S-wave Order Parameter Research Articles

What is the Symmetry of the High-Tc Order Parameter?

In recent years, there has been a raging controversy regarding the orbital symmetry of the superconducting order parameter (OP) in the high temperature superconductors. Many experiments were interpreted in terms of a dx2-y2-wave OP, but many others were interpreted in terms of a more conventional s-wave OP. We review the problems of both intrinsic and extrinsic natures with the phase-sensitive experiments on YBCO. We further show that the photoemission experiments of the purported superconducting gap in Bi2Sr2CaCu2O 8+δ are entirely consistent with charge- and/or spin-density wave formation in that material. The presence of such density waves greatly complicates the analysis of most experiments. Hence, we conclude that the orbital symmetry of the superconducting OP is still unknown in any of the high temperature superconductors.

Ginzburg-Landau Equations for (d+s)-Wave Superconductors in a Non-Fermi Liquid

The Ginzburg-Landau equations are microscopically derived for an anisotropic (d + s)-wave superconductor by assuming a non-Fermi-liquid normal state characterized by a nonvanishing exponent α. The transition temperature Tc as functions of α and the anisotropic parameter g which measures the ab anisotropy is studied. It is found that the anisotropic effect, which results in the enhancement of Tc and the nonzero bulk value of the s-wave order parameter, is strengthened due to the non-Fermi behavior.

Group Theoretical Analysis of the Vortex Lattice States in the Attractive Hubbard Model

This paper describes a group theoretical classification of superconducting states (vortex lattice states) in a unifom magnetic field of the extended Hubbard model with on-site attrac­ tion (U < 0), or nearest-neighbor attraction (V < 0) on a two-dimensional square lattice. Using symmetries of the magnetic translation and the tetragonal rotation of the system, we obtain invariance groups of vortex lattice states. When the magnetic flux <p through a unit cell of the crystal lattice is <PO/p2 (<Po = ch/2e is the flux quantum, p is an integer) we obtained three types of symmorphic tetragonal vortex lattice states, in which the center of point rotations is at a lattice site of the crystal lattice, corresponding to the irreducible representation of the tetragonal symmetry D 4 . For these three states, locations and winding numbers of vortex lines are determined by the symmetries of their invariance group. From these three symmorphic states we derived three (two) types of non-symmorphic vortex lat­ tice states in which the center of point rotations is at the center of a plaquette (link) of the crystal lattice. We show that d-wave, s-wave and p-wave local order parameters inevitably coexist due to the invariance magnetic translation group L of the vortex lattice state. From the irreducible projective representations (two-dimensional) of L, we show that all energy bands of the quasiparticles are at least dOD bly degenerate.

S + D Symmetry Order Parameters in Interband Superconductivity

The anisotropic order parameter case of the two-band superconductivity model with driving pair scattering between overlapping bands is investigated. The s + d-type interaction used can lead to mixed and also pure d- or s-wave order parameters. The behaviour of T c and order parameter symmetry change with chemical potential are illustrated. Also the dependences of gap symmetry components on temperature and chemical potential have been obtained. The singularities in the electronic spectra are reflected in T c .

Vortex Lattice Structure in adx2-y2-Wave Superconductor

The vortex lattice structure in a d_{x^2-y^2}-wave superconductor is investigated near the upper critical magnetic field in the framework of the Ginzburg Landau theory extended by including the correction terms such as the higher order derivatives derived from the Gor'kov equation. On lowering temperature, the unit cell shape of the vortex lattice gradually varies from a regular triangular lattice to a square lattice through the shape of an isosceles triangle. As for the orientation of the vortex lattice, the base of an isosceles triangle is along the a axis or the b axis of the crystal. The fourfold symmetric structure around a vortex core is also studied in the vortex lattice case. It is noted that these characteristic features appear even in the case the induced s-wave order parameter is absent around the vortex of the d_{x^2-y^2}-wave superconductivity. We also investigate the effect of the induced s-wave order parameter. It enhances (suppresses) these characteristic features of the d_{x^2-y^2}-wave superconductor when the s-wave component of the interaction is attractive (repulsive).

Supercurrents in c-axis Pb/Bi2Sr2CaCu2O8 tunnel junctions

We investigated Josephson junctions made on the surface of Bi2Sr2CaCu2O8 single crystals with Pb as the counterectrode. Junctions exhibit a clear lead gap at low temperatures as well as a nonzero supercurrent. In microwave fields Shapiro steps appear at multiples of hf/2e showing that the supercurrent is of first order. Consequences for a potential s-wave order parameter component in Bi2Sr2CaCu2O8 are discussed.

