Disorder in d-wave superconductors

I have been interested for many years in heavy fermion and, more recently, high temperature cuprate superconductivity. The heavy fermion materials are metals involving rare earth or actinide ions in which electrons behave as though they have masses much larger than their bare mass, sometimes as much as a proton mass. Transition temperatures are only about 1K, but the systems exhibit rather complex phase diagrams suggesting that the superconducting order is more complex than the old-fashioned superconductors. In fact there is strong evidence that superconductivity is unconventional, in the sense that the superconducting order parameter or pair wave function has symmetry less than the underlying crystal lattice. In very few cases is there a consensus on what the correct symmetry class is, however. The cuprate materials, with Tc's of order 100K or above, typically have a layered perovskite structure, and superconductivity seems to be nearly 2D. Here's a recent New York Times assessment of their technological potential. They are fascinating materials in part because in the "normal" state above the critical temperature they exhibit properties which deviate strongly from Landau's Fermi liquid state, the paradigm which seems to work well in almost all previously discovered classes of metals. It is now established that the cuprate materials have unconventional d-wave symmetry, meaning the wave function of Cooper pairs has orbital angular momentum L=2. Here is a recent review why we think so. The picture on the right shows that the d-wave gap goes to zero on the Fermi surface at 4 nodes, where low-energy excitations are possible. Note "+" and "-" means sign of order parameter for these directions of k. All materials contain defects, and d-wave superconductivity is particularly sensitive to disorder. Simple nonmagnetic defects can break pairs in a d-wave superconductor, in contrast to a conventional superconductor. In addition, the physics of disordered interacting electrons is a fundamental unsolved problem in condensed matter physics. This is why we study this problem, despite (or perhaps because of) Pauli's remark (right).

d-wave superconducting gap. Wolfgang Pauli: "Solid state physics is `dirty physics'"

 Recent projects

• Fourier power spectrum of many impurities in a d-wave superconductor and STM experiments

  FTDOS at w=14 meV for weak potential scatters (V0=0.67t1): (a) for one weak impurity, with a few important scattering wavevectors indicated; (b) for 0.15% weak scatterers. Cuts through the data of (a)(thick line) and (b)(thin line) along the (110) direction and scaled by 1/sqrt{N_I} are plotted vs. q_x in (c), while (d) shows the weak scattering response function.

  Recently scanning tunneling microscopy (STM) measurements of impurity states on superconducting surfaces have opened a new window on high temperature superconductivity. In principle, STM probes the local density of states (LDOS) on the surface of a material, and among the cuprates BSCCO-2212 is usually studied since one can cleave very clean surfaces. One way to analyze the LDOS data (Hoffman et al. Science 295, 466 (2002), Howald et al cond-mat 2002) has been to Fourier transform it to try to select out the important wavelengths of the Friedel oscillations driven by the disorder potential. As in a simple metal, these wavelengths tell you in principle about the Fermi surface properties of the pure material. The Davis group has used this technique to try to determine the Fermi surface and superconducting gap momentum space structure using STM! Exactly what kind of information is present in the quasiparticle interference patterns observed is the subject of our recent paper (cond-mat/0307288). We pointed out the importance of including the strong "native defect" scatterers in the analysis, and raise the possibility that the peaks observed in experiment do not, in fact, correspond necessarily to q-vectors connecting the tips of contours of constant quasiparticle energy as suggested by the experimental group and by 1-impurity analyses (see right).

  FTDOS data at 14meV from McElroy et al. on a BSCCO-2212 surface. Only 1st Brillouin zone is shown, x,y axes are Cu-O bond directions. Scattering vectors which maximize joint density of states connect the tips of the "bananas" which are contours of constant quasiparticle energy.

• Two impurities in a d-wave superconductor and STM experiments

  Recently STM has been able to image impurity bound states on superconducting surfaces. For the most part it has been assumed that these states could be treated theoretically within a one-impurity model, that is, the interference of the wave functions around the different impurities has been neglected. This would seem to be reasonable in dilute systems, but the d-wave superconductor is peculiar in that the impurity states have long-range "tails" in the direction of the gap nodes. We studied the simple "molecular" quantum mechanics problem of two potential scatterers in the presence of a d-wave host, and found that signatures of quantum interference were present for pairs of impurities with (110) orientations even 30 unit cells away. (cond-mat/0208008, to be published in Phys. Rev. B).

