Two-band Superconductor Research Articles

Collective excitations in three-band superconductors

We investigate equilibrium states, magnetic response and the normal oscillations of internal degrees of freedom (Higgs modes and Goldstone modes) of three-band superconductors accounting the terms of both internal proximity effect and the ``drag'' effect (intergradient interaction) in the Lagrangian. Both the Goldstone mode and the Higgs mode are split into three branches each: common mode oscillations and two modes of anti-phase oscillations, which are analogous to the Leggett mode in two-band superconductors. It is demonstrated that the second and third branches are nonphysical, and they can be removed by special choice of coefficients at the ``drag'' terms in Lagrangian. As a result, three-band superconductors are characterized by only single coherence length. Spectrum of the common mode Higgs oscillations has been obtained. The magnetic penetration depth is determined with densities of superconducting electrons in each band, although the drag terms renormalize the carrier masses.

High-order nonlinear terahertz probing of the two-band superconductor MgB2 : Third- and fifth-order harmonic generation

We report on high-order harmonic generation in the two-band superconductor ${\mathrm{MgB}}_{2}$ driven by intense terahertz electromagnetic pulses. Third- and fifth-order harmonics are resolved in time domain and investigated as a function of temperature and in applied magnetic fields crossing the superconducting phase boundary. The high-order harmonics in the superconducting phase reflects nonequilibrium dynamics of the superconducting order parameter in ${\mathrm{MgB}}_{2}$, which is probed via nonlinear coupling to the terahertz field. The observed temperature and field dependence of the nonlinear response allows to establish the superconducting phase diagram.

Reentrant superconductivity due to inter-band off-diagonal hopping in a two-band superconductivity model

Reentrant superconductivity due to inter-band off-diagonal hopping in a two-band superconductivity model

Multiband Superconductors: Two Characteristic Lengths for Each Contributing Condensate.

The interference of multiple condensates coexisting in one system may lead to unconventional coherent behavior. This is expected when the spatial lengths of the condensates are essentially different. Traditionally, the characteristic spatial length of a superconducting condensate is associated with the gap function. However, the broader readership is more familiar with the concept of the Cooper-pair wave function. For conventional single-band superconductors, the gap function coincides with the center-of-mass Cooper-pair wave function up to the coupling constant, and the corresponding gap and wave function characteristic lengths are the same. Surprisingly, we find that in two-band superconductors, these lengths are the same only near the critical temperature. At lower temperatures, they can significantly deviate from each other, and the fundamental question of which of these lengths should be preferred when specifying the spatial scale of a band-dependent condensate in multiband superconducting materials arises.

Contribution of collective excitations to third harmonic generation in two-band superconductors: The case of MgB2

Multiband superconductors can host collective excitations with marked differences with respect to their single-band counterpart. We first study the spectrum of collective amplitude fluctuations in a clean two-band superconductor, showing that the spectral weight of the Higgs mode rapidly deviates from the naive extension of the single band case as the interband coupling is turned on. These results are then used to critically analyze the nonlinear optical response in ${\mathrm{MgB}}_{2}$, providing an explanation for the apparently contradictory results of recent experiments, pointing towards a selective relevance either of the Leggett mode or of the amplitude fluctuations at twice the lower gap. By using exact numerical simulations and realistic estimates of disorder we compute the relative contribution of the quasiparticle, amplitude, and phase fluctuations to the nonlinear optical response. We show that at low pumping frequency only the resonance at twice the smaller gap emerges, as due to the BCS response, while the Leggett mode dominates only in a narrow range of higher pumping frequencies matching its low-temperature value. Our findings provide a fresh perspective on the potential of nonlinear THz spectroscopy to detect collective modes in other multiband systems, such as iron-based superconductors.

Magneto-topological transitions in multicomponent superconductors

Multi-component spin-singlet superconductors with competing 0- and $\pi$-pairing couplings, as in $s_{++}$ and $s_{\pm}$ phases, are close to instabilities with a spontaneous breaking of time-reversal symmetry. We demonstrate that the modification of the kinetic energy of superconducting electrons in a doubly connected superconducting cylinder, determined by the applied flux, generally drives transitions from chiral superconducting states to configurations that are time-reversal symmetric. This magneto-topological-induced changeover is investigated by means of a Ginzburg-Landau approach for a two-band superconductor with interband interactions and impurity scattering investigated for the case of a sample in the form of a mesoscopically thin-walled cylinder. We find that the application of a magnetic flux can convert a chiral $s_{\pm}+is_{++}$ state into a $s_{\pm}$ configuration and vice versa or tune the energy splitting of chiral states having inequivalent pairing amplitudes. We discuss signatures for the detection of these phases and of the corresponding transitions in mesoscopic superconducting loops.