Temperature and angular dependence of the upper critical field of UPd 2Al 3 thin films

Temperature and angular resolved measurements of the upper critical field B c2 of UPd 2Al 3 thin films are presented and compared with new theoretical calculations for weak-coupling s- and d-wave superconductors in the clean limit. In order to take account of the observed anisotropy of the initial slopes near T c , B′ c2 = −4.9 T/K for B‖ c and B′ c2 = −5.8 T/K for B⊥ c , and the crossing of the phase boundaries in the temperature dependence of B c2 near T ∗ ≈ 0.75 T/T c , the calculations include anisotropic orbital and paramagnetic pair breaking. Recent speculations upon the occurence of a non-uniform superconducting “Fulde-Ferrell-Larkin-Ovchinnikov”-state are also considered and incorporated in the calculations. In the region around T ∗ a minimum between B‖ c and B⊥ c in the out-of-plane angular dependence of the upper critical field B c2( Θ) is found experimentally. This cannot be explained by an s-wave order parameter and reduces drastically the possible d-wave functions. To explain the overall data, a mixing of s-wave and a particular d-wave order parameter is proposed.

Superconducting Gap Anisotropy vs Doping Level in High-Tc Cuprates.

We report the results of electronic Raman scattering in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ (Bi2212) and ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ (Tl2201) high-${T}_{c}$ superconductors with variations in the oxygen content. Near optimal doping, both materials show gap anisotropy, with $2\ensuremath{\Delta}/{k}_{B}{T}_{c}$ values of 7.2 ( ${B}_{1g}$) vs 5.8 ( ${A}_{1g}$) in Tl2201 and 8.5 ( ${B}_{1g}$) vs 6.2 ( ${A}_{1g}$) in Bi2212. However, overdoped samples exhibit a symmetry independent gap with $2\ensuremath{\Delta}{/k}_{B}{T}_{c}$ ranging from 5.2 for Bi2212 ( ${T}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}57\mathrm{K}$) to 3.9 in Tl2201 ( ${T}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}37\mathrm{K}$). We compare the data with calculations using both isotropic s-wave and d-wave order parameters.

Vortex state in a d-wave superconductor.

We discuss the physics of the vortex state in a d-wave superconductor, using the phenomenological Ginzburg-Landau theory, where many unusual phenomena arise from the small admixture of the s-wave component induced by spatial variations in the dominant d wave. Properties of an isolated vortex and of the Abrikosov vortex lattice are studied by means of analytic and numerical methods. An isolated vortex has a considerable structure, with four ``extra'' nodes in the s-wave order parameter symmerically placed around the core and an amplitude forming a four-lobe profile decaying as 1/${\mathit{r}}^{2}$ at large distances. The supercurrent and magnetic-field distributions are also calculated. The Abrikosov lattice is in general oblique with the precise shape determined by the magnetic field and s-d mixing parameter ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{v}}$. The magnetic-field distribution in the Abrikosov state has two nonequivalent saddle points resulting in the prediction of a double peak line shape in \ensuremath{\mu}SR and NMR experiments as a test of a d-wave symmetry. Detailed comparison is made with existing experimental data and experiments are proposed to test for the predicted effects. \textcopyright{} 1996 The American Physical Society.

Renormalization of the electron-phonon interaction by strong electronic correlations in high-Tc superconductors.

The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt\ensuremath{'} model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard's X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt\ensuremath{'} models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\ensuremath{\alpha}}}^{2}$F in general: The full-symmetric component ${\mathrm{\ensuremath{\alpha}}}^{2}$${\mathit{F}}_{1}$ is in the t model somewhat, in the tt\ensuremath{'} model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\ensuremath{\alpha}}}^{2}$${\mathit{F}}_{\mathit{i}}$ with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\ensuremath{\alpha}}}^{2}$${\mathit{F}}_{1}$ near half-filling.Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt\ensuremath{'} models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-${\mathit{T}}_{\mathit{c}}$ compounds. \textcopyright{} 1996 The American Physical Society.

Spontaneous generation of s-wave component in a purely d-wave superconductor

We have derived microscopically the Ginzburg-Landau equations in a superconductor with d x 2 − y 2 symmetry. We have found that s-wave order parameter is spontaneously generated in a purely d-wave superconductor whenever d-wave order parameter is inhomogeneous. Examples presented are the structure of a single vortex and superconductor boundary. We have also obtained the time dependent Ginzburg-Laudau equations for such a superconductor.