  Bound state wave functions and local dos for 2 strong impurities with separation (6,6).

• Quantum interference with many impurities: a real space perspective

  Local density of states of 2% random potential scatterers of infinite strength for tight-binding band at half-filling. Impurity sites on A sublattice are black circles, those on B are white circles.

  One mystery we would like to understand is why STM experiments seem to measure similar impurity resonances around different impurities in different disorder environments. The two-impurity considerations, taken naively, might tell us that different impurities might "light up" (become resonant) at different energies, and spatial patterns might be distorted by interference. At first glance, this is borne out by studies of a model with infinitely strong scatterers and a half-filled tight-binding band (left). Note impurities on one sublattice are not resonant at all! However study of a generic model without the special sublattice symmetries shows that the STM measurement averages over a range of states and the 4-fold symmetry of all individual impurity sites is recovered, with the width of the resonance given by the impurity bandwidth. (cond-mat/0301630). We are developing numerical techniques to make direct predictions of real-space and Fourier transform spectra for a variety of one-impurity potentials.

• Density of states in disordered 2D d-wave superconductor

  SCTMA: Density of states Exact DOS (but no order parameter supression): Exact amplitude of d-wave order parameter around strongly scattering impurities.

  In a d wave superconductor, there is no Anderson theorem for nonmagnetic impurities: dirt breaks Cooper pairs . The simplest theory of disorder in unconventional systems includes multiple scattering processes from a single impurity site, but then uses a self energy which neglects interference processes which arise when electrons scatter from different sites. This is called the self-consistent T-matrix approximation (SCTMA), and it predicts for example that the density of states (DOS) should be finite at zero energy, a result which appears to be confirmed by experiment. In 2D, this perturbative approach breaks down, and a variety of nonperturbative approaches have made wildly different predictions for the DOS and other properties. Using exact solutions of the Bogoliubov-de Gennes equations (mean field theory), Bill Atkinson, Allan MacDonald, Klaus Ziegler and I have been able to reconcile many of these results. We have shown a) that the "details" of the disorder distribution and particle-hole symmetry of the normal state band are crucial for the low-energy behavior ( cond-mat/0005487 , Phys. Rev. Lett. 85, 3926 (2000)); and b) that the self-consistently determined supression of the order parameter around the impurity site is crucial, leading to a strong pseudogap behavior even for strong disorder ( cond-mat/0002333 , Phys. Rev. Lett. 85, 3922 (2000)). Figures: Top: SCTMA scenario. Middle: breakdown of Pepin-Lee prediction of divergence in DOS for small deviations from unitarity limit scattering. Using weak localization arguments, we were able to show that a "Pi-diffusion mode" is responsible for this effect and at what scales it breaks down (cond-mat/0102310, Phys. Rev. Lett. 86,5982 (2001).) A review of recent developments is given in cond-mat/0108487 (J. Low Temp Phys. 126, 881 (2002)), and some recent work on transport properties in cond-mat 0108519 (Phys. Rev. Lett. 88, 187003 (2002)). Bottom: exact solution of the Bogoliubov-de Gennes equations for the amplitude of the d-wave order parameter in the presence of strong potential scatterers. Note the supression nearly to zero over a range of order the unit cell size.

• Transport properties

  There are several discrepancies between experiment and the SCTMA picture of low-temperature transport in the cuprates. One includes a finite-frequency peak in the low temperature conductivity of all disordered cuprates, see left. Bogoliubov-de Gennes calculations with W. Atkinson (Phys. Rev. Lett. 88, 187003 (2002)) showed this can be understood in terms of weak localization effects. The low-temperature limit of the microwave conductivity is found to vary as T, not T2, the prediction of the SCTMA theory. The BdG calculations recover the T behavior when self-consistency, i.e. the supression of the order parameter around impurity sites, is included.

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