Self-consistent calculation of physical quantities for the two-dimensional hubbard model and comparison with cuprate high-T c superconductors

The conserving and self-consistent fluctuation-exchange approximation is used to calculate the quasi-particle self-energy and the dynamic spin susceptibility for the two-dimensional Hubbard model at band-fillings n somewhat below half-filling (n =1). The spectral density of the spin susceptibility exhibits incommensurate peaks similar to those observed by neutron scattering in La_2–xSr_xCuO₄. No spin-density-wave instability is found down to the lowest temperatures T (40 K) and up to the largest n (0.85) and couplings U/t y(5) treated in this paper. For the same parameter space we calculate the eigenvalues of the superconducting eigen-solutions withd-wave and extendeds-wave symmetry. These eigenvalues increase as T is decreased and n is increased, however, they are always far below unity which means that one is far from a superconducting transition. The calculated normal-state quantities are in qualitative agreement with the measured quantities of the cuprates, for example, the electrical resistivity is nearly a linear function of T like that of YBa₂Cu₃O₇.     

Spin dynamics and implications for superconductivity: some problems with thed-wave scenario

We review the spin dynamics of the normal state of the cuprates with special emphasis on neutron data in both the YBa₂Cu₃O_7?δ and La_2?x Sr _x CuO₄ systems. When realistic models of the Fermi surface shapes are incorporated, along with a moderate degree of spin fluctuations, we find good semiquantitative agreement with experiment for both cuprates. Building on the success of this Fermi-liquid-based scheme, we explore the implications ford-wave pairing from a number of vantage points. We conclude that our present experimental and theoretical understanding is inadequate to confirm or refute thed-wave scenario.     

Comparison of Superconductivity and Structure for Lanthanum Replacing for Barium and Copper in YBa2Cu3Od

X-ray diffraction studies and the resistivity measurements are used to characterize the structure and the superconductivity of the nominal composition of YBa₂Cu_{3 – x} La _x O _d (YBCLO) cuprates with x 0.30. There was a BaCuO₂ impurity phase detected with x 0.10. The structure of the main phase (123) has the orthorhombic form with Pmmm symmetry in the whole doping range. With increasing content of lanthanum, x, the lattice constants increase for x < 0.04, and decrease for x 0.04. Rietveld refinements for X-ray diffraction show that the dopant of lanthanum substitutes for copper in the lower doping level, and replaces for both barium and copper in the high doping level. The zero-resistance temperature T _c0 first increases with the increase of the content of lanthanum in YBCLO as x 0.04 and then decreases with x as x 0.04. We compared the results with those of La-doped YBa_{2 – z} La _z Cu₃O _y cuprates. The different relationship in superconductivity dependence of lanthanum content may result from the strains due to the different occupancy of lanthanum in the unit cell of YBa₂Cu₃O _d .     

Spin Properties of Electrons in Low-Dimensional SiGe Structures

Using ESR, we investigate g-factor and spin coherence time of electrons confined in 2D Si_1–xGe_x {channels} (x < 0.1) by barriers with x > 0.2 and in SiGe quantum dots grown on prepatterned Si substrates. The quantum wells exhibit 2D-anisotropy of both g and which can be explained in terms of the Bychkov–Rashba field. The latter increases with increasing Ge content in the well indicating that the increasing spin-orbit coupling is more important than interface properties. The narrow ESR permits selective spin manipulation already for x > 0.02. Large, regular arrays of Ge quantum dots (about 10⁹) were grown on prepatterned substrates. Strain in the Si capping layer lowers the conduction band relative to that of Ge causing confinement. The g-shift observed implies the possibility of g-tuning by confinement. The line width shows substantial inhomogeneous broadening whereas the longitudinal spin lifetime is hardly changed with respect to 2D structures.     