Effect of impurity scattering on a (d+s)-wave superconductor.

Recent measurements of the surface impedance in ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ single crystals show large in-plane anisotropy which implies that orthorhombicity due to the Cu-O chains is not small. In addition, the controversy between the d-wave and the s-wave order parameter persists. In the framework of the orthorhombic symmetry, an order parameter with (d+s)-wave symmetry is considered in an attempt to understand current experimental results. We provide a detailed study of the characteristics of such an order parameter in the low-temperature behavior of the penetration depth, the density of states, and the ${\mathit{T}}_{\mathit{c}}$ suppression in the presence of nonmagnetic impurity scattering. The results are discussed in comparison with the d-wave and the anisotropic s-wave cases.

Penetration-depth calculations in the ab and c directions in a layered S/N superconductor.

We present the results of calculations of the penetration depths ${\ensuremath{\lambda}}_{\mathit{a}\mathit{b}}$ and ${\ensuremath{\lambda}}_{\mathit{c}}$ (the subscripts refer to the direction of the screening currents). Our model is a layered superconducting/normal metal (S/N) model, in which the two types of layers are stacked in alternating fashion. The S and N layers are coupled in a coherent fashion and the N layer is driven superconducting by a proximity effect. We calculate the penetration depths for both d-wave and s-wave order parameters for a range of interlayer coupling strengths, and we discuss the effect that the interlayer coupling has on the temperature dependence of the penetration depths. We finish by comparing our results with experimental observations of ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$.

Effect of pair-breaking on the superconducting fluctuation contribution to the NMR relaxation rate

We investigate the effect of pair-breaking on the contribution of s-wave order parameter fluctuations to the NMR relaxation rate for clean 2D superconductors. For Gaussian fluctuations we evaluate the Maki-Thomson diagram taking into account pairing breaking in both the fluctuation propagator and the Green's functions. The fluctuation peak in 1/T1 near Tc is strongly reduced by pair-breaking as is the case for the Hebel-Slichter peak.

Local density of states near an impurity in unconventional superconductors

A nonmagnetic impurity can break Cooper pairs in unconventional superconductors and perturb the density of states out to the distance of order the coherence length. We compute the density of states as a function of position for the d-wave, p-wave and anisotropic s-wave order parameters which have often been considered in the high temperature superconductors. We find that the local density of states near an impurity oscillates with the Tomasch wavelength which depends strongly on the frequency and the symmetry of order parameters. This is, in principle, detectable perhaps through the scanning tunneling microscopy and could provide a crucial information on the symmetry of the underlying order parameters.

Can impurity effects help to identify the symmetry of the order parameter of the cuprates?

The effects of non-magnetic impurities on the properties of superconductors with order parameters (OP) of different symmetry are discussed. In particular, we contrast the case of a d x 2− y 2 with various forms of an anisotropic s-wave (ASW) gap. The biggest qualitative difference occurs if the phase of the s-wave OP does not change sign.

Superconducting gap and order parameter in Bi2Sr2CaCu2O8+x.

Using angle-resolved photoemission, we observed a nonzero superconducting gap and the presence of a superconducting condensate along all three major symmetry directions. We find that the gap is highly anisotropic, with a gap as small as 1--2 meV along \ensuremath{\Gamma}-Y, 4--8 meV along \ensuremath{\Gamma}-X, and 14--20 meV along \ensuremath{\Gamma}-M. We argue that our data imply either an anisotropic s-wave order parameter or an unconventional, two-component order parameter, and that measuring the size of the gap does not by itself distinguish between the two possibilities. We propose a phenomenological unconventional order parameter, and note the quantitative agreement between theory and experiment.

Superconductivity in Two-Band Model

Superconductivity in two-band models involving non s·wave pairing is investigated. We con­ sider the following case: one band has s-wave pairing interaction and the other has non s-wave pairing interaction, and only s-wave Cooper pairs can be transferred between bands. Possible appearance of non s-wave superconducting order parameter in an s·wave superconducting state and its onset temperature are discussed. The non s·wave order parameter is suppressed by s-wave supercon­ ductivity, and cannot appear unless the non s·wave pairing interaction is stronger than critical value. We also consider the appearance of s-wave superconducting order parameter in a non s-wave superconducting state. In this case, the s-wave order parameter can always appear on lowering temperature. Temperature dependence of the order parameters and the specific heat are shown in a simple case.

Phenomenological model of the copper oxide superconductors.

