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arXiv:0708.0451 · Physical Review B

s -wave superconductivity phase diagram in the inhomogeneous two-dimensional attractive Hubbard model

Karan Aryanpour, Thereza Paiva, Warren E. Pickett, Richard T. Scalettar

Published Nov 30, 2007• 23 citations
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Abstract

We study $s$-wave superconductivity in the two-dimensional square-lattice attractive Hubbard Hamiltonian for various inhomogeneous patterns of interacting sites. Using the Bogoliubov--de Gennes mean-field approximation, we obtain the phase diagram for inhomogeneous patterns in which the on-site attractive interaction ${U}_{i}$ between the electrons takes on two values ${U}_{i}=0$ and $\ensuremath{-}U∕(1\ensuremath{-}f)$ (with $f$ the concentration of noninteracting sites) as a function of average electron occupation per site, $n$, and study the evolution of the phase diagram as $f$ varies. In certain regions of the phase diagram, inhomogeneity results in a larger zero-temperature average pairing amplitude (order parameter) and also a higher superconducting critical temperature ${T}_{c}$, relative to a uniform system with the same mean interaction strength (${U}_{i}=\ensuremath{-}U$ on all sites). These effects are observed for stripes, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. The phase diagrams also include regions where superconductivity is obliterated due to the formation of various charge-ordered phases. The enhancement of ${T}_{c}$ due to inhomogeneity is robust as long as the electron doping per site, $n$, is less than twice the fraction of interacting sites $[2(1\ensuremath{-}f)]$ regardless of the pattern. We also show that for certain inhomogeneous patterns, when $n=2(1\ensuremath{-}f)$, increasing temperature can work against the stability of existing charge-ordered phases for large $f$ and, as a result, enhance ${T}_{c}$.

