Disorder in unconventional superconductors

d-wave superconducting gap Wolfgang Pauli: "Solid state physics is `dirty physics'" magnetic droplet induced around impurity

2D gap map of the magnitude of the tunneling gap on the surface of Bi-2212, a high-Tc cuprate superconductor. The red dots represent O dopant atoms which are also imaged. I have been interested for many years in unconventional superconductivity in e.g., heavy fermion, cuprate, and Fe-based materials. 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  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 review  of 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. The Fe-based superconductors were discovered in 2008 and have caused something of a stir in the condensed matter community because they are high-temperature superconductors without Cu or -- apparently -- extremely strong electronic correlations. The comparisons with the cuprates will certainly prove useful. 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 (left). In recent years, I have been interested in the role of residual interactions in the superconducting state and their interplay with disorder.  In particular, many unconventional superconducting materials have strong antiferromagnetic correlations which give rise to magnetic droplets  around impurities of size equal to the antiferromagnetic correlation length.  When these droplets overlap, long range order can be created.  Many of these phenomena were reviewed in a paper I wrote with the Orsay group of Henri Alloul, Julien Bobroff, and Marc Gabay. A picture of a magnetic droplet is shown on the left.

 Recent projects

• Disorder induced magnetism

  There is much evidence to suggest that the cuprate high temperature superconductors are unusual because of their proximity to a magnetic state.  In the past several years, several kinds of experimental probes have reported seeing long-range antiferromagnetic order in the superconducting state at low temperatures in the material La1-xSrxCuO4.   One well-known neutron study [1] which observed field observed order also noted a residual staggered magnetic signal in an underdoped sample in zero field.  This effect does not occur in the cleaner cuprate materials like YBa2Cu3O7.  We proposed that each impurity induces a small magnetic droplet around itself, and that quasi-long range magnetic order occurs when these bubbles overlap.  To test this hypothesis, we performed simulations of a d-wave superconductor in the presence of local Coulomb interactions and nonmagnetic disorder. This problem was studied by solving the Bogoliubov-de Gennes equations for a d-wave superconductor including a Hartree treatment of the Hubbard interaction, known to produce a good description of impurity-induced magnetism in NMR experiments in Zn-substituted YBCO[2].   We modeled dopant Sr impurities in LSCO with weak potentials.  A schematic of results is shown in Fig. 1a.  Magnetic droplets created around individual nonmagnetic impurities in the correlated system interact via an effective collective exchange, which may be understood by the need to maintain Neel coherence among the different droplets.  In 1b, a typical magnetization map of the disordered superconductor is shown for U=3.2t, and in 1c) the Fourier transform of m(r) yields the structure factor S(q), compared to the same quantity measured in the Lake et al. experiment (1d). Figure 1.  The effect of such static magnetism on scattering by quasiparticles was also studied[3], where it was found that universal transport coefficients such as that predicted for thermal conductivity are strongly suppressed by magnetic scattering of this type.  This explains several recent experiments on underdoped cuprates where strong deviations from the expected universal value were seen, and suggests that the conclusion of large gap slopes near the d-wave node deduced from these results is incorrect. This work strongly suggests that the physics of underdoped, intrinsically doped cuprates is dominated by magnetic fluctuations pinned by dopant disorder.  Such tendencies are strongly enhanced in both the superconducting and pseudogap states due to incipient bound state formation around nonmagnetic impurities.  References [1]   B. Lake et al., Nature (London) {\bf 415}, 299 (2002). [2]  B.M. Andersen et al. Phys. Rev. Lett. 99, 147002 (2007). [3]  B.M. Andersen and P.J. Hirschfeld, arXiv:0711.2294v2.  The conventional picture of this phenomenon assumes that antiferromagnetism (AF) and superconductivity (SC) are competing orders, and that suppressing SC automatically induces SC.  More recently, we showed instead  that the formation of the magnetic state is concomitant with the splitting of an Andreev bound state around both impurities and vortex cores, and is thus characteristic of d-wave superconductivity only. Fig. 2 shows the basis for our argument that the magnetic state induced by strong spatial perturbations in the presence of AF correlations[1] is directly associated with an Andreev bound state in the d-wave superconductor.   