# D-wave superconductivity: Functional forms?

csmallw
Two questions, really:

I’m finding it hard to wrap my head around the connections between k-space and real-space for d-wave 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 d-wave pairing in the Cuprate Superconductors” by Doug Scalapino (1995), and am confused by Appendix B, which--at least from the tone of the appendix--should be pretty universally-agreed-upon stuff. The purpose of the appendix is to precisely explain what is meant by d-wave pairing. Scalapino writes at one point that:

“Physically, in a superconductor, the quasi-particles interact with the pair condensate so that the gap Δk in the quasi-particle spectrum is related to ψk [a notation for k-space amplitudes of the orbital wave function, which was earlier on described in real space]. The BCS theory tells us that

ψk = Δk / Ek.”

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 Δk that can be quantitatively written down as a “pure” d-wave, i.e., as

Δk = Δ0 [Cos(kx) – Cos(ky)]?

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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/Classi...ns from BCS, Sov-Phys JETP 9, 1364 (1959).pdf

Staff Emeritus
Also, can anyone explain why, out of all the various quantities related to orbital symmetry, it is Δk that can be quantitatively written down as a “pure” d-wave, i.e., as

Δk = Δ0 [Cos(kx) – Cos(ky)]?

I'm a bit unclear what you are asking here. Are you asking why, out of all the available symmetries, that we pick the $d_{x^2-y^2}$ symmetry here?

Zz.

csmallw
Thanks. As I understand it, Gorkov writes (see Eq. 15) that

ψGL(r) = Δ(r) * constant.

In the Scalapino reference, he writes that the "relative coordinate wave function ψ(x1-x2) describes the orbital symmetry of the pairs" and is given by

ψ(x1-x2) = ∑k ψk eik(x1-x2),

with ψk defined as in my earlier post. So I have two orbital "wave functions," ψ and ψGL, 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 "x1-x2" are fundamentally different quantities.

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csmallw
Are you asking why, out of all the available symmetries, that we pick the dx2−y2 symmetry here?

ZapperZ, sorry, I'm still struggling for the right way to word this. I guess what I'm trying to ask is this: Let's assume that we have a dx2-y2 superconductor in a square-lattice crystal structure. (I'm imagining a cuprate superconductor, but it might not necessarily have to be the case).

What I want to know is, why is it that Δk = Δ0[cos(kx)-cos(ky)], and not, for example, Δk = Δ0 (kx2 - ky2)?

And why is it that Δk = Δ0[cos(kx)-cos(ky)] and not ψ(x1-x2) = Δ0[cos(x)-cos(y)]?

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.

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csmallw
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.

Good point. I could be a little more creative with functional forms and still get around this, however. For example, Δk would still be smooth and continuous at the Brillouin zone boundary if it had the following form, wouldn't it?:

Δk = EkΔ0[cos(kx)-cos(ky)], with Ek=√[Δ022(k)]

Gold Member
It is perhaps worth remembering that the reason for why we "pick" the d-wave 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 p-wave can be elininated due to Knigth-shift measurements) but there is still not universally accepted that e.g. YBCO is a "pure" d-wave superconductor: there are experiment suggesting that it might be d+is or d+s; i.e. with a small s-wave admixture
Hence, there is no microscopic theory that can tell you why it should be d-wave.

csmallw
According to theory (i.e. symmetry analysis) several possible forms are possible based on the underlying crystal symmetry of the cuprates

Can you refer me to any sources where this symmetry analysis is discussed at a "textbook" level? Surely somewhere out there, there must exist a source that can explain why the d-wave form of the gap Δk should be given by

$$\Delta_k = \Delta_0 \left( \cos k_x - \cos k_y \right)$$

as opposed to some other function, what the above functional form might have to do with nearest-neighbor hopping, and why it is more natually expressed in terms of k-space rather than real space, don't you think?