D-wave Pairing State Research Articles

Theory on Superconductivity of CeIn3in Heavy Fermion System

On the basis of a three-dimensional Hubbard model, the superconducting mechanism of CeIn3 under high pressure is investigated by the third-order perturbation theory with respect to the on-site Coulomb interaction U. Here we propose the d-wave pairing state induced by antiferromagnetic spin fluctuations. The estimated superconducting transition temperature is lower by one order than that in the two-dimensional system for the same value of U/W (W=bandwidth). This result is consistent with the difference between CeIn3 and CeRhIn5.

The superconducting mechanism of the heavy-fermion system CeIn3with three-dimensional properties

On the basis of the three-dimensional Hubbard model, the superconducting (SC) mechanism ofCeIn3under high pressure is investigated by means of third-orderperturbation theory with respect to the on-site Coulomb interactionU.Here we propose a d-wave pairing state induced by antiferromagneticspin fluctuations. The estimated SC transition temperature is lower byone order than that in the two-dimensional system for the same value ofU/W (W = bandwidth).This result is consistent with the difference between CeIn3 andCeRhIn5.

Staggered flux state of electrons in a two-dimensionalt−Jmodel

The competition between the staggered flux state, or the d-density wave state, and the d-wave pairing state is analyzed in a two-dimensional $t\ensuremath{-}J$ model based on the U(1) slave boson mean-field theory. Not only staggered flux of spinons but also staggered flux of holons are considered. In this formalism, the hopping order parameter of physical electron is described by the product of hopping order parameters of spinons and holons. The staggered flux amplitude of electrons is the difference of staggered flux amplitude of spinons and that of holons. In the $\ensuremath{\pi}$-flux phase of spinons, staggered fluxes of spinons and holons cancel completely and the staggered flux order of electrons does not exist. However, in the staggered flux phase of spinons whose staggered flux amplitude is not $\ensuremath{\pi},$ fluxes does not cancel completely and the staggered flux amplitude of electron remains. Thus the phase transition between these two phases, the $\ensuremath{\pi}$-flux phase and the staggered flux phase of spinons, becomes a second-order transition in the physical electron picture. The order parameter which characterizes this transition is the staggered flux order parameter of electrons. A mean-field phase diagram is shown. It is proved analytically that there is no coexistence of staggered flux and d-wave pairing. The temperature dependence of Fermi surface and excitation gap at $(0,\ensuremath{\pi})$ are shown. These behaviors are consistent with angle-resolved photoemission spectroscopy experiments.

Anisotropy on the Fermi surface of the two-dimensional Hubbard model

We investigate anisotropic charge fluctuations in the two-dimensional Hubbard model at half filling. By the quantum Monte Carlo method, we calculate a momentum-resolved charge compressibility $\kappa (\bm{k}) = {d < n(\bm{k}) >}/{d \mu}$, which shows effects of an infinitesimal doping. At the temperature $T \sim {t^2}/{U}$, $\kappa (\bm{k})$ shows peak structure at the $(\pm \pi/2,\pm \pi/2)$ points along the $|k_x| + |k_y| = \pi$ line. A similar peak structure is reproduced in the mean-filed calculation for the d-wave pairing state or the staggered flux state.

Perturbation Theory on the Superconductivity of Heavy Fermion Superconductors CeIrxCo1-xIn5

We reformulate the Eliashberg's equation for the superconducting transition within the quasi-particle description, which takes account of the heavy electron mass. We discuss the superconductivity of CeIr_{x}Co_{1-x}In_5 by using such a renormalized formula. On the basis of an effective two-dimensional Hubbard model which represents the most heavy quasi-two dimensional Fermi surface with f-character of CeIr_{x}Co_{1-x}In_5, both normal and anomalous self-energies are calculated up to third order with respect to the renormalized on-site repulsion U between quasi-particles. The superconducting transition temperature is obtained by solving the Eliashberg's equation. Reasonable transition temperatures are obtained for moderately large U. It is found that the momentum and frequency dependence of spin fluctuations given by RPA-like terms gives rise to the d-wave pairing state, while the vertex correction terms are important for obtaining reasonable transition temperatures and for differentiating n-dependence of T_c. These obtained results seem to explain the x-dependence of T_c in CeIr_{x}Co_{1-x}In_5 and confirm the usefulness of the perturbation theory with respect to U.

