not surprisingly, the situation is not as straight forward as I suggested. What I was referring to was the case of CuO2 ladder systems which have recently been found to superconduct under application of high pressure. These systems are composed of CuO2 ladders (yes they actually look like ladders) and neighbouring insulating charge reservoirs. It has been suggested that applying pressure causes a transfer of holes onto the ladders from the reservoir, which then superconduct.
As for case of CuO2 planes; well pressure for one reduces the separation between existing holes by simply reducing the lattice parameter, thereby increasing the hole concentration N. It has also been suggested that there is an additional hole transfer, presumably the same mechanism as the ladders. Another possible effect of pressure is that the c-axis tunneling between sets of CuO2 planes increases. So there are at least 3 changes with pressure which generally always cause Tc to increase.
"Also why is it difficult to make thicker slices of CuO2?"
Actually 4 slice systems have been made, but their quality are somewhat dubious. It is believed that the holes don't distribute themselves evenly, or "homogeneously", amoung the 4 CuO2 planes. As a result Tc saturates, or stops increasing. This is more of a materials problem than a physics problem, and remains a challenge.
"And if a voltage is pulsed through a superconductor how does this affect the
number of cooper pairs? "
Well, if one flows a strong enough DC supercurrent in a superconductor, superconductivity can be lost, or Cooper pairs break. This is known as critical current density Jc. Typically, if one has a 1cm thick wire of say YBCO, at the lowest temperature one can flow 10^7 Amps without breakdown. This is an enormous current !
"Finally, could photons be injected into a superconductor to "squeeze" cooper pairs together?"
Superconductors have a frequency dependent conductivity. If one applies electromagnetic radiation at low temperatures, photons with frequencies less than the superconducting gap will go straight through the material. This is typically in the infrared range. So superconductors are transparent to microwaves for instance. If you apply optical waves, then they break the Cooper pairs and one just gets back the normal state properties.
kurious
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If you increase the pressure on a liquid it stays as a liquid at a higher temperature.
Perhaps something similar applies to high temperature superconductors - under pressure they would still be in a superconducting phase at a higher temperature than normal.
If I rotated a superconducting disc this would force the atoms on the disc further apart.Would it stop superconducting vortices from being destroyed (as regular conducting regions get closer together) until a higher temperature is reached.
Or are superconductors generally too brittle to be made into a disc that could rotate quickly enough?
Fermi surfaces are used in superconductor theory.An electron that has a higher energy than the minimum can move onto a fermi surface.
Do cooper pairs move onto fermi surfaces or only unpaired conduction electrons?
Perhaps something similar applies to high temperature superconductors - under pressure they would still be in a superconducting phase at a higher temperature than normal.
If I rotated a superconducting disc this would force the atoms on the disc further apart.Would it stop superconducting vortices from being destroyed (as regular conducting regions get closer together) until a higher temperature is reached.
Or are superconductors generally too brittle to be made into a disc that could rotate quickly enough?
Fermi surfaces are used in superconductor theory.An electron that has a higher energy than the minimum can move onto a fermi surface.
Do cooper pairs move onto fermi surfaces or only unpaired conduction electrons?
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clint
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" Fermi surfaces are used in superconductor theory.An electron that has a higher energy than the minimum can move onto a fermi surface.
Do cooper pairs move onto fermi surfaces or only unpaired conduction electrons?"
Three necessary ingredients to Cooper paring are; a well defined Fermi surface of electrons, some finite attractive interaction (independent of the mechanism eg, phonons, magnons, etc.), and the Coulomb interaction. In Cooper theory the pairs are formed by electrons of opposite momentum (opposite sides of the Fermi surface), and opposite spin (singlet). Cooper first started by considering pairs in the vicinity of Ef (~10,000's K), which in conventional superconductors is given by the Debye interaction ~100's K. Then BCS theory included all the conduction electrons in the paired ground state.
Do cooper pairs move onto fermi surfaces or only unpaired conduction electrons?"
Three necessary ingredients to Cooper paring are; a well defined Fermi surface of electrons, some finite attractive interaction (independent of the mechanism eg, phonons, magnons, etc.), and the Coulomb interaction. In Cooper theory the pairs are formed by electrons of opposite momentum (opposite sides of the Fermi surface), and opposite spin (singlet). Cooper first started by considering pairs in the vicinity of Ef (~10,000's K), which in conventional superconductors is given by the Debye interaction ~100's K. Then BCS theory included all the conduction electrons in the paired ground state.
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