Observation of Three Characteristic Energy Scales (Energy Gaps) in the Conductance of Underdoped YBa2Cu3O6+x Junctions

Conductance measurements in underdoped YBa₂Cu₃O_6+x (YBCO) tunneling junctions in the a–b plane are reported which reveal that in addition to the dominant d_x ^{2-y 2}-wave component of the order parameter (OP), a small is (or id_xy) component is also present, and both overlay a broad pseudogap feature. Extensive measurements by other groups from Urbana, Tel-Aviv and Jerusalem. in optimally or overdoped YBCO films showed a splitting of the zero bias peak. This was interpreted as due to the existence of a small imaginary component in the OP of YBCO. We show that also in underdoped YBCO there is an additional small is component in the OP. In ramp type (edge) junctions, oriented along different directions in the a–b plane, we find that the _x ^{2-y 2}-wave gap is suppressed in the region near the node, but has a non zero value of =3.0±0.5 mV at the node itself. This result supports the existence of a sub-dominant is component in the order parameter of underdoped YBCO.     

Magnetic order and evolution of the electronic state aroundx=0.12 in La2?xBaxCuO4 and La2?xSrxCuO4

Muon spin rotation ( ⁺SR) measurement provides clear evidence of the antiferromagnetic order of Cu moments below 35 K for La_2–x Ba _x CuO₄ and below 15 K for La_2–x Sr _x CuO₄ in the narrow range ofx where the high-T _c superconductivity (SC) is suppressed remarkably. The results suggest that the change of the electronic state coupled with the lattice instability is relevant to the local suppression of SC and freezing of spin fluctuations of the Cu moment.     

Point contact spectroscopy in oriented La2?xSrxCuO4 superconductors; energy gap and Fermi velocity

Point contact measurements in oriented La_2–xSr_xCuO₄ samples were performed using metal tips. The current-voltage curves measured along the CuO plane direction (ab) are characteristic of the Andreev reflection phenomenon. The superconducting energy gap in theab plane is determined, _ab = 6± 1 meV, with a possibility for the existence of a lower subgap in the plane. A lower limit for the Fermi velocity in the CuO planes is also set by the measurements;V _F6×10⁷ cm/sec, which is significantly higher than the average velocity obtained by band calculations. The results are discussed in the context of different models for superconductivity in the layered oxides. In particular, we raise the possibility of an anisotropic gap parameter which may indicate a nonstandards-wave pairing in La_2–xSr_xCuO₄. A comparison with previous results obtained on YBa₂Cu₂O_7– is made.     

Critical Behavior in the Low-Energy Spin Fluctuations in High-Tc Superconductors

We study the spin fluctuations of pure and Zn-substituted La_{2 – x} Sr _x CuO₄ using the muon spin relaxation technique. Superconductivity is found to coexist with low-frequency spin fluctuations over a large region of the superconducting phase diagram. The characteristic temperature of spin fluctuations detected by SR decreases with increasing x and vanishes above a critical doping x _c 0.19. This value of x _c coincides with the doping at which the normal state pseudogap extrapolates to zero. These results are discussed in terms of a quantum transition that separates the superconducting phase diagram of high-temperature superconductors into two distinct ground states.     

Comparison of Superconductivity and Structure for YBa2Cu3Oy with Potassium and Sodium Doping

The nominal composition of YBa_2–x M _x Cu₃O _y (M = K, Na) cuprates with x 0.30 were synthesized by the standard solid-state reaction technique. X-ray diffraction (XRD) and the resistivity measurements were used to characterize the structure and the superconductivity of the doped cuprates. There was no impurity phase detected within the whole doping range. The structure of the main phase (123) has the orthorhombic with P _mmm symmetry. With increasing the content of dopants, the lattice constants and some other structural parameters are almost unchanged for M = K, whereas they changed for M = Na. The refined contents of dopants are consistent with that of the nominal ones. The zero resistance temperature T _c0 decreases sharply with the increase of the content of potassium in potassium-doped samples as x 0.20. For sodium-doped YBa_2–z Na _x Cu₃O _y cuprates, T _c0 varies very little. The difference in superconductivity depression may result from the shift oxygen, which transfers conducting carriers from Cu-O chains to Cu-O₂ sheets or the structural stress effect.     