Perturbation theory of the superconductivities in heavy fermion systems

We investigate the superconducting mechanism and the transition temperature T c of heavy fermion superconductor UPd 2Al 3 and CeIr x Co 1− x In 5 on the basis of a single band two-dimensional Hubbard model, which represents the most heavy band, respectively. Both normal and anomalous self-energies are calculated up to third order with respect to the Coulomb repulsion U. Reasonable transition temperatures are obtained for moderately large U by solving the Éliashberg's equation. It is found that the momentum and frequency dependence of spin fluctuations given by RPA-like terms gives rise to the d-wave pairing state, while the vertex correction terms are important for obtaining reasonable transition temperatures. We point out that it is reasonable to treat the electron correlation by perturbation theory based on Fermi liquid theory, in calculating T c of typical heavy fermion superconductors.

Correlation effects on the Fermi surface of the two-dimensional Hubbard model

Effects of electron correlation on the Fermi surface (FS) is investigated for the two-dimensional Hubbard model by the quantum Monte Carlo method. At first, an infinitesimal doping from the half-filling is focused on and the momentum-dependent charge susceptibility κ( k)=d n( k)/d μ is calculated at a finite temperature. At the temperature T∼ t 2/ U, it shows peak structure at (±π/2,±π/2) on the FS (line). It is consistent with the mean-field prediction of the d-wave pairing state or the staggered flux state. This momentum-dependent structure disappears at the high temperature T≈ U. After summarizing the results of the half-filling case, we also discuss the effects of the doping on the momentum-dependent charge susceptibility. The anisotropic structure at half-filling fades out with sufficient doping.

Third-order perturbation analysis for p- and d-wave pairing states in two-dimensional repulsive Hubbard model

We study instability of a singlet and triplet pairing states in a quasi-two-dimensional Hubbard model on the basis of the third-order perturbation theory. Comparing the result obtained here with that is obtained by using only the second-order effective interaction, a stable p-wave state spreads from low to intermediate density region. The vertex correction of the third-order effective interaction makes a d x 2− y 2 -wave state for the high density near half-filling. On the other hand, the third-order term makes the p-wave state dominant.

Stripes and superconductivity in high- Tc superconductors

The ground state of the three-band Hubbard model with d and p orbitals is investigated by using a quantum variational Monte Carlo method (VMC) with focus on the underdoped region of high- T c superconductors. The antiferromagnetic (AF) state is considerably stable compared to the d-wave pairing state in the underdoped region, and is further stabilized by introducing spin modulations and stripes. A possibility of pure d-wave pairing state in the underdoped region is small since this state is overwhelmed by the AF ordering for small doping. The VMC evaluations show that the superconducting state is realized as a coexistence state of d-wave and AF states in the low-doping region. In particular, the non-uniform pairing state along with stripes is possible with the oscillating superconducting order parameter in space.

Perturbation Theory on the Superconductivity of Heavy Fermion Superconductor UPd2Al3

We investigate the superconducting mechanism and the transition temperature of heav y fermion superconductor UPd_2Al_3 on the basis of a single band two-dimensional Hubbard model on triangular lattice, whi ch represents the most heavy band of UPd_2Al_3. Both normal and anomalous self-energies are calculated up to third order with respect to the Coulomb repulsion U between itinerant electrons. The superconducting transition temperature is obtained by solving the Eliashberg's equation. Reasonable transition temperatures are obtained for moderately large U. It is fou nd that the momentum and frequency dependence of spin fluctuations given by RPA-like terms gives rise to the d-wave pairing state, while the vertex correction terms are important for obtaining reasonable transition tempe ratures. These results seem to show that the superconductivity in UPd_2Al_3 can be explained by the perturbation theory with respect to U.

Impurity-induced gap modification in anisotropic superconductors: mixed-state specific heat of La 2− xSr x(Cu 1− yZn y)O 4 and Y(Ni 1− xPt x) 2B 2C

We have examined the effect of nonmagnetic impurity on the superconducting gap of La 2− x Sr x (Cu 1− y Zn y )O 4 and Y(Ni 1− x Pt x ) 2B 2C from the mixed-state specific heat C( T, H). In La 2− x Sr x CuO 4, impurity scattering leads to an appearance of a finite density of states (DOS) at the Fermi level, consistent with d-wave pairing state with nonmagnetic impurity scattering. In contrast, nonmagnetic impurity scattering in YNi 2B 2C leads to a smearing of the gap anisotropy and the opening of a gap in the DOS over the whole Fermi surface, though C( T, H) for pure compound suggests the presence of anisotropic gap. This indicates YNi 2B 2C is a highly anisotropic s-wave superconductor.

