Mixed Order Parameter Research Articles

Singlet-triplet mixing in the order parameter of the noncentrosymmetric superconductor Ru7B3

One of the key effects which is predicted to arise in superconductors without a centre of inversion is the mixing of singlet and triplet order parameters, which are no longer good quantum numbers on their own due to parity. We have probed the gap structure in the noncentrosymmetric superconductor Ru$_7$B$_3$, through small-angle neutron diffraction from the vortex lattice, in order to search for the proposed mixed order parameter. We find that the measured temperature dependence of the vortex-lattice form factor is well characterised by a model constructed to describe the effects of broken inversion symmetry on the superconducting state, indicating the presence of a mixed singlet-triplet gap and confirming the theoretical predictions.

Temperature dependence of London penetration depth anisotropy in superconductors with anisotropic order parameters

We study the effects of anisotropic order parameters on the temperature dependence of London penetration depth anisotropy $\gamma_\lambda(T)$. After MgB$_2$, this dependence is commonly attributed to distinct gaps on multi-band Fermi surfaces in superconductors. We have found, however, that the anisotropy parameter may depend on temperature also in one-band materials with anisotropic order parameters $\Delta(T,k_F)$, a few such examples are given. We have found also that for different order parameters, the temperature dependence of $\Delta(T)/\Delta(0)$ can be represented with good accuracy by the interpolation suggested by D. Einzel, J. Low Temp. Phys, {\bf 131}, 1 (2003), which simplifies considerably the evaluation of $\gamma_\lambda(T)$. Of particular interest is mixed order parameters of two symmetries for which $\gamma_\lambda(T)$ may go through a maximum for a certain relative weight of two phases. Also, for this case, we find that the ratio $\Delta_{max}(0)/T_c$ may exceed substantially the weak coupling limit of 1.76. It, however, does not imply a strong coupling, rather it is due to significantly anisotropic angular variation of $\Delta$.

Near-degeneracy of extendeds+dx2−y2anddxyorder parameters in quasi-two-dimensional organic superconductors

The symmetry of the superconducting order parameter in quasi-two-dimensional BEDT-TTF organic superconductors is a subject of ongoing debate. We report ab initio density functional theory calculations for a number of organic superconductors containing $\kappa$-type layers. Using projective Wannier functions we derive parameters of a common low-energy Hamiltonian based on individual BEDT-TTF molecular orbitals. In a random phase approximation spin-fluctuation approach we investigate the evolution of the superconducting pairing symmetry within this model and point out a phase-transition between extended $s + d_{x^2-y^2}$ and $d_{xy}$ symmetry. We discuss the origin of the mixed order parameter and the relation between the realistic molecule description and the widely used dimer approximation. Based on our ab initio calculations we position the investigated materials in the obtained molecule model phase diagram and simulate scanning tunneling spectroscopy experiments for selected cases. Our calculations show that many $\kappa$-type materials lie close to the phase transition line between the two pairing symmetry types found in our calculation, possibly explaining the multitude of contradictory experiments in this field.

Superconductivity and magnetism in noncentrosymmetric RhGe

RhGe synthesized at high pressure crystallizes in the noncentrosymmetric B20 cubic structure. Measurements of electrical resistivity and magnetization demonstrate a superconducting state below Tc ∼ 4.3 K and weak ferromagnetism below Tm ∼ 140 K. Specific heat data confirm the bulk nature of superconductivity in this ferromagnetic superconductor. Point contact spectra in the superconducting state exhibit the complex Andreev reflection signal which may indicate a mixed singlet-triplet order parameter. The superconducting region forms a dome on the P-T diagram with a maximum of Tc near 4 GPa. It was also found that the superconductivity of RhGe4 is suppressed at high pressure in contrast with RhGe. Ab initio simulations suggest that the observed weak magnetization emerges from the pronounced toroidal-like spin polarization with magnetic quadrupole and toroidal moments located at Rh and Ge sites.

Domain walls in superconductors: Andreev bound states and tunneling features

Domain walls can be formed in superconductors with a discrete degeneracy of the ground state, for instance, due to the breaking of time reversal symmetry. We study all cases where the formation of domain walls is possible in a tetragonal superconductor with the point group ${D}_{4h}$. We discuss both triplet and mixed singlet order parameters. It is found that in all cases the domain walls support subgap Andreev bound states, whose energies strongly depend on the direction of semiclassical propagation. The bound state contribution to the density of quasiparticle states exhibits peculiar features, which can be observed in tunneling experiments.