Figure 2: a) LDOS on nn site of impurity with (U=2.2t) and without (U=0) correlations; b)induced magnetization by impurity, U=2.2t c) LDOS in vortex core with and without correlations; d) induced magnetization in vortices, U=2.2t; e) near p,p)  magnetic order vs. T(K) with and without applied magnetic field in 7.5 \% doped LSCO [2]; f) Simulations of T dependence of disordered magnetic state with and without two flux quantum field in 40x40 system. This work strongly suggests that the physics of underdoped, intrinsically doped cuprates is dominated droplets induced by disorder; a magnetic field can induce the same state.   The origin of this state is peculiar to the d-wave ground state of the cuprate superconductors, which allows Andreev bound states near strong local perturbations.   Such tendencies are strongly enhanced in both the superconducting and pseudogap states due to incipient bound state formation around nonmagnetic impurities and in vortex cores.. This magnetic order scatters low-energy quasiparticles strongly, and leads to a breakdown of universal transport predicted for d-wave systems in the presence of ordinary disorder[3], and to upturns in the resistivity in the normal state[4] References [1]  H. Alloul et al., Rev. Mod. Phys. 81, 45 (2009).     [2]   B. Lake et al, Nature  415, 299 (2002). [3]   B.M. Andersen and P.J. Hirschfeld,    Phys. Rev. Lett. 100, 257003 (2008). [4]   W. Chen, B.M. Andersen and P.J. Hirschfeld, Phys. Rev. B 80, 134518 (2009) The low-temperature ``universal” thermal conductivity in cuprates has been extremely important to theoretical approaches to the superconducting state.  For example the slope of the d-wave gap near the node v2 is often extracted from it using the relation k0~vF/v2 but recent measurements on underdoped samples show strong violations of this apparently fundamental property of  d-wave nodal quasiparticles[1].  Using analytical arguments and Bogoliubov-de Gennes simulations, we showed that the breakdown of universality is a consequence of disorder-induced magnetic states in the presence of increasing antiferromagnetic correlations in the underdoped state[2].  These same correlations protect the low energy density of states [2,3].  Figure 3 (a) Results of Bogoliubov-de Gennes calculations for d-wave superconductor with Hubbard-type correlations indicating that k is suppressed by the growth of magnetic droplets, as shown in (b).  Also shown in insert to (b) is suppression of plateau in DOS by magnetic correlations.  (c)  Experimental dependence of  low-T thermal conductivity on doping in Bi-2212 [2].  (d) Conflict with results of photoemission studies if naïve universal k formula is assumed [2]. An  interesting aspect of the interplay of disorder and   correlations, which become stronger in underdoped materials, is that the low-energy DOS is suppressed (Fig 3b), leading to the notion of nodal states being “protected” from disorder by interactions[3].  Nevertheless the interactions have a dramatic effect on transport, leading to a violation of the cancellations between scattering lifetime and DOS in the thermal conductivity. This work strongly suggests that the physics of underdoped, intrinsically doped cuprates is dominated by magnetic fluctuations pinned by dopant disorder.  Such tendencies are strongly enhanced in both the superconducting and pseudogap states due to incipient bound state formation around nonmagnetic impurities. This magnetic order scatters low-energy quasiparticles strongly, and leads to a breakdown of universal transport predicted for d-wave systems in the presence of ordinary disorder.   The universal thermal conductivity expression may therefore not be used to determine the doping dependence of the gap velocity in the underdoped regime of the phase diagram. References [1]   X.F. Sun et al., Nature Phys. Rev. Lett. 96, 017008 (2006). [2]  B.M. Andersen and P.J. Hirschfeld, Phys. Rev. Lett. 100, 257003 (2008). [3]   A. Garg et al., Nat. Phys. 4, 762 (2008). [4]   B.M. Andersen et al. Phys. Rev. Lett. 99, 147002 (2007).

• Atomic scale modulation of pairing interactions

  Fig. 1: Left: 90x90 Bogoliubov-de Gennes calculation of LDOS in d-wave superconductor with 7.5% out-of-plane O defects modeled by local enhancements of pair potential g with range 0.5a.  Note highly particle-hole symmetric modulation of coherence peak positions, unlike results obtained with conventional Coulomb potential. Right: expermimental data on Bi-2212 from Lang et al (2002).    Fig. 2: top left: structure of BSCCO-2212.  Top right: above: structure of crystal from x-ray diffraction; below: from DFT calculations in collaboration with group of H.-P. Cheng.  Bottom: schematic of buckling of CuO8 half-octahedra which may modulate pairing interaction.