Quasi-one-dimensional band in the t- J model

We study the two-dimensional (2d) t- J model in a mean filed approximation. Assuming uniform charge distribution, we find that the 2d t- J model has an instability toward forming a quasi-one-dimensional (q-1d) Fermi surface. This q-1d state competes with a d-wave pairing state, which overcomes the q-1d state for a realistic parameter region. However, we can show that the q-1d state coexists with the d-wave pairing state when we introduce a small (about 3%) spatial anisotropy in t and J in the original t- J model.

Impurity-induced gap renormalization in anisotropic superconductors: Mixed-state specific heat of La 2− xSr x(Cu 1− yZn y)O 4 and Y(Ni 1−xPt x) 2B 2C

The effect of nonmagnetic impurity on the superconducting gap is examined on La 2− x Sr x (Cu 1− y Zn y )O 4 and Y(Ni 1− x Pt x ) 2B 2C from the T and H dependence of the electronic specific heat C e ( T, H) at low temperatures. In La 2− x Sr x CuO 4, impurity scattering leads to an appearance of a finite density of states (DOS) at the Fermi energy, consistent with d-wave pairing state with nonmagnetic impurity scattering in the unitarity limit. In contrast, nonmagnetic impurity scattering in YNi 2B 2C leads to a smearing of the gap anisotropy and the opening of a gap in the DOS over the whole Fermi surface, though C e ( T, H) for pure compound suggests the presence of anisotropic gap. This indicates YNi 2B 2C is a highly anisotropic s-wave superconductor.

Instability toward Formation of Quasi-One-Dimensional Fermi Surface in Two-Dimensionalt-JModel

We show within the slave-boson mean field approximation that the two-dimensional t-J model has an intrinsic instability toward forming a quasi-one-dimensional (q-1d) Fermi surface. This q-1d state competes with, and is overcome by, the d-wave pairing state for a realistic parameter choice. However, we find that a small spatial anisotropy in t and J exposes the q-1d instability which has been hidden behind the d-wave pairing state, and brings about the coexistence with the d-wave pairing. We argue that this coexistence can be realized in La_{2-x}Sr_{x}CuO_4 systems

Surface states and tunneling spectroscopy of high-Tcsuperconductors

Recent studies of high- Tcsuperconductors have clarified new aspects of tunneling spectroscopy. The unconventional pairing states, i.e. d-wave symmetry in these materials have been established through various measurements. Differently from isotropic s-wave superconductors, d-wave pairing states have an internal phase of the pair potential. The internal phase modifies the surface states due to the interference effect of the quasiparticles. Along these lines, a novel formula of tunneling spectroscopy has been presented that fully takes into account of the anisotropy of the pair potential. The most essential difference of this formula from conventional ones is that it suggests the phase-sensitive capability of tunneling spectroscopy. The formula suggests that the symmetry of the pair potential is determined by the orientational dependence measurements of tunneling spectroscopy. Along these lines, several experiments have been performed on high-Tc superconductors. The observation of the zero-bias conductance peaks (ZBCP) on Y Ba2Cu3O7−δstrongly suggests the dx2−y2-wave pairing states of hole-doped high-Tc superconductors. On the other hand, the absence of ZBCP on (electron-doped)Nd1.85Ce0.15CuO4−δindicates that the pair potential of this material is a nodeless state. In this paper, recent developments of tunneling spectroscopy for anisotropic superconductors are reviewed both on theoretical and experimental aspects.

