Vortex State Phase Transitions In High Tc Superconductors Based On Bogoliubov-de Gennes Theory

The vortex structure in high temperature superconductors (HTSCs) has attracted significant interest for many years. Since the parent compounds are antiferromagnetic Mott insulators, novel physical properties of HTSCs including those in the vortex state would be expected due to the competition between spin magnetism and superconductivity in these systems. Unlike what occurred to a metallic superconductor, the Thomas-Fermi screening effect was substantially reduced for underdoped, or even slightly overdoped HTSCs since the short wave length part of the Coulomb interaction between conducting electrons, which has been omitted in the TF approximation, becomes important. Interesting phenomenas would be expected due to the inducing of the long-rang Coulomb interaction, which may help us to well understand recent experiments.The influence of the long-rang Coulomb interaction to the charged vortex, vortex structure transitions and the local density of states of HTSCs are investigated theoretically in the framework of the Bogoliubov-de Gennes theory. Main results are summarized as follows:(1) The effect of the long-rang Coulomb interaction on the antiferromagnetism and vortex charges for doped HTSCs is investigated by numerically solving the Bogoliubov-de Gennes equations based on a model Hamiltonian with competing antiferromagnetic (AFM) and d-wave superconductivity interactions. We find that for slightly overdoped HTSCs, a sign change between positive to negative may occur with the enhancement of the AFM order. For underdoped HTSCs, the AFM is reduced because of the increasing of the strength of the long-rang Coulomb interaction. Consequently, the positive vortex charge could be found for underdoped samples of large AFM order when a reasonable large long-rang Coulomb interaction had been introduced into the model Hamiltonian. We show that the charged vortex can induce a spin-orbit coupling. Fractional flux quanta are possible for HTS.(2) The effect of model ordering on the vortex structures for underdoped HTSCs is investigated by numerically solving the Bogoliubov-de Gennes equations based on a model Hamiltonian with competing antiferromagnetic (AFM) and d-wave superconductivity interactions. Firstly, the effect of next-nearest-neighbor (nnn) hopping on the spin and charge structures around vortices is studied. We show that one-dimensional (1D) y- or x-axis–oriented stripes for the SDW and the associated charge density wave (CDW) exist for small nnn hopping strengths. A coexistence of both y- and x-axis–oriented stripes may occur for a relatively large nnn hopping strength. With further enlarging nnn hopping strength, the field-induced AFM core is surrounded by the x- and y-axis–oriented stripe-like structures, accompanied with a checkerboard-like distribution for CDW order. Then, the effect of long-range Coulomb interaction is studied. We show that transition from a checkerboard pattern to stripe structure for spin density wave, charge density wave, and d-wave orderings may occur by enhancing the strength of the on-site repulsion U in the absence of the long-range Coulomb interaction. The long-range Coulomb interaction provides an intrinsic mechanism for electron depletion inside an AF-like vortex core, and the field induced AF order was screened and confined onto the near core region. Consequently, two-dimensional modulations or the checkerboard patterns of vortex charge distribution and spin density wave order recovered for underdoped samples which favered the strip structure when introducing a large long-range Coulomb interaction into the model Hamiltonian.(3) The effect of the long-rang Coulomb interaction on the local density of states (LDOS) for HTSCs is investigated by numerically solving the Bogoliubov-de Gennes equations based on a model Hamiltonian with competing antiferromagnetic (AFM) and d-wave superconductivity interactions. Since the LDOS probed in the STM experiments usually provided the information of excited states of quasiparticles, while the calculated spatially CDW order represents the charge density configuration in the ground states. It is therefore important to discuss the local density of states in the presence of vortex core. Our results shows that, the vortex-core state with energies slightly above and below the Fermi level shows a fourfold symmetry with two-dimensional modulations to one-dimensional stripe structure transition with the increasing of the AFM order. For the long-range Coulomb interaction, the LDOS maps not only change from the 1D stripe-like to a 2D structure, but also change the periodicity from 5 lattice sites to 4 lattice sites. Furthermore we have revealed that long-range Coulomb interaction can both induce a spin-flip which gives rise to a hopping between spin-up and spin-down electrons and can split the spin degeneracy of core states as well, As a result, the two featured peaks of the core states may move more close to the Fermi level, but remain distinctive. Recent scanning tunneling microscopy experiments may be understood in terms of the present results.