Two-band Superconductor Research Articles

Fulde–Ferrell–Larkin–Ovchinnikov States in Two-Band Superconductors

We examine the possible phase diagram in an $H$-$T$ plane for Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states in a two-band Pauli-limiting superconductor. We here demonstrate that, as a result of the competition of two different modulation length scales, the FFLO phase is divided into two phases by the first-order transition: the $Q_1$- and $Q_2$-FFLO phases at the higher and lower fields. The $Q_2$-FFLO phase is further divided by successive first order transitions into an infinite family of FFLO subphases with rational modulation vectors, forming a {\it devil's staircase structure} for the field dependences of the modulation vector and paramagnetic moment. The critical magnetic field above which the FFLO is stabilized is lower than that in a single-band superconductor. However, the tricritical Lifshitz point $L$ at $T_{\rm L}$ is invariant under two-band parameter changes.

Influence of s± symmetry on unconventional superconductivity in pnictides above the Pauli limit – two-band model study

The theoretical analysis of the Cooper pair susceptibility shows the two-band Fe-based superconductors (FeSC) to support the existence of the phase with nonzero Cooper pair momentum (called the Fulde--Ferrel--Larkin--Ovchinnikov phase or shortly FFLO), regardless of the order parameter symmetry. Moreover this phase for the FeSC model with $s_{\pm}$ symmetry is the ground state of the system near the Pauli limit. This article discusses the phase diagram $h-T$ for FeSC in the two-band model and its physical consequences. We compare the results for the superconducting order parameter with s-wave and $s_{\pm}$-wave symmetry -- in first case the FFLO phase can occur in both bands, while in second case only in one band. We analyze the resulting order parameter in real space -- showing that the FeSC with $s_{\pm}$-wave symmetry in the Pauli limit have typical properties of one-band systems, such as oscillations of the order parameter in real space with constant amplitude, whereas with s-wave symmetry the oscillations have an amplitude modulation. Discussing the free energy in the superconducting state we show that in absence of orbital effects, the phase transition from the BCS to the FFLO state is always first order, whereas from the FFLO phase to normal state is second order.

Two-Fluid Description of Two-Band Superconductors

We present a systematic study of the response properties of two-band (multi-gap) superconductors with spin-singlet (s-wave) pairing correlations, which are assumed to be caused by both intraband (\lambda_{ii}, i=1,2) and interband (\lambda_{12}) pairing interactions. In this first of three planned publications we concentrate on the properties of such superconducting systems in global and local thermodynamic equilibrium, the latter including weak perturbations in the stationary long-wavelength limit. The discussion of global thermodynamic equilibrium must include the solution (analytical in the Ginzburg-Landau and the low temperature limit) of the coupled self-consistency equations for the two energy gaps \Delta_i(T), i=1,2. These solutions allow to study non-universal behavior of the two relevant BCS-M\"uhlschlegel parameters, namely the specific heat discontinuity \Delta C/C_N and the zero temperature gaps \Delta_i(0)/\pi k_B T_c, i=1,2. The discussion of a local equilibrium situation includes the calculation of the supercurrent density as a property of the condensate, and the calculation of both the specific heat capacity and the spin susceptibility as properties of the gas of thermal excitations in the spirit of a microscopic two-fluid description. Non-monotonic behavior in the temperature dependences of the gaps and all these local response functions is predicted to occur particularly for very small values of the interband pair-coupling constant \lambda_{12}.

Ginzburg-Landau Analysis of the Critical Temperature and the Upper Critical Field for Three-Band Superconductors

Using the microscopic formalism of Eilenberger equations, a three-band Ginzburg-Landau theory for the intraband dirty limit and clean interband scattering case is derived. Within the framework of this three-band Ginzburg-Landau theory, expressions for the critical temperature Tc and the temperature dependence of the upper critical field H_c2 are obtained. Based on some special cases of the matrix of interaction constants, we demonstrate the influence of the sign of the interband interaction on the critical temperature and the upper critical field as compared with a two-band superconductor where it plays no role. We study also analytically and numerically the effect of its magnitude.

