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Dwave superconductivity: Functional forms? 
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#1
Mar3114, 12:09 PM

P: 18

Two questions, really:
I’m finding it hard to wrap my head around the connections between kspace and realspace for dwave symmetry, as well as the connections between “order parameter,” “gap,” “Cooper pair wave function,” and “superconducting wavefunction,” which are all mentioned at various points in some of the other threads I have been looking through. Right now I’m reading the article “The Case for dwave pairing in the Cuprate Superconductors” by Doug Scalapino (1995), and am confused by Appendix B, whichat least from the tone of the appendixshould be pretty universallyagreedupon stuff. The purpose of the appendix is to precisely explain what is meant by dwave pairing. Scalapino writes at one point that: “Physically, in a superconductor, the quasiparticles interact with the pair condensate so that the gap Δ_{k} in the quasiparticle spectrum is related to ψ_{k} [a notation for kspace amplitudes of the orbital wave function, which was earlier on described in real space]. The BCS theory tells us that ψ_{k} = Δ_{k} / E_{k}.” The same equality appears in other places, for example, Tsuei and Kirtley, RMP 72, 969 (2000), but so far, I haven't felt like I have gotten a sufficiently satisfactory explanation of its origin. Does anyone know where this equation comes from, how I might derive it, or better yet, how I might understand it intuitively? I have not (yet) been able to locate in the original BCS paper, or in Michael Tinkham’s book “Introduction to Superconductivity.” Also, can anyone explain why, out of all the various quantities related to orbital symmetry, it is Δ_{k} that can be quantitatively written down as a “pure” dwave, i.e., as Δ_{k} = Δ_{0} [Cos(k_{x}) – Cos(k_{y})]? Thanks in advance. 


#2
Mar3114, 01:59 PM

Sci Advisor
P: 3,555

Tinkhams book is quite centered on phenomenology, and not so much upon microscopic description. I think you should get accquainted to Ginzburg Landau theory. A derivation of the Ginzburg Landau equation from microscopic theory was given first by Gorkov and I think it is still one of the best papers ever written on superconductivity:
http://www.w2agz.com/Library/Classic...4%20(1959).pdf 


#3
Mar3114, 02:43 PM

Emeritus
Sci Advisor
PF Gold
P: 29,239

Zz. 


#4
Mar3114, 02:57 PM

P: 18

Dwave superconductivity: Functional forms?
Thanks. As I understand it, Gorkov writes (see Eq. 15) that
ψ_{GL}(r) = Δ(r) * constant. In the Scalapino reference, he writes that the "relative coordinate wave function ψ(x_{1}x_{2}) describes the orbital symmetry of the pairs" and is given by ψ(x_{1}x_{2}) = ∑_{k} ψ_{k} e^{ik(x1x2)}, with ψ_{k} defined as in my earlier post. So I have two orbital "wave functions," ψ and ψ_{GL}, expressed in real space, which clearly seem not to be equal to each other. How do I reconcile this? Edit: Ah, I realize after writing this that "r" and "x_{1}x_{2}" are fundamentally different quantities. 


#5
Mar3114, 03:55 PM

P: 18

What I want to know is, why is it that Δ_{k} = Δ_{0}[cos(k_{x})cos(k_{y})], and not, for example, Δ_{k} = Δ_{0} (k_{x}^{2}  k_{y}^{2})? And why is it that Δ_{k} = Δ_{0}[cos(k_{x})cos(k_{y})] and not ψ(x_{1}x_{2}) = Δ_{0}[cos(x)cos(y)]? 


#6
Apr114, 03:05 AM

Sci Advisor
P: 3,555

I suppose k_x and k_y are measured in units of the reciprocal basis vectors, so that k_x=2π at the edge of the Brillouin zone. I.e. the wavefunctions have to be compatible with the symmetry of the lattice. Something like the Bloch theorem.



#7
Apr114, 08:20 PM

P: 18

Δ_{k} = E_{k}Δ_{0}[cos(kx)cos(ky)], with E_{k}=√[Δ_{0}^{2}+ε^{2}(k)] 


#8
Apr314, 04:50 AM

Sci Advisor
PF Gold
P: 2,241

It is perhaps worth remembering that the reason for why we "pick" the dwave symmetry is because of experimental evidence. According to theory (i.e. symmetry analysis) several possible forms are possible based on the underlying crystal symmetry of the cuprates; most of them can quickly be eliminated using fairly straightforward measurement (e.g pwave can be elininated due to Knigthshift measurements) but there is still not universally accepted that e.g. YBCO is a "pure" dwave superconductor: there are experiment suggesting that it might be d+is or d+s; i.e. with a small swave admixture
Hence, there is no microscopic theory that can tell you why it should be dwave. 


#9
Apr1714, 02:14 PM

P: 18

[tex]\Delta_k = \Delta_0 \left( \cos k_x  \cos k_y \right)[/tex] as opposed to some other function, what the above functional form might have to do with nearestneighbor hopping, and why it is more natually expressed in terms of kspace rather than real space, don't you think? 


#10
Apr2314, 05:52 AM

Sci Advisor
PF Gold
P: 2,241

I have no specific reference (the last time I read about this was when I was writing my thesis, some ten years ago). However, if you look up one of the standard references on this (e.g. Tsuei and Kirtley's Rev. Mod. Physics paper from 2000) you will find a discussion (and more references) about this.
Also, the "real vs. kspace" question is a bit odd; things like this (electron states etc) are ALWAYS discussed in terms of kspace in condensed matter physics (in the B.Z.), I am not even sure how you would think about it in real space. 


#11
Apr2414, 10:38 AM

P: 18

Thanks. Funny you should mention writing your thesis...I'm getting ready to do that myself in a month or two, which is why I am interested in getting a better handle on these concepts ;)



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