Pseudospin S=1 description of the cuprate complexity: the charge triplets model

We introduce a minimal model for 2D cuprates with the on-site Hilbert space reduced to only three effective valence centers CuO47-,6-,5- (nominally Cu1+,2+,3+) and make use of the S=1 pseudospin formalism. Despite its seeming simplicity the model is believed to capture the salient features both of the hole- and electron-doped cuprates. The pseudospin formalism elucidates an unique fermion-boson duality of the doped cuprates, does provide an unified standpoint for classification of the "myriad" of electronic phases in cuprates and the evolution of the CuO2 planes under a nonisovalent doping, introduces the on-site mixed valence quantum superpositions and order parameters to be novel features of the cuprate physics, does provide a comprehensive description of the correlated one- and two-particle transport, coexistence of p- and n-type carriers, electron-hole asymmetry, anticorrelation of conventional spin and superconducting order parameters. Concept of the electron and hole centers, differing by a composite local boson, and electron-hole pairing are shown to explain central points of the cuprate puzzles, in particular, the HTSC itself, pseudogap, and Fermi surface reconstruction.

Observation of a Bulk Nodal-Gap in Overdoped Y $$_{0.9}$$ 0.9 Ca $$_{0.1}$$ 0.1 Ba $$_{2}$$ 2 Cu $$_{3}$$ 3 O $$_{7-\delta }$$ 7 - δ Thin Films

Complex conductivity of overdoped Y\(_{0.9}\)Ca\(_{0.1}\)Ba\(_{2}\)Cu\(_{3}\)O\(_{7-\delta }\) thin films was measured in Terahertz frequency using frequency and time domain methods. The films were measured in the frequency range of 3–100 cm\(^{-1}\) and in the temperature range of 20–300 K. Results show a possible deviation from a pure \(d_{x^{2}-y^{2}}\)-wave superconductor, indicated by the existence of an energy sub-gap in overdoped Y\(_{0.9}\)Ca\(_{0.1}\)Ba\(_{2}\)Cu\(_{3}\)O\(_{7-\delta }\) films. Evidence for this sub-gap appears as non-monotonic behavior of \(\sigma _{1}(\omega ,T)\) as a function of frequency followed by a sharp decrease at low frequencies, and a dip in the imaginary part of the optical conductivity multiplied by frequency, \(\omega {{\sigma }_{2}}(\omega ,T)\). The mentioned features were observed at energy of about 1.2 meV in 10 % Ca-doped YBCO thin films. Our complex conductivity spectra are in agreement with the theoretical prediction obtained by using a mixed symmetry order parameter within the Born limit, shown by Schurrer et al. We suggest that these observations are direct evidence for a nodal gap obtained in a \(d_{x^{2}-y^{2}}\)-wave superconductor and can be theoretically clarified by adding an imaginary component as \(is\) or \(i{{d}_{xy}}\) to the main \(d_{x^{2}-y^{2}}\)-wave order parameter.

Low-energy singularities in the ground state of fermionic superfluids

We analyze the effects of order parameter fluctuations on the ground state of fully gapped charge-neutral fermionic superfluids. The Goldstone mode associated with the spontaneously broken symmetry leads to a problem of coupled singularities in $d \leq 3$ dimensions. We derive a minimal set of one-loop renormalization group equations which fully captures the interplay of the singularities. The flow equations are based on a symmetry conserving truncation of a scale dependent effective action. We compute the low energy behavior of longitudinal, transverse and mixed order parameter correlations, and their impact on the fermionic gap. We demonstrate analytically that cancellations protecting the Goldstone mode are respected by the flow, and we present a numerical solution of the flow equations for the two-dimensional attractive Hubbard model.

An atomistic description of the nematic and smectic phases of 4-n-octyl-4′ cyanobiphenyl (8CB)

We report the results of atomistic molecular dynamics simulations of 4-n-octyl-4' cyanobiphenyl (8CB) on samples of 750 and 3000 molecules showing the spontaneous formation of the nematic phase and then of smectic layers by gradually cooling down from the isotropic phase. Orientational, positional, and mixed order parameters, layer spacing, translational diffusion tensor components and their temperature dependence are reported. A detailed comparison with available experimental data validates the model and force field employed and clarifies the molecular organization of this important liquid crystal often used as reference smectic material.