The Critical Temperature of Two-Band Superconductors with Pseudogap

The exact formula of Tc’s equation of two-band s-wave superconductors with Pseudogap in weak-coupling limit are derived by considering the influence of interband interaction. The pairing interaction in each band consisted of 2 parts the electron-phonon interaction and non-electron-phonon interaction are included in our model. It was found that the critical temperature is increased as Pseudogap, phonon coupling of 1st band constant and interband phonon coupling constant increased.

The Specific Heat Jump of Hybridized Two-Band Superconductor

The specific heat jump of the two-band hybridized superconductor was studied. The two-band model is consist of conduction electron band and other-electron band with the upper and lower band of quasi-particle energy spectra occurred by hybridization. The specific heat jump was determined by making the assumption that the conduction electron band having the same energy as other-electron band, and the other-electron band having the energy near the Fermi energy. The analytic formula of specific heat jump of the two-band hybridized superconductor was derived and the effect of hybridization coefficient on the specific heat jump was investigated.

Coexistent symmetries in a two-band organic superconductor

We have revisited the phase diagram of perchlorate Bechgaard salts under magnetic field at small temperature, using a spin ladder model with a Hubbard Hamiltonian. Using a non-perturbative Gor'kov method, we have calculated the free energies of all possible phases, including a phase with coexisting singlet and triplet superconducting pairs. With our choice of coupling parameters, this coexisting phase proved not favorable. Nevertheless, this study opens new fields of investigation, while the search of such a coexisting phase seems a realistic and promising goal.

Rotation of Triangular Vortex Lattice in the Two-Band Superconductor MgB2

To identify the contributions of the multiband nature and the anisotropy of a microscopic electronic structure to a macroscopic vortex lattice morphology, we develop a method based on the Eilenberger theory near \(H_{\text{c2}}\) combined with the first-principles band calculation to estimate the stable vortex lattice configuration. For a typical two-band superconductor MgB2, successive transitions of vortex lattice orientation that have been observed recently by small angle neutron scattering [Das et al.: Phys. Rev. Lett. 108 (2012) 167001] are explained by the characteristic field-dependence of two-band superconductivity and the competition of sixfold anisotropy between the σ- and π-bands. The reentrant transition at low temperature reflects the Fermi velocity anisotropy of the σ-band.

Temperature dependence of the anisotropy parameter of the LiFeAs upper critical field in the two-band Ginzburg-Landau theory

An analytical formula is obtained for the temperature dependence of the anisotropy parameter of the upper critical field of a two-band superconductor in the context of the generalized Ginzburg-Landau theory for layered superconductors. The theoretical results are compared with the experimental data for new superconductor LiFeAs.

Half-flux quanta and vortex-antivortex pairs in two-band superconductors

Vortex states of two-band superconductors are investigated based on the microscopic Ginzburg-Landau model. We show that the nonlocal interband coupling can lead to the breaking of translation symmetry and the axial rotation symmetry of the conventional Abrikosov vortex states, resulting in a novel vortex structure comprising a mixture of triangle and square-like vortex lattices that have no counterparts in the single-band superconductors. Half-quantized vortices and stable vortex-antivortex pairs are found. These results may relate to the experimental observations of anomalous vortex structures in the two-band magnesium diboride superconductor.

On Some Factors Affecting the Critical Temperature in a Two-Band Model of Superconductivity

The influence of a few factors on the critical temperature in a two-band superconducting system is investigated. The system contains conventional Cooper pairs from a wider band (the d-band) as well as local pairs (local bipolarons) from a narrower band (the f-band), which are induced via a pair-exchange potential. These factors are the Coulomb repulsion between f-electrons, the position of the f-band with respect to the bottom of the d-band and two kinds of hopping in the f-band: a single-polaron hopping and a pair hopping. The Coulomb potential turns out to lower the critical temperature from higher values to the pure BCS one. Each of the kinds of hopping is treated by making use of perturbation theory. Pair hopping is incorporated as the first order correction but the hopping of single polarons enters as the second order one. Each of them increases this temperature on its own, however, the hopping of single polarons makes it stronger. The position of the f-band that corresponds to the site energy of a f-electron poses a very interesting case. There appear two peaks at two values of that energy meaning a rapid increase of the critical temperature due to the strong effect of the presence of local electron pairs. In this case one has to do with a purely chemical mechanism of the increase of the critical temperature.

