Quasiparticles in the vortex lattice of strongly type-II superconductors are investigated by means of a singular gauge transformation applied to the tight binding lattice Bogoliubov-de Gennes Hamiltonian. We present a detailed derivation of the gauge invariant effective low energy Hamiltonian for the quasiparticle-vortex system and show how the physics of the "Doppler shift" and "Berry phase" can be incorporated at the Hamiltonian level by working in the singular gauge. In particular, we show that the "Berry phase" effect manifests itself in the effective Hamiltonian through a half-flux Aharonov-Bohm scattering of quasiparticles off vortices and stress the important role that this effect plays in the quasiparticle dynamics. Full numerical solutions in the regime of intermediate fields Hc1 ≪ B ≪ Hc2 are presented for model superconductors with s-, p-and d-wave symmetries and with square and triangular vortex lattices. For s-and p-wave cases we obtain low energy bound states in the core, in agreement with the existing results. For d-wave case only extended quasiparticle states exist. We investigate in detail the nature of these extended states and provide comparison to the previous results within linearized "Dirac fermion" model. We also investigate internodal interference effects when vortex and ionic lattices have high degree of commensurability and discuss various specific choices for the singular gauge transformation.
A comparison of recent experimental STM data with single-impurity and many-impurity Bogoliubov-de Gennes calculations strongly suggests that random out-of-plane dopant atoms in cuprates modulate the pair interaction locally. This type of disorder is crucial to understanding the nanoscale electronic inhomogeneity observed in BSCCO-2212, 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 of the coherence peaks. Our results imply that each dopant atom modulates the pair interaction on a length scale of order one lattice constant.
The local density of states power spectrum of optimally doped Bi2Sr2CaCu2O8+x (BSCCO) has been interpreted in terms of quasiparticle interference peaks corresponding to an "octet" of scattering wave vectors connecting k-points where the density of states is maximal. Until now, theoretical treatments have not been able to reproduce the experimentally observed weights and widths of these "octet" peaks; in particular, the predominance of the dispersing "q1" peak parallel to the Cu-O bond directions has remained a mystery. In addition, such theories predict "background" features which are not observed experimentally. Here, we show that most of the discrepancies can be resolved when a realistic model for the out-of-plane disorder in BSCCO is used. Weak extended potential scatterers, which are assumed to represent cation disorder, suppress large-momentum features and broaden the low-energy "q7"-peaks, whereas scattering at order parameter variations, possibly caused by a dopant-modulated pair interaction around interstitial oxygens, strongly enhances the dispersing "q1"-peaks.
We present a theory of quasiparticle Hall transport in strongly type-II superconductors within their vortex state. We establish the existence of integer quantum spin Hall effect in clean unconventional d x 2 Ϫy 2 superconductors in the vortex state from a general analysis of the Bogoliubov-de Gennes equation. The spin Hall conductivity xy s is shown to be quantized in units of ប/8. This result does not rest on linearization of the BdG equations around Dirac nodes and therefore includes inter-nodal physics in its entirety. In addition, this result holds for a generic inversion-symmetric lattice of vortices as long as the magnetic field B satisfies H c1 ӶB ӶH c2 . We then derive the Wiedemann-Franz law for the spin and thermal Hall conductivity in the vortex state. In the limit of T→0, the thermal Hall conductivity satisfies xy ϭ(4 2 /3)(k B /ប) 2 T xy s . The transitions between different quantized values of xy s as well as relation to conventional superconductors are discussed.
We study the mixed state in an extreme type-II lattice d x 2 −y 2 -wave superconductor in the experimentally most relevant regime of intermediate magnetic fields Hc1 ≪ H ≪ Hc2. We analyze the low energy spectrum of the problem dominated by nodal Dirac-like quasiparticles with momenta near kF = (±kD, ±kD) and find that the spectrum exhibits characteristic oscillatory behavior with respect to the product of kD and magnetic length l. The Simon-Lee scaling, predicted in this regime, is satisfied only on average, with the magnitude of the oscillatory part of the spectrum displaying the same l −1 dependence as its monotonous "envelope" part. In general, the spectrum obeys a scaling law E nk = vF l En(kl, t/∆, kDl), where E is a dimensionless universal 2π-periodic function of kDl. The oscillatory behavior of the spectrum is due to the inter-nodal interference enhanced by the singular nature of the low energy eigenfunctions near vortices. Our results constitute an example of a finite size scaling of the Dirac-type quantum criticality. We also study a separate problem of a single vortex piercing an isolated superconducting grain of size L × L. Here we find that the periodicity of the quasiparticle energy oscillations with respect to kDL is doubled relative to the case where the field is zero and the vortex is absent, both such oscillatory behaviors being present at the leading order in L −1 . Finally, we review the overall features of the tunneling conductance experiments in YBCO and BSCCO, and suggest an interpretation of the peaks at 5 − 20 meV observed in the tunneling local density of states in these materials. We find that in the case of a pure d-wave superconducting order parameter with featureless vortex cores, the zero bias conductance peak (ZBCP) appears only on the sites that are the immediate nearest neighbors of vortex locations, while all the other sites in the close vicinity of vortices exhibit no such ZBCP, and instead display pronounced peaks at sub-gap energies, typically at about a half or less of the coherence peak energy. Furthermore, we find that the on-site ZBCP can be strongly suppressed by enhanced local pairing near a vortex.
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