1. Phase diagram of s-wave SC Introduction
The Fulde-Ferrell-Larkin-Ovchinnikov states of a 2D square-lattice in s and d-wave superconductor with bandwidth imbalance
Yonghui Wang and Yan Chen
Department of physics and Lab of Advanced Materials, Fudan University, Shanghai,
Abstract
We present a numerical study of the phase diagram for a two-dimensional square-lattice s and d-wave superconducting system with bandwidth imbalance by solving the Bogoliubov-de Gennes equations. It is found that, the spatial configuration of the order parameter should include both the uni-directional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, and the two-dimensional FFLO state by varying the bandwidth imbalance. Local density of states spectra could be used to distinguish these exotic phases.
Phase diagram of s-wave SC
Introduction
We consider the superconducting systems with effective bandwidth imbalance between different spins. We numerically observe the order parameter for increasing the degree of bandwidth mismatch increases, the system is transformed from the uniform BCS state to the FFLO state, and finally to the spin polarized normal state.
β-T phase diagram for uniform s-wave, uni-directional FFLO, 2D FFLO and FM states as calculated in s-wave superconductors. Here, Tc0 ≈ 0.19 is the SC transition temperature.
FFLO state:
Cooper pairs have non-zero momentum;
Pairing gap becomes a spatially varying function;
Translational invariance broken
Local density of states
The local density of states (LDOS) are calculated and suggested to be signatures to distinguish different phases . The LDOS can be calculated from:
Model
We study a phenomenological tight-binding model on a two-dimensional square lattice, The mean-field Hamiltonian can be written as:
For uni-directional FFLO in s-wave SC
(a) The contour plot of the uni-directional FFLO order parameter.
(b) The spin-up LDOS at three different location corresponding to a (nodal line), b (middle line), and c (maximum line) are shown in (a). Similarly, (c) shows total spin LDOS spectrum. (d) represents the LDOS map for spin quasiparticles at energy-peak [E = ] in (c).
(a)
(b)
(c)
(d)
The Hamiltonian can be diagonalized by solving the Bogoliubov-de Gennes (BDG) equation:
All the order parameters are self-consistently determined by:
(a)
S-wave SC
For 2D FFLO in s-wave SC Δi mi ni
β=0.15
β=0.35
β=0.39
β=0.65
(a) shows the contour plot of the SC order parameter, where a-c points denote the saddle point, middle point, and maximum point, respectively. (b) and (c) exhibit the spin-up
and total spin spectrum, respectively. The (black) solid line, the (red) dashed line, and the (green) dotted line corresponds to a, b, c points, respectively. (d) and (e) represent the LDOS maps at two specific energy-peaks (E = -0.08, ) in (c).
(c)
(b)
(d)
(e)
Summary
We have found that, the spatial configuration of the order parameter should include the uni-directional FFLO state, and the two-dimensional FFLO state by varying the bandwidth imbalance. By increasing the degree of bandwidth mismatch of spin-up and spin-down, a transition between uni-directional FFLO states and 2D FFLO states may appear both in s and d-wave superconductor. The LDOS spectrum can be used to distinguish different FFLO phases.
Spatial variations of the order parameter for various bandwidth imbalance in s-wave superconductor.
D-wave SC
Spatial profiles of the order parameter for varying β. The three columns correspond to d-wave pairing order, local Magnetization, and filling factor,
respectively.
β=0.40
β=0.45 Δi mi ni
Reference
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(2006).
[2] T. Zhou, and C. S. Ting, Phys. Rev. B. 80, (2009)
[3] Y. Chen, Z. D. Wang, F. C. Zhang, and C. S. Ting, Phys. Rev. B.79,054512
(2009).
[4] Y. H. Wang, Z. D. Wang, C. S. Ting, and Y. Chen, to be submitted.