Erratum

Recent studies of high- Tcsuperconductors have clarified new aspects of tunneling spectroscopy. The unconventional pairing states, i.e. d-wave symmetry in these materials have been established through various measurements. Differently from isotropic s-wave superconductors, d-wave pairing states have an internal phase of the pair potential. The internal phase modifies the surface states due to the interference effect of the quasiparticles. Along these lines, a novel formula of tunneling spectroscopy has been presented that fully takes into account of the anisotropy of the pair potential. The most essential difference of this formula from conventional ones is that it suggests the phase-sensitive capability of tunneling spectroscopy. The formula suggests that the symmetry of the pair potential is determined by the orientational dependence measurements of tunneling spectroscopy. Along these lines, several experiments have been performed on high-Tc superconductors. The observation of the zero-bias conductance peaks (ZBCP) on Y Ba2Cu3O7 − δstrongly suggests the dx2 − y2-wave pairing states of hole-doped high-Tc superconductors. On the other hand, the absence of ZBCP on (electron-doped)Nd1.85Ce0.15CuO4 − δindicates that the pair potential of this material is a nodeless state. In this paper, recent developments of tunneling spectroscopy for anisotropic superconductors are reviewed both on theoretical and experimental aspects.

New gap equation for strongly correlated metals

Assuming a phenomenological self-energy Im Σ( ω)∼| ω| β , ( β=1), which becomes gapped below T c, we derived a new gap equation. The new gap equation contains the effect of the kinetic energy gain upon developing a superconducting order parameter. However, this new kinetic energy gain mechanism works only for a repulsive pairing potential leading to a s-wave state. In this case, compared to the usual potential energy gain in the superconducting state as in the BCS gap equation, the kinetic energy gain is more effective to easily achieve a high critical temperature T c, since it is naturally Fermi energy scale. In view of the experimental evidences of a d-wave pairing state in the hole-doped copper-oxide high- T c superconductors, we discuss the implications of our results.

Low Field Anomaly in the Specific Heat ofs-Wave Superconductors due to the Expansion of the Vortex Cores

The magnetic field dependence of the electronic specific heat C(H) in the s-wave superconductor NbSe_2 shows curvature at low fields, resembling the near H^1/2 term in C(H) which has been reported in high-T_c superconductors and attributed to a d-wave pairing state. In NbSe_2 we find that the low field behavior in C(H) is described quantitatively by the expansion of vortex cores and the field dependence of the magnetic induction B above H_c1. The associated change in the density of quasiparticle states localized in the vortex cores provides a simple explanation for the "low-field anomaly" in C(H) observed in s-wave superconductors.

Surface states and tunneling spectroscopy of high-Tcsuperconductors

Recent studies of high- Tcsuperconductors have clarified new aspects of tunneling spectroscopy. The unconventional pairing states, i.e. d-wave symmetry in these materials have been established through various measurements. Differently from isotropic s-wave superconductors, d-wave pairing states have an internal phase of the pair potential. The internal phase modifies the surface states due to the interference effect of the quasiparticles. Along these lines, a novel formula of tunneling spectroscopy has been presented that fully takes into account of the anisotropy of the pair potential. The most essential difference of this formula from conventional ones is that it suggests the phase-sensitive capability of tunneling spectroscopy. The formula suggests that the symmetry of the pair potential is determined by the orientational dependence measurements of tunneling spectroscopy. Along these lines, several experiments have been performed on high-Tc superconductors. The observation of the zero-bias conductance peaks (ZBCP) on Y Ba2Cu3O7−δstrongly suggests the dx2−y2-wave pairing states of hole-doped high-Tc superconductors. On the other hand, the absence of ZBCP on (electron-doped)Nd1.85Ce0.15CuO4−δindicates that the pair potential of this material is a nodeless state. In this paper, recent developments of tunneling spectroscopy for anisotropic superconductors are reviewed both on theoretical and experimental aspects.

Unconventional pairing in anomalous superconductors

The existence of unconventional pairing ( p- or d-wave) in the high T c cuprate and heavy fermion superconductors raises many important questions. The unusual pairing state suggests novel pairing mechanisms, and may also be related to other `anomalous' normal state effects such as heavy fermion behaviour, magnetic fluctuations or the normal state pseudogap. In this context, it is interesting to ask whether other classes of anomalous superconductors also may have p- or d-wave pairing states. Indeed there is now strong evidence for p-wave pairing in Sr 2RuO 4, and some indications of d- or p-wave pairing in some organic superconductors. This article summarises some of the evidence indicating either conventional s-wave, or unconventional p- or d-wave pairing in various anomalous systems including the ruthenates, organics, boro-carbides, bismuthates, fullerenes, and C15s.