Analysis of the Pairing Interaction in the Cuprates

Thermal difference reflectance (TDR) spectra taken on a large number of superconducting cuprates has enabled us to determine the energy dependence and strength of the pairing interaction in each. All show strong contributions from the phonons and a smaller, but significant contribution from an electronic transition near 1.7 eV. Recent improvements in the signal-to-noise ratio have revealed that the electronic excitation is accompanied in all cases by a weaker companion about 0.5 eV lower in energy. No other contributions are found. We identify these transitions as the d _z2 to d _x2–y2 excitations of the Cu ion in the 3d ⁹ and 3d ⁸ states, respectively, and have calculated the coupling strength mediated by each. The calculated values agree in order of magnitude with the observed strengths. We find that, in addition to the direct electronic coupling that corresponds to an electron–phonon-like term, the overlap between the oxygen and the copper orbitals leads to an exchange term. The direct term couples only to an s-wave gap, as for phonons, while the exchange term couples s-wave to d-wave, and vice versa. We discuss the consequences of this result.     

Symmetries of Vortex Lattice Solutions of the Bogoliubov–de Gennes Equation in a Two-Dimensional Square Lattice

This report describes symmetry properties of tetragonal vortex lattice solutions of the Bogoliubov–de Gennes equation in a two-dimensional square lattice in a uniform magnetic field. The invariance group of a tetragonal vortex lattice solution is expressed in a form of G _(l) = (e + tC _2x ) (l = 0, 2, ± 1), where tC _2x is a space rotation around the x-axis accompanied with time reversal, is a kind of fourfold rotation group, and L is the magnetic translational group of the vortex lattice state. We give a new, refined definition of local symmetric order parameters (OPs) (s-wave, d-wave, and p-wave), which have a well-defined nature such that the OP (e.g., s-wave OP) at the translated site by a lattice vector (of the vortex lattice) from a site (m, n) is expressed by the OP (e.g., s-wave) at the site (m, n) times a phase factor. Winding numbers around the origin of s-wave and d-wave OPs are obtained for four types of solutions G _(l) (l = 0, 2, ± 1). It is shown that all energy bands of quasiparticles of a vortex lattice state are doubly degenerate.     

Stripe correlations of spins and holes in cuprate superconductors

It is argued that the stripe order of spins and holes found in La_2–x Sr _x NiO₄+ is a useful model for spin and charge correlations in La_2–x Sr _x CuO₄. A direct connection with the cuprates is now established by neutron diffraction evidence for stripe order in La_1.6–x Nd_0.4Sr _x CuO₄ withx=0.12. The experimental work has stimulated new theoretical ideas.It is a pleasure to acknowledge my collaborators, especially J. D. Axe, D. J. Buttrey, N. Ichikawa, J. E. Lorenzo, Y. Nakamura, V. Sachan, B. J. Sternlieb, P. Wochner, and S. Uchida. Work at Brookhaven is supported by the Division of Materials Sciences, U.S. Department of Energy, under Contrast No. DE-AC02-76CH00016.     

d-Hole localization and the suppression of superconductivity in YBa2(Cu1-xZnx)3O7?y

The temperature and Zn concentration dependence of the electrical resistivity, specific heat, magnetic susceptibility, and electron paramagnetic resonance (EPR) spectra of YBa₂(Cu_1–x Zn _x )₃O_7–y withy0.1 has been measured forx0.16. In addition, the temperature and field dependence of the magnetization has been measured for 2<T<300K and 0<H<9.0T, along with the temperature and quasihydrostatic pressure dependence of the electrical resistivity for selected samples for 0<P<13 GPa. The substitution of Zn for Cu in YBa₂Cu₃O_7–y causes a rapid and nearly linear depression of the superconducting transition temperature,T _c , withT _c going to 0 K forx 0.10. YBa₂(Cu_1–x Zn _x )₃O_7–y retains the YBa₂Cu₃O_7-y orthorhombic structure forx0.16 for both the superconducting and nonsuperconducting samples. Initially, the unit cell volume increases nearly linearly with Zn content; however, an abrupt change occurs in the vicinityx=0.8–0.10. Forx<0.10, the temperature dependence of the electrical resistivity,(T), is metallic-like (d/dT>0) and increases gradually with increasing Zn content. However, forx 0.10,(T) becomes semiconductor-like, with a very rapid increase of the resistivity with increasingx. The electrical resistivity, magnetic susceptibility, EPR spectra, and specific heat all indicate that thed-holes associated with the Cu ions become localized in the nonsuperconducting phase,x>-0.10.     

