Victor Galitski: Mesoscopic disorder fluctuations in a d-wave superconductor

Thursday, Aug 23th

Victor Galitski started our Patio Discussion by returning to the recent STM experiments by Ali Yazdani showing an inhomogeneous spatial gap distribution above the superconducting transition temperature Tc in the cuprates. Taking these as motivation for his today's presentation, he first pointed out that the important features seen by Ali Yazdani are that the experimental gapmaps are static and reproducible when varying temperature. In particular this means no phase separation takes place.

Victor went on by stressing the break-down of Anderson's theorem in d-wave superconductors in the presence of disorder potentials, leading to a dependence of Tc on the disorder. As the density of impurities is random, there are fluctuations in real space. These are according to Victor associated with a local Tc larger than the Tc for a corresponding homogeneous state. A picture of paddles of superconductivity within a normal background emerges, where each of the paddles have their private Tc. Victor now continued with an overview over what is known from s-wave superconductors, in which case Tc does not depend on disorder in leading order in accordance to Anderson's theorem. In this case fluctuations are not important.
Victor proceeded by reminding us that in s-wave superconductors with magnetic impurities there is an Abrikosov-Gorkov formula
ln (Tc0/Tc) = \Psi(1/2 + \Gamma/[2\pi Tc]) - \Psi(1/2)
that determines the actual Tc in terms of the critical temperature for a system without disorder, Tc0. The crucial parameter in this formula is the pair breaking parameter Gamma. In a magnetic field and in the diffusive limit it is proportional to D*H, where D is the diffusion constant and H the magnetic field. This leads to the well known Hc2(T) curve. Victor draw our attention to the fact that for s-wave superconductors this theoretical curve is smooth at low temperatures, whereas experimentally often an upturn of the Hc2-curve is observed. A possible explanation would then be that Tc depends on disorder via the diffusion constant D, and thus Hc2(0)~n_imp. Dan Sheehy asked the question what happens for n_imp=0, and Victor stressed that he restricts his discussions to the dirty limit, so that Tc0 \tau <<>

Next Victor draw a picture of superconducting islands connected by the Josephson effect and mentioned the works about Josephson networks by Spivak/Zhou PRL '95 and by Larkin/Galitski PRL 2002. At this point a specific model in terms of a Ginzburg-Landau action followed, in which spatial fluctuations of the order parameter where taken into account.

Victor mentioned in passing that in cuprates in principle Tc0 depends on doping, such that Tc is determined by an interplay between the intrinsic x-dependence of Tc0 and the induced one by the spatial disorder. This leads to a superconducting dome resembling very roughly that of the cuprates.

The spatial randomness of the gaps introduces via the eigenvalue equation

(1/v) \int C(r,r') \Delta(r') = (Tc/Tc0) \Delta(r)

also a random Tc. The random operator C(r,r') is the Cooperon. The statistics of C(r,r') can be expressed diagrammatically, and leads to a distribution of Tc's as function of coherence length, mean free path and (Tc-Tc0)/Tc0. Victor finished his talk with developing a picture of underdoped cuprates in terms of superconducting islands separated by normal regions, however with a fluctuation gap. This also implies a reduced local density of states in the normal regions.

In the discussion part, Phil Anderson commented that all this does not seem to be related to high-Tc cuprates, but to d-wave BCS superconductors. The nature of the phase transition in cuprates is that of an x-y model, where Tc~\rho_s, not ~\Delta. Thus, fluctutating gaps are not related to fluctuation Tc's. Victor basically agreed and mentioned that he studied a BCS model, not an x-y model. Andrei Chubukov commented that Tc in the calculations should be related to the pseudogap temperature T*.

Dirk Morr asked how the distribution of local Tc's is related to the global Tc. Victor answered that the distribution of Tc's is related to disorder, but that there were no direct relation to a global Tc. Claudio Castellani commented at this point that he thinks Tc as a local quantity is only a technical parameter of the BCS model, any real Tc has to be global. Victor disagreed in the sense that if the puddles are in size larger that a coherence volume, it makes sense to talk about a local Tc for each puddle.