Stabilizing fractional vortices in multiband superconductors with periodic pinning arrays

Multiband superconductors support the excitation of vortices with fractional quantum flux, known as fractional vortices. In the ground state, the fractional vortices in different bands bond together to form a composite vortex with the standard flux quantum ${\ensuremath{\Phi}}_{0}$; thus it is difficult to stabilize the fractional vortices. Here we show that fractional vortex lattices can be stabilized in multiband superconductors with a periodic pinning array at the half-matching field of the composite system, when full matching of the fractionalized vortices occurs. In the presence of a high current, the fractional vortices in different bands decouple and move at different velocities. When the current is turned off suddenly, the fractional vortices in different bands may be trapped at different pinning sites. This system also exhibits rich dynamic behavior, and for fractional vortices in two-band superconductors we find phases where the vortices in one band are pinned and the vortices in the other band are moving. These different phases can be observed in transport measurements and produce a new type of self-induced Shapiro steps that can arise under the application of only a dc drive.

Dissociation Transition of a Composite Lattice of Magnetic Vortices in the Flux-Flow Regime of Two-Band Superconductors

In multiband superconductors, each superconducting condensate supports vortices with fractional quantum flux. In the ground state, vortices in different bands are spatially bounded together to form a composite vortex, carrying one quantum flux Φ(0). Here we predict dissociation of the composite vortices lattice in the flux flow state due to the disparity of the vortex viscosity and flux of the vortex in different bands. For a small driving current, composite vortices start to deform, but the constituting vortices in different bands move with the same velocity. For a large current, composite vortices dissociate and vortices in different bands move with different velocities. The dissociation transition shows up as an increase of flux flow resistivity. In the dissociated phase, Shapiro steps are developed when an ac current is superimposed with a dc current.

Enhanced stability of vortex-antivortex states in two-component mesoscopic superconductors

Using the Ginzburg-Landau (GL) theory, we calculate the stability of sample symmetry-induced vortex-antivortex molecules in a mesoscopic superconducting bilayer exposed to a homogeneous magnetic field. We demonstrate the conditions under which the two condensates cooperatively broaden the field-temperature stability range of the composite (joint) vortex-antivortex state. In cases when such broadening is not achieved, a reentrance of the vortex-antivortex state is found at lower temperatures. In a large portion of the phase diagram noncomposite states are possible, in which the antivortex is present in only one of the layers. In this case, we demonstrate that the vortex-antivortex molecule in one of the layers can be pinned and enlarged by interaction with a vortex molecule in the other. Using analogies in the respective GL formalisms, we map our findings for the bilayer onto mesoscopic two-band superconductors.

Theory of nonequilibrium dynamics of multiband superconductors

We study the nonequilibrium dynamics of multiband BCS superconductors subjected to ultrashort pump pulses. Using density-matrix theory, the time evolution of the Bogoliubov quasiparticle densities and the superconducting order parameters are computed as a function of pump pulse frequency, duration, and intensity. Focusing on two-band superconductors, we consider two different model systems. The first one, relevant for iron-based superconductors, describes two-band superconductors with a repulsive interband interaction V12 which is much larger than the intraband pairing terms. The second model, relevant for MgB2, deals with the opposite limit where the intraband interactions are dominant and the interband pair scattering V12 is weak but attractive. For ultrashort pump pulses, both of these models exhibit a nonadiabatic behavior which is characterized by oscillations of the superconducting order parameters. We find that for nonvanishing V12, the superconducting gap on each band exhibits two oscillatory frequencies which are determined by the long-time asymptotic values of the gaps. The relative strength of these two frequency components depends sensitively on the magnitude of the interband interaction V12.