Direct evidence of a bulk nodal gap in the overdoped regime of Y0.9Ca0.1Ba2Cu3O7−δ thin films from THz spectroscopy

We measured the terahertz (THz) complex conductivity of Ca-doped YBa2Cu3O7−δ thin films in the frequency range of 0.1 to 3 THz (3 to 100 cm−1) and in the temperature range of 20 to 300 K. The films were measured using time domain and frequency domain THz spectroscopy methods. We show evidences for the existence of an additional energy sub-gap in overdoped Y0.9Ca0.1Ba2Cu3O7−δ samples. The evidences for this gap appear as a sharp decrease in the real part of the optical conductivity, , and as a dip in the imaginary part of the optical conductivity multiplied by frequency, . These features are observed at a frequency equivalent to an energy of 1.2 meV. Our complex conductivity spectra are in agreement with the theoretical prediction assuming a mixed symmetry order parameter (Schürrer I . et al., Physica C, 303 (1998) 287). We suggest that these observations are direct evidence of a nodal gap obtained in a -wave superconductor when adding an imaginary s or dxy bulk component in the overdoped regime.

Superconductivity without Local Inversion Symmetry; Multi-layer Systems

While multi-layer systems can possess global inversion centers, they can have regions with locally broken inversion symmetry. This can modify the superconducting properties of such a system. Here we analyze two dimensional multi-layer systems yielding spatially modulated antisymmetric spin-orbit coupling (ASOC) and discuss superconductivity with mixed parity order parameters. In particular, the influence of ASOC on the spin susceptibility is investigated at zero temperature. For weak inter-layer coupling we find an enhanced spin susceptibility induced by ASOC, which hints the potential importance of this aspect for superconducting phase in specially structured superlattices.

Multiple interfaces in diffusional phase transitions in binary mesogen-nonmesogen mixtures undergoing metastable phase separations

Theory and simulations of simultaneous chemical demixing and phase ordering are performed for a mixed order parameter system with an isotropic-isotropic (I-I) phase separation that is metastable with respect to an isotropic-nematic (I-N) phase-ordering transition. Under certain conditions, the disordered phase transforms into an ordered phase via the motion of a double front containing a metastable phase produced by I-I demixing, a thermodynamically driven mechanism not previously reported. Different kinetic regimes are found depending on the location of the initial conditions in the thermodynamic phase diagram and the ratio between diffusional and nematic phase-ordering mobilities. For a diffusional process, depending if the temperature is above or below the critical codissolution point, an inflection point or a phase separation takes place in the depletion layer. This phase separation leads to the formation of a second interface where the separation of the two metastable isotropic phases grows monotonically with time. The observed deviations from the typical Fickian concentration profiles are associated with strong positive deviations of the mixture from ideality due to couplings between concentration and nematic ordering. Although systems of interest include liquid-crystalline nanocomposites, this mechanism may apply to any mixture that can undergo an order-disorder transition and demix.

Locally Non-centrosymmetric Superconductivity in Multilayer Systems

Although multilayer systems possess global inversion symmetry, some of the layers lack local inversion symmetry because no global inversion centers are present on such layers. Such locally non-centrosymmetric systems exhibit spatially modulated Rashba spin–orbit coupling. In this study, the superconductivity in multilayer models exhibiting inhomogeneous Rashba spin–orbit coupling is investigated. We study the electronic structure, superconducting gap, and spin susceptibility in the superconducting state with mixed parity order parameters. We show the enhancement of the spin susceptibility by Rashba spin–orbit coupling and interpret it on the basis of the crossover from a centrosymmetric superconductor to a non-centrosymmetric superconductor. It is also shown that the spin susceptibility is determined by the phase difference of the order parameter between layers and is nearly independent of the parity mixing of order parameters. An intuitive understanding is given on the basis of the analytic expression of superconducting order parameters in the band basis. The results indicate that not only a broken global inversion symmetry but also a broken local inversion symmetry leads to unique properties of superconductivity. We discuss the superconductivity in artificial superlattices involving CeCoIn 5 and multilayer high- T c cuprates.

Possible secondary component of the order parameter observed in London penetration depth measurements

We discuss the effect of a secondary component of the superconducting order parameter on the superfluid density in the cuprates. If we assume a main ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ gap, the most stable realization of a mixed order parameter has a time-reversal breaking ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+\ifmmode \imath \else \i \fi{}{d}_{xy}$ symmetry. In this state the nodes are removed and the temperature dependence of the superfluid density changes from the linear behavior of a pure $d$ wave to a more rounded shape at low temperature. The latter is compatible with the behavior experimentally observed in the in-plane magnetic field penetration depth of optimally doped ${\text{La}}_{2\ensuremath{-}x}{\text{Sr}}_{x}{\text{CuO}}_{2}$ and ${\text{YBa}}_{2}{\text{Cu}}_{3}{\text{O}}_{7\ensuremath{-}y}$.

