BCS Theory Questions - Seeking Explanation & Resources

[leoant]

Hello Everyone here:
I am reading the very paper of BCS theory by J.Bardeen, L.N.Cooper and J.R.Schrieffer (Phys.Rev.108, 1175,1957). But I find so many things hard to understand for me, though I have finished my Quantum Mechanics and Solid State Theory, especially a paragraph list below:
"Excited states are treated in much the same way as the ground state. One must distinguish between singly excited particles,in which one and only one of a pair (k,-k)is occupied, and excited or "real" pair states. We treat singly excited particles in the Bloch scheme, as in the normal metal. They contribute a negligible amount to the interaction energy directly, but reduce the amount of phase space available for real and virtual pairs..."
Here is my question:
1 what is the exact singly excited particles? and excited pair states? and "real" pair states? and virtual pairs (in the next paragraph of above in the very paper)?
2 what's Bloch scheme mean?
btw: this is just a small part of the original paper, I am afraid there would be mountains of questions contained in the paper, thus would some one be kind to refer me to some courses which explicit the BCS Theory specificly? And is there any schematic explanation of the theory( pls give me some step by step illustration on the paper or refer me to some close related papers or books which would make me feel free to read the theory)?

Thanks in advance.
My email: leoant21@hotmail.com

 

[ZapperZ]

Hello Everyone here:
I am reading the very paper of BCS theory by J.Bardeen, L.N.Cooper and J.R.Schrieffer (Phys.Rev.108, 1175,1957). But I find so many things hard to understand for me, though I have finished my Quantum Mechanics and Solid State Theory, especially a paragraph list below:
"Excited states are treated in much the same way as the ground state. One must distinguish between singly excited particles,in which one and only one of a pair (k,-k)is occupied, and excited or "real" pair states. We treat singly excited particles in the Bloch scheme, as in the normal metal. They contribute a negligible amount to the interaction energy directly, but reduce the amount of phase space available for real and virtual pairs..."
Here is my question:
1 what is the exact singly excited particles? and excited pair states? and "real" pair states? and virtual pairs (in the next paragraph of above in the very paper)?
2 what's Bloch scheme mean?
btw: this is just a small part of the original paper, I am afraid there would be mountains of questions contained in the paper, thus would some one be kind to refer me to some courses which explicit the BCS Theory specificly? And is there any schematic explanation of the theory( pls give me some step by step illustration on the paper or refer me to some close related papers or books which would make me feel free to read the theory)?

Thanks in advance.
My email: leoant21@hotmail.com

Boy, does this sounds familiar! :)

"Singly-excited state" as in the BCS paper implies that only one of the electron in a cooper pair occupies the state available for the pair - meaning it is in the right state, but no partner. This is to be distinguised with the "real" excited state that has the "complete" pair.

While it is good to really study this paper, if you really want to study the BCS theory, I recommend, for example, Micheal Tinkham's book on Superconductivity. Not only does he explains the derivation carefully, he also does it in two different ways! Only after you understood that is it advisable for you to go back to the original BCS paper and restudy it. You'll find that it will be A LOT clearer what they mean.

Zz.

 

[leoant]

Boy, does this sounds familiar! :)

I recommend, for example, Micheal Tinkham's book on Superconductivity. Not only does he explains the derivation carefully, he also does it in two different ways!
Zz.

Thank you very much! However, I don't think there's the book in our library thus I want to buy it. Then comes the problem:
Maybe there are two edition of the very book which were published in 1975 and 1996, respectively, which one should I get?
And is the title of the book exactly "Introduction to superconductivity", or "Introduction to the theory of superconductivity"(b/c I want someone else to buy it for me, thus I want to make it more clear)

 

[ZapperZ]

Thank you very much! However, I don't think there's the book in our library thus I want to buy it. Then comes the problem:
Maybe there are two edition of the very book which were published in 1975 and 1996, respectively, which one should I get?
And is the title of the book exactly "Introduction to superconductivity", or "Introduction to the theory of superconductivity"(b/c I want someone else to buy it for me, thus I want to make it more clear)

Just buy the latest edition - in fact Amazon is listing a Dover edition, which should be dirt cheap.

Zz.

 

[leoant]

Boy, does this sounds familiar! :)

"Singly-excited state" as in the BCS paper implies that only one of the electron in a cooper pair occupies the state available for the pair - meaning it is in the right state, but no partner. This is to be distinguised with the "real" excited state that has the "complete" pair.

Zz.

Here I list my interpretation and may you be kind to tell me they are right or wrong:
Ground State of superconductivity--all electrons are in cooper pair states, which let the total energy be lower with respect to "free electrons"-which distribute in Fermi-Dirac method. However,here I get a question: I think that cooper pair are not real pair, but a pair of state whose interaction is attractive, comparing to independent electrons. Am I right?
Exited State--Suppose my understandin of ground state is true, then I think excited state may be some electrons get out of the restraint of cooper pair and behave like independent particles. Thus, Single-excited state means that it happens a electron in k state without another in -k, and this electron is a free electron. Then, real and complete pair is nothing but two states(k,-k), which are both occupied by free electrons, but they are not in cooper pair state. At last, "virtual pairs" are those possible state that electrons may be excited to from cooper pair state, but there're nothing--without electrons occupied but when cooper pairs are excited, electrons may occupy those states.

Thanks in advance. Pls tell me more.

 

[ZapperZ]

Here I list my interpretation and may you be kind to tell me they are right or wrong:
Ground State of superconductivity--all electrons are in cooper pair states, which let the total energy be lower with respect to "free electrons"-which distribute in Fermi-Dirac method. However,here I get a question: I think that cooper pair are not real pair, but a pair of state whose interaction is attractive, comparing to independent electrons. Am I right?
Exited State--Suppose my understandin of ground state is true, then I think excited state may be some electrons get out of the restraint of cooper pair and behave like independent particles. Thus, Single-excited state means that it happens a electron in k state without another in -k, and this electron is a free electron. Then, real and complete pair is nothing but two states(k,-k), which are both occupied by free electrons, but they are not in cooper pair state. At last, "virtual pairs" are those possible state that electrons may be excited to from cooper pair state, but there're nothing--without electrons occupied but when cooper pairs are excited, electrons may occupy those states.

Thanks in advance. Pls tell me more.

Here is one of the major difficulties in interpreting what is being said in the BCs theory:

When we look at the conduction electrons in the Fermi sea, what we are using is the single-particle description. The came out of the Fermi Liquid theory that allows us to remove all the electron-electron interactions and renormalize the electrons as "quasiparticles". But the single-electron picture remains, and the bands and energy gap, etc. that you get are based on the single-electron description.

However, once you have coupling, such as the formation of cooper pairs, then to describe such pairs, one can no longer use a single-electron picture anymore. The cooper pairs requires 2-particle description. This is definitely more complicated - for example, the Green's function for the 2-particle channel is a nightmare.

What saves us in this case is that we can still use the single-particle description (i.e. the single-particle Green's function) by "imagining" the excitation of ONE of the electrons in the cooper pair. For example, the energy gap that we observe in the density of states is the energy state of a single electron to break away from its partner. So there are still some usefulness of the single-particle picture.

Now why am I wasting your time here in not answering your question directly? It is because the ground state wavefunction that is incorporated in the BCS theory is the single particle wavefunction. Thus, all the so-called "excitation" are single-particle excitation. This may not explain or answer your question, but it should give you a "frame of reference" when you read the BCS paper. Again, it isn't easy to understand and I recommend studying the BCS theory from another source first before going back to this paper. At least, that was what I did.

Zz.