Study of the Order Parameter Symmetry in Overdoped Y1−x Ca x Ba2Cu3O7−δ by Measurement of Andreev Reflection, Evidence for Enhancement of the Sub Dominant Order Parameter Amplitude in High Transparency Contacts

We have measured the differential conductance of Au/Y_1–x Ca _x Ba₂Cu₃O_7– point contacts in the regime where it is dominated by Andreev reflection, which enhances its value at low bias. We find that the characteristics can not be fitted by a pure d-wave Order Parameter (OP). Using the formalism developed by Kashiwaya and Tanaka, the best fits are obtained by adding a sub-dominant imaginary OP, whose amplitude appears to depend on the transparency of the contact. At high transparencies it can reach up to 60% of the total amplitude of the OP, for lower transparencies it is substantially smaller. We attribute this enhancement, at high transparencies to a proximity effect between the Au tip and the superconducting electrode.     

Synthesis and Superconductivity of New Cuprates (Pb0.8W0.2)Sr2(Nd1 ? xCax)Cu2O7 ? δ

New cuprates with nominal composition (Pb_0.8W_0.2)Sr₂(Nd_{1 – x} Ca _x )Cu₂O_{7 –} (0x1) were synthesized by solid-state reaction in N₂. The crystal structure was characterized by X-ray powder diffraction (XRD) as tetragonal. Direct current electrical resistance measurements were applied to check the existence of superconductivity in these cuprates. Superconductivity with T _c (onset) up to 82 K is observed when x = 0.6. Synthesis in N₂ is necessary in obtaining superconductivity in these cuprates. Preparation in air or post-treatment in flowing oxygen destroys superconductivity. A comparison is made with previous Pb-based 1212 superconducting oxides. The valence of Pb and the possible position of W in the lattice are discussed.     

Structure, Phonon Vibration, and Spin Correlation of LaBa2Cu3?xCoxOy

Polycrystalline samples LaBa₂Cu_3–xCo _x O _y (0 x 1.0) were synthesized by solid state reaction method. The structure, phonon vibration, conduction, and spin correlation were investigated by means of X-ray diffraction, infrared spectra, resistivity, and electron spin resonance. It is found that there are orthorhombic–tetragonal and tetragonal–orthorhombic structural transitions with Co doping, and the conduction behavior changes from metallic to semiconducting. With the increase of Co content, the Cu(1)—O(1) phonon mode around 531 cm^–1 softens, the Cu(2)—O(2) phonon mode around 657 cm^–1 hardens, and the Cu(1)—O(4) mode around 583 cm^–1 is nearly unchanged. The Cu²⁺ spins tend to localize with Co doping. The changes in structure, phonon vibration, and spin correlation with Co doping are analyzed and discussed.     

Cuprates and Ruthenates: Similarities and Differences

There is growing evidence that the p-wave superconductivity of Sr₂RuO₄ occurs primarily in the planar -band. Thus the minimum model for both cuprates and ruthenates is a single active band with onsite Coulomb interactions. Recent renormalization group analysis shows that such a model can show singlet d-wave or triplet p-wave pairing. The energy of the van Hove singularity in the band and the shape of the Fermi surface are the decisive factors at weak to moderate interaction strengths.     

Spin Properties of Electrons in Low-Dimensional SiGe Structures

Using ESR, we investigate g-factor and spin coherence time of electrons confined in 2D Si_1–xGe_x {channels} (x < 0.1) by barriers with x > 0.2 and in SiGe quantum dots grown on prepatterned Si substrates. The quantum wells exhibit 2D-anisotropy of both g and which can be explained in terms of the Bychkov–Rashba field. The latter increases with increasing Ge content in the well indicating that the increasing spin-orbit coupling is more important than interface properties. The narrow ESR permits selective spin manipulation already for x > 0.02. Large, regular arrays of Ge quantum dots (about 10⁹) were grown on prepatterned substrates. Strain in the Si capping layer lowers the conduction band relative to that of Ge causing confinement. The g-shift observed implies the possibility of g-tuning by confinement. The line width shows substantial inhomogeneous broadening whereas the longitudinal spin lifetime is hardly changed with respect to 2D structures.