Spin current in topologically trivial and nontrivial noncentrosymmetric superconductors

We study theoretically the surface of time-reversal-symmetric, noncentrosymmetric superconductor with mixed singlet and triplet order parameters. A pair of counterpropagating subgap quasiparticle surface bound states with opposite spin projections are obtained in the nontrivial Z$_2$ case where the triplet component is larger than the singlet one, contributing to a spin current. In contrast to the pure p-wave cases, these subgap states do not have a fixed spin projections but depend on the momenta along the surface. In the trivial Z$_2$ case where the singlet order parameter is larger, no subgap surface bound states show up. In both cases, there is also a finite contribution to the spin current from the continuum states with energies between the two gaps. The method for obtaining the quasiclassical Green's functions associated with the noncentrosymmetric superconductors is also presented.

Evidence of s-wave subdominant order parameter in YBa2Cu3O7−δ from break-junction tunneling spectra

The tunneling spectra of YBa2Cu3O7−δ break junctions have been investigated for tunneling close to the node direction. The behavior of the zero-bias conductance peak (ZBCP) and the Josephson current have been studied as a function of the temperature and magnetic field. The deep splitting of the ZBCP is observed starting at TS<20–30K in agreement with the dx2−y2±is order parameter theory [Y. Tanuma, Y. Tanaka, S. Kashiwaya, Phys. Rev. B 64, 214519 (2001)]. We have observed that a low (0.04T) magnetic field significantly suppresses such splitting. The 1∕T temperature dependence of the maximum Josephson current going to saturation at T<TS also confirms mixed order parameter formation.

Mixed pairing symmetry inκ−(BEDT-TTF)2Xorganic superconductors from ultrasonic velocity measurements

Discontinuities in elastic constants are detected at the superconducting transition of layered organic conductors \kappa-(BEDT-TTF)_{2}X by longitudinal and transverse ultrasonic velocity measurements. Symmetry arguments show that discontinuities in shear elastic constants can be explained in the orthorhombic compound only if the superconducting order parameter has a mixed character that can be of two types, either A_{1g}+B_{1g} or B_{2g}+B_{3g} in the classification of irreducible representations of the orthorhombic point group D_{2h}. Consistency with other measurements suggests that the A_{1g}+B_{1g} (d_{xy}+d_{z(x+y)}) possibility is realized. Such clear symmetry-imposed signatures of mixed order parameters have not been observed in other superconducting compounds.

Type-IV Superconductivity: Can Superconductivity be more Exotic than Unconventional?

Abrikosov vortex phase in s-wave layered superconductor in a parallel magnetic field is shown to be absolutely unstable with respect to the appearance of a triplet component which breaks inversion (parity) and spin-rotational symmetries of Cooper pairs. Symmetry breaking paramagnetic effects are demonstrated to be of the order of unity if the orbital upper critical field, Hc2 (0), is of the order of Clogston paramagnetic limiting field, Hp. We suggest a generic phase diagram of such type-IV superconductor, which is singlet one at H=0 and in the Meissner phase and characterized by mixed singlet-triplet order parameter with broken symmetries of Copper pairs in vortex phase. A possibility to observe type-IV superconductivity in clean organic, high-Tc, and MgB2 superconductors is discussed.

Type-IV superconductivity: Cooper pairs with broken inversion and time-reversal symmetries in conventional superconductors

The vortex phase in a singlet superconductor in the absence of impurities is shown to be absolutely unstable with respect to the appearance of a triplet component that breaks both the inversion and time-reversal symmetries of Cooper pairs. The symmetry breaking paramagnetic effects are demonstrated to be of the order of unity if the orbital upper critical field, H c2(0), is of the order of the Clogston paramagnetic limiting field, H p. We suggest a generic phase diagram of such a type-IV superconductor that is a singlet one at H = 0 and characterized by a mixed singlet-triplet order parameter with broken time-reversal symmetry in the vortex phase. The possibility to observe type-IV superconductivity in clean organic, high-T c, MgB2, and other superconductors is discussed.

Multipolar Moments in Pr-based Filled-Skutterudite Compounds with Singlet–Triplet Crystal-Field Levels

Low-lying singlet–triplet crystal-field levels are realized in Pr-based filled-skutterudite compounds. The properties of multipolar moments in such systems are studied theoretically to elucidate the microscopic origin of their anomalous ordered states. First, we introduce a proper set of multipoles to express the active degrees of freedom in the singlet–triplet subspace. Then, a group-theoretical classification of multipoles is presented to discuss the nature of mixed order parameters in zero and finite magnetic fields. Using a pseudo-spin representation, we construct a generic form of pair interactions and identify the sources of anisotropy for directions of magnetic field. The interactions and the phase diagram of PrOs 4 Sb 12 , which is a typical singlet–triplet system, are discussed in connection with the T h symmetry specific to the filled-skutterudite systems.