The Study on Hybridized Two-Band Superconductor

The two-band hybridized superconductor which the pairing occurred by conduction electron band and other-electron band are considered within a mean-field approximation. The critical temperature, zero-temperature order parameter, gap-to-Tcratio, and isotope effect coefficient are derived. We find that the hybridization coefficient shows a little effect on the superconductor that conduction electron band has the same energy as other-electron band but shows more effect on the superconductor that conduction electron band coexists with lower-energy other-electron band. The critical temperature is decreased as the hybridization coefficient increases. The higher value of hybridization coefficient, lower value of gap-to-Tcratio, and higher value of isotope effect coefficient are found.

Theoretical Analysis of Isotope Effect Exponent of Diboride and Iron-Based Superconductors: Three-Square-Well Model Approach

Analytical calculations of isotope effect exponent of both s-wave and π-phase-shifted s-wave two-band superconductors are presented within the framework of Bogoliubov–Valatin formalism using three-square-well model approach. The effective pairing interaction consists of contributions from the attractive acoustic electron–phonon, optical electron–phonon contributions in addition to repulsive electron–electron or antiferromagnetic fluctuation mediated pairing contributions for the diboride or pnictide superconductors, respectively. These are used to obtain analytical results that are applied in studying the isotope effect exponent in the diboride and pnictide superconductors, and the results are compared with experimental data.

Fluctuation current in superconducting loops

A superconducting loop that encloses noninteger flux holds a permanent current. On the average, this current is also present above Tc, and has been measured in recent years. We are able to evaluate the permanent current within the TDGL or the Kramer-Watts-Tobin models for loops of general configuration, i.e., we don't require uniform cross section, material or temperature. We can also consider situations in which the width is not negligible in comparison to the radius. Our results agree with experiments. The situations with which we deal at present include fluctuation superconductivity in two-band superconductors, equilibrium thermal fluctuations of supercurrent along a weak link, and ratchet effects.

Topological Defect-Phase Soliton and the Pairing Symmetry of a Two-Band Superconductor: Role of the Proximity Effect

We suggest a mechanism which promotes the existence of a phase soliton--a topological defect formed in the relative phase of superconducting gaps of a two-band superconductor with s(+-) type of pairing. This mechanism exploits the proximity effect with a conventional s-wave superconductor which favors the alignment of the phases of the two-band superconductor which, in the case of s(+-) pairing, are π shifted in the absence of proximity. In the case of a strong proximity such an effect can be used to reduce the soliton's energy below the energy of a soliton-free state, thus making the soliton thermodynamically stable. Based on this observation we consider an experimental setup, applicable for both stable and metastable solitons, which can be used to distinguish between ss(+-) and s(++) types of pairing in the iron-based multiband superconductors.

Two-band superconductors: Extended Ginzburg-Landau formalism by a systematic expansion in small deviation from the critical temperature

We derive the extended Ginzburg-Landau (GL) formalism for a clean $s$-wave two-band superconductor by employing a systematic expansion of the free-energy functional and the corresponding matrix gap equation in powers of the small deviation from the critical temperature $\ensuremath{\tau}=1\ensuremath{-}T/{T}_{c}$. The two lowest orders of this expansion produce the equation for ${T}_{c}$ and the standard GL theory. It is shown that in agreement with previous studies, this two-band GL theory maps onto the single-band GL model and thus fails to describe the difference in the spatial profiles of the two-band condensates. We prove that this difference appears already in the leading correction to the standard GL theory, which constitutes the extended GL formalism. We derive linear differential equations that determine the leading corrections to the band order parameters and magnetic field, discuss the validity of these equations, and consider examples of an important interplay between the band condensates. Finally, we present numerical results for the thermodynamic critical magnetic field and temperature-dependent band gaps for recent materials of interest, which are in very good agreement with those obtained from the full BCS approach in a wide temperature range. To this end, we emphasize the advantages of our extended GL theory in comparison with the often used two-component GL-like model based on an unreconstructed two-band generalization of the Gor'kov derivation.