DOI
10.1103/physrevb.76.184521

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We study sss-wave superconductivity in the two-dimensional square-lattice attractive Hubbard Hamiltonian for various inhomogeneous patterns of interacting sites. Using the Bogoliubov--de Gennes mean-field approximation, we obtain the phase diagram for inhomogeneous patterns in which the on-site attractive interaction Ui{U}_{i}Ui​ between the electrons takes on two values Ui=0{U}_{i}=0Ui​=0 and \ensuremath−U∕(1\ensuremath−f)\ensuremath{-}U∕(1\ensuremath{-}f)\ensuremath−U∕(1\ensuremath−f) (with fff the concentration of noninteracting sites) as a function of average electron occupation per site, nnn, and study the evolution of the phase diagram as fff varies. In certain regions of the phase diagram, inhomogeneity results in a larger zero-temperature average pairing amplitude (order parameter) and also a higher superconducting critical temperature Tc{T}_{c}Tc​, relative to a uniform system with the same mean interaction strength (Ui=\ensuremath−U{U}_{i}=\ensuremath{-}UUi​=\ensuremath−U on all sites). These effects are observed for stripes, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. The phase diagrams also include regions where superconductivity is obliterated due to the formation of various charge-ordered phases. The enhancement of Tc{T}_{c}Tc​ due to inhomogeneity is robust as long as the electron doping per site, nnn, is less than twice the fraction of interacting sites [2(1\ensuremath−f)][2(1\ensuremath{-}f)][2(1\ensuremath−f)] regardless of the pattern. We also show that for certain inhomogeneous patterns, when n=2(1\ensuremath−f)n=2(1\ensuremath{-}f)n=2(1\ensuremath−f), increasing temperature can work against the stability of existing charge-ordered phases for large fff and, as a result, enhance Tc{T}_{c}Tc​.
Inhomogeneous sss-wave superconductivity is studied in the two-dimensional, square lattice attractive Hubbard Hamiltonian using the Bogoliubov-de Gennes (BdG) mean field approximation. We find that at weak coupling, and for densities mainly below half-filling, an inhomogeneous interaction in which the on-site interaction Ui{U}_{i}Ui​ takes on two values, Ui=0,2U{U}_{i}=0,\phantom{\rule{0.3em}{0ex}}2UUi​=0,2U results in a larger zero temperature pairing amplitude, and that the superconducting Tc{T}_{c}Tc​ can also be significantly increased, relative to a uniform system with Ui=U{U}_{i}=UUi​=U on all sites. These effects are observed for stripe, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. Monte Carlo calculations which reintroduce some of the fluctuations neglected within the BdG approach see the same effect, both for the attractive Hubbard model and a Hamiltonian with ddd-wave pairing symmetry.
The phase diagram of the attractive Hubbard model with spatially inhomogeneous interactions is obtained using a single-site dynamical mean-field theory-like approach. The model is characterized by three parameters: the interaction strength, the active fraction (fraction of sites with the attractive interaction), and electron filling. The calculations indicate that in a parameter regime with intermediate values of interaction strength (compared to the bare bandwidth of the electrons), and intermediate values of the active fraction, unconventional superconductivity is obtained. The results of this work are likely to be relevant to many systems with spatially inhomogeneous superconductivity such as strongly correlated oxides, systems with negative UUU centers, and, in future, cold atom optical lattices.
Inhomogeneities on various length scales suggesting phase separation seem to be a generic feature of hole-doped high-temperature superconducting cuprates. Also firmly established is the d-wave symmetry of the superconducting state. The interplay of superconductivity and phase separation has been noted in previous mean-field studies of correlated electron models. To further explore this issue in a more rigorous approach we have conducted quantum Monte Carlo calculations of a two-dimensional Hubbard model with nearest-neighbor attractive interaction. For a vanishing on-site repulsion, the result agrees with that of Micnas, Ranninger, and Robaszkiewicz [Phys. Rev. B 39, 11 653 (1989)], and indicates that for a given strength of the attractive interaction, homogeneous d-wave superconducting phase can exist only above a critical doping concentration, below which phase separation occurs. A finite on-site repulsion modifies the phase separation behavior. A stronger d-wave pairing correlation is seen in the inhomogeneous phase compared to the homogeneous phase.
We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction fff of sites, while keeping a finite UUU on the remaining ones. Through quantum Monte Carlo simulations for several values of fff and UUU and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor Ps{P}_{s}Ps​ and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for Ps{P}_{s}Ps​ have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration fc{f}_{c}fc​, which increases with UUU, at least up to U=4U=4U=4 (in units of the hopping energy). Also, the normalized zero-temperature gap as a function of fff shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of fff for U=3U=3U=3, 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest Tc{T}_{c}Tc​ increasing with UUU in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.
Our previous mean-field study concerning superconducting pairings in the extended Hubbard model with on-site repulsive and intersite attractive interaction and arbitrary electron density is extended to include the effects of nearest-neighbor exchange (JJJ) and correlated hopping (KKK) interaction. Detailed numerical analysis of sss-, ppp-, and ddd-wave pairing solutions is performed for the two-dimensional square lattice with nearest- (ttt) and next-nearest- (t2{t}_{2}t2​) neighbor hopping, and the resulting mutual stability phase diagrams are given as a function of band filling and the interaction parameters. Antiferromagnetic exchange can enhance both the ddd-wave and the extended sss-wave pairing. If it prevails over the repulsive intersite density-density interaction, such a term can be a leading mechanism for superconductivity. However, its effects strongly depend on the band filling and the t2t\frac{{t}_{2}}{t}tt2​​ ratio, giving in particular for t2=0{t}_{2}=0t2​=0 sss-wave (ddd-wave) pairing as most favorable in the low (high) density limit, but increasing t2{t}_{2}t2​ can even reverse this tendency. The correlated hopping term which breaks the electron-hole symmetry affects sss-wave pairing mostly. A condensation transition (phase separation), possible in the presence of the attractive density-density interaction, is also analyzed within the random-phase approximation, and stability conditions of such an electron droplet phase with respect to other types of ordering are determined as a function of the band filling for the two-dimensional square lattice.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.