Cu d_x²-y² Wannier function isosurfaces a) in CuO₂ plane; b) extending to BiO surface. Arrows point to apical O p states.

Theoretical calculations of STM images on cuprate superconductors have been confined almost exclusively to models of electrons hopping on a square lattice corresponding to the Cu sites in the CuO2 plane; this is a clear disadvantage, for example when the elusive charge order is seen to correspond to local density of states modulations primarily on the O sites. We developed a method [1] that started with the lattice calculations that we and others had already performed, and simply used the Wannier functions generated in the process of downfolding a renormalized DFT band structure onto a tight-binding model to calculate the continuum instead of the lattice Green's function. The crucial aspect was found to be the tails of the Cu Wannier functions, which include significant weight on apical O sites in neighboring unit cells. The results compare nearly perfectly to experimental findings that have puzzled the community for more than 10 years.

1. Choubey et al, Phys. Rev. B 90, 134520 (2014); A. Kreisel et al, Phys. Rev. Lett. 114, 217002 (2015).
2. S. Pan et al, Nature 403, 746 (2000).
3. K. Fujita et al. Science 344, 612 (2014).

Calculation of Zn impurity LDOS in BSCCO-2212: a) traditional Bogoliubov-de Gennes (BdG) solution; (b) BdG+W [1] approach; (c) experiment from Ref. [2].

QPI patterns for E =24meV for weak scattererer: (a) BdG; (b) BdG+W; [2] (c) experiment from Ref. [3].

One of the fascinating phenomena observed in high-Tc cuprates is the so-called charge order, developed in the underdoped phase and partially, but not totally coincident with the "pseudogap" identified in the effective 1-particle spectrum. Here, the charge carriers tend to form a regular pattern of stripes in the copper oxygen planes. Being placed in a regular arrangement make the carriers less mobile and competes with the formation of the superconducting state: charge order is antagonistic to superconductivity, making it important to understand limits of superconductivity itself. Charge order was observed in Nd-doped LSCO classes already in 1995. It took, however, quite some time to reveal that many other classes of cuprates exhibit similar behavior, and only in recent years evidence for an ubiquitous phenomenon was accumulated, with the important observation of charge order in YBCO in 2012 via x-ray experiments. It is generally believed that the bulk charge order is related to, or identical to, the fascinating ordered LDOS patterns that had been seen in STM starting in the mid-2000's. A photo of this so-called checkerboard pattern as observed in the Z-map, a ratio of tunneling conductance maps at +/-E, for E near the pseudogap energy, is shown on the right. The image shows bright spots on a subset of all oxygen sites. Since most models of high-Tc cuprates are based on square lattices of Cu atoms, they a priori miss these features. The Hirschfeld group is attempting to determine the symmetry and state of charge order from the theory of STM on cuprates discussed above, coupled with treatments of strong-coupling lattice models of electrons.

Z-map of BSCCO-2212, courtesy J.C. Davis.

Older projects: quasiparticles in the Vortex State

---

 

Thermodynamics and transport: semiclassical theory

In 1993, Volovik argued that the thermodynamic properties of a nodal superconductor in the vortex state would be governed by the nodal quasiparticle states rather than any vortex bound states. To a good approximation, the effect is described by the Galilean shift in the quasiparticle energy in the presence of the superflow field around the vortex; this is now known as the Volovik effect. In 1997, Kuebert and I showed that this concept, together with an elementary treatment of disorder, could explain existing low-T specific heat data on the cuprates ( Sol. St. Commun. 105, 459 (1998)). In Phys. Rev. Lett. 80, 4963 (1998), we generalized this notion to quasiparticle transport, predicting the increase of thermal conductivity with applied field at low temperatures, which had never been observed. This prediction was verified by Taillefer's group on Zn-substituted YBCO in data shown at right. Scaling properties in H/T were derived but shown to break down in the presence of sufficient disorder. A generalization to nodal Fe-based superconductors was given in Phys. Rev. B 80, 224525 (2009).

Comparison of experiment of Chiao et. al. on YBCO with semiclassical theory of Kuebert & PH.

Related work within the same semiclassical model regarding anisotropy of field dependence for thermodynamic properties with field in the ab plane with Vekhter, Nicol, & Carbotte appears in Phys. Rev. B 59, R9023 (1999) and subsequent papers. These works laid the groundwork for what is now the simplest bulk probe of gap nodal structures, which are typically difficult to pin down. The phase space for excitation of nodal quasiparticles dues to the Volovik effect is largest for those quasiparticles with momenta perpendicular to the field. Thus as the field rotates, different nodes contribute with different weights (see figure to right), leading to oscillations in thermodynamic and transport quantities (far right).

DOS dependence leads to oscillations in specific heat.

With the field in the plane of the "2D" superconductor, different d-wave nodes contribute different Doppler shift energies according to their orientation.

In optical pump-probe experiments, the system is pumped and its quasiequilibrium reflectance measured with a probe pulse after a time delay.

Nonequilibrium Superconductivity

In conventional superconductors, the study of non-equilibrium QP relaxation was used successfully to extract information on residual particle--particle interactions, as well as to pin down QP and phonon lifetimes. The typical time-resolved experiment is a measurement of the change in the system's dielectric constant as a function of time following a pump pulse which creates a non-equilibrium QP distribution. The excited QPs decay to equilibrium over a series of timescales involving several steps, including at least: (i) a cascade of pair production until a quasi-equilibrium is reached between ``hot" QPs of roughly the gap energy and phonons of energy twice the gap, and (ii) slow recombination of QPs into Cooper pairs. The timescales involved in step (i) are O(ps), but can be much longer in step (ii) O(ns--mus)] since energy is continually exchanged between the electron and phonon systems until heat is removed at the sample surfaces; this long decay is sometimes referred to as the ``phonon bottleneck".

A priori, several differences are to be expected in nodal superconductors. The very strong interactions in the normal state and the larger gap scale suggest that electron--electron rather than electron--phonon scattering is the dominant relaxation mechanism. Furthermore, the gap anisotropy and the equilibration with thermal QPs at the nodes leads to a new type of slow decay, which we call the ``antinodal bottleneck". This is the inability of antinodal QPs to decay rapidly by pair-breaking while simultaneously satisfying energy and momentum conservation, due to the large difference in velocities at the nodal and antinodal points of the Fermi surface. This picture is valid only in systems so clean that the momentum distribution is not equilibrated by disorder.

Relaxation of hot d-wave quasiparticles

Together with P. Howell and A. Rosch, I have presented a theory (PRL 92, 037003) of the single aspect of the complicated non-equilibrium physics of pump--probe experiments most peculiar to the d-wave superconductor, namely the mechanism intermediate between steps (i) and (ii) whereby hot QPs scatter through the antinodal bottleneck before recombination. Within a model where QPs are scattered by a simple local interaction, we show that at high T there is a fast relaxation due to Umklapp scattering, but below some crossover temperature relaxation is dominated by diffusion in momentum space along the Fermi surface from the antinode to the nodes. This idea is a momentum-space analog of real-space ``QP traps" in conventional superconductors, in the sense that there is an intermediate stage of relaxation in which QPs diffuse to a region of lower gap. New directions to be explored include: 1) direct calculation of the reflectivity change in a pump-probe experiment; 2) numerical solutions to quantum kinetic equation including realistic collision models; 3) analysis of spatially inhomogeneous situations.

 

 

Momentum space trajectory of an antinodal quasiparticle as it relaxes towards the node