  Comparison of recent experimental STM data with single-impurity and many-impurity Bogoliubov-de Gennes calculations strongly suggests that out-of-plane dopant atoms in cuprates modulate the pair interaction on the atomic scale.  This type of pairing disorder is crucial to  understanding the nanoscale electronic structure inhomogeneity observed in the BSCCO 2212 system, and can reproduce observed correlations between the positions of impurity atoms and various aspects of the local density of states  such as the gap magnitude and the height and weight of the coherence  peaks.   In addition, off-diagonal scattering of this type is shown to account for some heretofore unexplained aspects of  Fourier  transform STM on these systems.   This type of analysis suggests a) that the famous nanoscale inhomogeneity observed in many cuprates is a consequence of  dopant disorder; and b) the validity of a new approach to determining the pair interaction in these materials,  by attempting to understand changes in local electronic structure and correlating them with measured changes in the superconducting gap. References  1)  K. McElroy et al., Science 309, 1048 (2005). 2)  T.S. Nunner,    B.M. Andersen, A. Melikyan and P.J. Hirschfeld,  Phys. Rev. Lett. 95, 177003 (2005).   This  suggests the possibility of a new paradigm for investigating pairing and its origins in the cuprates.  Measurements tell us how the gap responds to a spatial perturbation on the atomic scale. Density functional theory (DFT) or  x-ray analysis can then help determine the exact atomic displacements which occur. With this information in hand, one can use it to calculate local changes in electronic structure and constrain pairing theories of high-temperature superconductors.  Very recently, the Cornell STM group has measured a 10% variation of the superconducting gap directly correlated with the phase of the structural supermodulation known to exist in BSSCO-2212 [3] which were shown to be entirely consistent with the magnitude of the pairing interaction modulations expected from the dopant work [4].  Results of a DFT structural calculation involving 5x2 complete BSCCO-2212 unit cells[5]  are in excellent agreement with those of x-ray analyses, as shown, e.g. in 2nd panel of Fig. 2, where structure of modulated Bi-2212 (nominal crystal structure shown on left)  as found by DFT (top center) and x-ray diffraction (bottom center) are compared.   Furthermore the atomic displacements found correlated in exactly the same way with increased pairing strength as in the impurity prbblem!    The most important effect on the CuO2 plane of both the structural supermodulation and the O dopant appears to be the tilting of the CuO2 half-octahedra and concomitant buckling of the plane; this corresponds locally to increased superconducting gap.  The local electronic structure from these calculations will now be downfolded and the effect on the pair interaction within various models will now be determined.   References [1]   K. McElroy et al. Science  309, 1048 (2005). [2]   T. S. Nunner et al,  Phys. Rev. Lett.   95, 177003 (2005). [3]   J.A. Slezak et al.,  submitted to Proc. Nat. Acad. Sci. [4]   B.M. Andersen et al.,  Phys. Rev. B Phys. Rev. B 76, 020507 (2007). [5]   Y. He et al, arXiv:0709.0662 [6]   A.A. Levin et al, J. Phys.: Condens. Matter 6, 3539 (1994).

• 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  Phys. Rev. B 69, 060503 (2004). 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).   To understand why certain q-vectors were still too weak compared to experiment in these patterns, we proposed in Phys. Rev. B 73, 104511 (2006)  that scattering processes could be classified according to whether or not they connected points on the Fermi surface with same or opposite sign of the order parameter.  In particular, so-called "tau1" processes were found to enhance several q's including q1.

  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.  (Phys. Rev. B 67, 094508 (2003)).

  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.  We are developing numerical techniques to make direct predictions of real-space and Fourier transform spectra for a variety of one-impurity potentials. (Phys. Rev. B 68, 054501 (2003).)

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

  SCTMA: Density of states Exact DOS (but no order parameter supression):

  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.                                                                   Exact amplitude of d-wave order parameter around strongly scattering impurities.

• Transport properties due to strong disorder

  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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