We propose a phenomenological model for the copper oxide superconductors in which one complex s-wave order parameter (OP) is associated with each of the N conducting layers per unit cell, with N-1 equal spacings d and one different spacing d'; the c axis repeat distance s=d'+(N-1)d. The layers are coupled by Josephson-like tunneling, with parameters ${\ensuremath{\zeta}}_{1}$ and ${\ensuremath{\zeta}}_{2}$, respectively. The Gaussian fluctuation free energy is diagonalized, yielding N distinct ${T}_{c}$ values. Just above the highest ${T}_{c}$, the fluctuations are usually dominated by the three-dimensional (3D) regime of a single cellular OP. In the 2D regime further above ${T}_{c}$, more of the OP's contribute to the fluctuations, their relative contributions depending upon the ${\ensuremath{\zeta}}_{1}$ and ${\ensuremath{\zeta}}_{2}$ values. The temperature (T) and angular (\ensuremath{\theta}) dependence of the resulting fluctuation magnetization M(\ensuremath{\theta},T) is calculated.In a weak magnetic field B [B${B}_{0}$=${\ensuremath{\varphi}}_{0}$/(${\mathrm{sv}}_{F}$${\ensuremath{\tau}}_{\ensuremath{\varphi}}$), where ${\ensuremath{\varphi}}_{0}$, ${v}_{F}$, and ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ are the flux quantum, intralayer Fermi velocity, and phase coherence lifetime, respectively], the susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is diagonal in the crystal representation, resulting in an ordinary (anisotropic mass) \ensuremath{\theta} dependence at fixed T near ${T}_{c}$. At very high fields in very high-quality single crystals (${\ensuremath{\omega}}_{c}$${\ensuremath{\tau}}_{\ensuremath{\varphi}}$\ensuremath{\gg}1, where ${\ensuremath{\omega}}_{c}$ is the pair cyclotron frequency), M is best evaluated in the field representation. The component ${M}_{\mathrm{B}}$(\ensuremath{\theta},T) exhibits anomalous oscillations in its \ensuremath{\theta} dependence, arising from degenerate multiple minima in the pair potential, provided that the effective high-momentum cutoff ${q}_{c}^{\mathrm{*}}$ is sufficiently large. In this field regime, ${q}_{c}^{\mathrm{*}}$ is either on the order of \ensuremath{\pi}s/a, where a is the intralayer oxygen site repeat distance, or equal to \ensuremath{\pi}[1+(B/${B}_{0}$)sin\ensuremath{\theta}].The oscillations are similar to de Haas--van Alphen oscillations in the B dependence of the normal state M. They are broadened by local, clean-limit dynamic effects but should be observable for ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$${k}_{B}$${T}_{c}$/\ensuremath{\Elzxh}\ensuremath{\gtrsim}\ensuremath{\gtrsim}\ensuremath{\gtrsim}1 and N=2, which we argue is the case in the best samples. In addition, the fluctuation specific heat (FSH) and Aslamazov-Larkin conductivity are calculated for B=0, including dynamic effects. For arbitrary N, the FSH above ${T}_{c}$ is found to be proportional to ${\ensuremath{\chi}}_{\mathrm{cc}}$(T)/T. For fits of the theoretical FSH to experimental data, it is necessary to modify the mean-field expressions for T${T}_{c}$, such that the entropy of the superconducting transition remains zero. An example of such a modification is given. All fluctuation quantities exhibit dimensional crossover (DCR) from 3D behavior near ${T}_{c}$ to 2D behavior further from ${T}_{c}$. For the fluctuation diamagnetism the DCR temperature ${T}_{0}$(\ensuremath{\theta}) depends strongly upon \ensuremath{\theta} in the vicinity of \ensuremath{\pi}/2, where it diverges. Away from ${T}_{c}$ dynamic effects are found to be important and can be so strong as to mimic DCR behavior, even for \ensuremath{\theta}=\ensuremath{\pi}/2, for which DCR should not occur. The dynamic effects make quantitative corrections to the fluctuation quantities for T as close as 1 K to ${T}_{c}$. For N=2 detailed plots of the above fluctuation quantities for a range of the microscopic parameters are presented.

Fluctuation specific heat and vortex configurations in model high-temperature superconductors

We calculate the fluctuation specific heat and diamagnetic susceptibility of a single layer of model high-temperature superconductor with parameters resembling YBa2Cu3O7−δ, as a function of perpendicular magnetic field, assuming an s-wave order parameter and fluctuations about a Ginzburg-Landau mean-field theory. Snapshots of the vortex configurations in a perpendicular field are also shown.