Suppression of fluctuations in a two-band superconductor with a quasi-one-dimensional band

Chain-like structured superconductive materials (such as A_2Cr_3As_3, with A = K, Rb, Cs) exhibit the multiband electronic structure of single-particle states, where coexisting quasi-one-dimensional (Q1D) and conventional higher-dimensional energy bands take part in the creation of the aggregate superconducting condensate. When the chemical potential approaches the edge of a Q1D band in a single-band superconductor, the corresponding mean-field critical temperature increases significantly but the superconductivity is quenched by fluctuations. However, recent investigation has revealed that when a Q1D band is coupled to a higher dimensional one by the interband Cooper-pair transfer, the thermal superconductive fluctuations can be suppressed so that the resulting critical temperature can be close to its mean-field value. In the present work, we calculate the mean-field T_c0 and fluctuation-shifted T_c critical temperatures for a two-band superconductor where a Q1D band coexists with a higher-dimensional band, and investigate how the thermal fluctuations are sensitive to the system parameters. We find that T_c is close to T_c0 in a wide range of microscopic parameters, and even the dimensionality of the higher-dimensional band does not play an essential role. Thus, the screening mechanism for suppressing fluctuations via the pair-exchange coupling between the bands is indeed relevant for a large class of Q1D multiband superconducting materials, encouraging further experiments aimed at reaching larger critical temperatures in such multiband superconductors.

The dynamics of current-driven vortex in two-band superconductor with s+d wave pairing

Two-band mesoscopic superconductors are known to exhibit specific vortex states. In this paper, finite element method (FEM) is used to numerically solve the time-dependent Ginzburg-Landau (TDGL) equation with s+d wave coupling for a 3D mesoscopic superconducting strip. We discuss the effect of external magnetic field H and applied current I on s+d wave coupling at the temperatures Ts<T<Td, where Ts and Td are the critical temperatures of s- and d-bands, respectively. The obtained results show that this coupling is enhanced by the applied current and reflected by the vortex dynamics in two bands. A small amount of Cooper pairs in the s-band form a series of different vortex and antivortex states. Mutual attraction of vortex and antivortex induces simultaneous motion of s- and d-vortices at applied current I. This work provides theoretical background for the design of mesoscopic superconducting devices.

Dynamics of Fractional Vortices in Two-Band Superconductors

Dynamics of Fractional Vortices in Two-Band Superconductors

Topological nature of the transition between the gap and the gapless superconducting states

Recently it was demonstrated that the long-known transition between the gap and gapless superconducting states in the Abrikosov-Gor'kov theory of superconducting alloy with paramagnetic impurities is of the Lifshitz's type, i.e., at zero temperature this is the order phase transition. Since transitions of this kind in a normal metal are always associated to certain topological changes, then below we clarify the topological nature of the transition under consideration. Namely, we demonstrate that the topological invariant which in process of the transition undergoes the change is nothing but the Euler characteristic. Alternatively, in terms of the theory of catastrophes one can relate this transition to appearance of the cuspidal edge at the corresponding surface of the density of states as the function of energy and superconducting order parameter. The concept of experiments for the confirmation of order topological phase transition is proposed. The obtained theoretical results can be applied for the explanation of recent experiments with lightwave-induced gapless superconductivity, for the interpretation of the disorder-induced transition states via gapless phase in two-band superconductors, and the emergence of gapless color superconductivity in quantum chromodynamics.

Peak effect in a superconductor/normal-metal strip in a vortex-free state

We theoretically predict that the critical current ${I}_{c}$ and magnetization $M$ of a hybrid superconductor/normal-metal (SN) strip may have nonmonotonous dependence on a perpendicular magnetic field---so-called peak effect. In contrast to familiar peak effect, which is connected either with vortex entry to the superconductor or with peculiarities of vortex pinning, the found phenomenon exists at low fields, in the vortex-free (Meissner) phase. We argue that the effect appears at specific parameters of the studied hybrid structure when its in-plane current-supervelocity relation has two maxima. We expect that the same peak effect may exist in two-band superconductors (like ${\mathrm{MgB}}_{2}$) where similar current-supervelocity dependence was predicted at low temperatures.

Two-band superconductivity through structural and electronic reconstruction on interface: YBa2Cu3O7/LaAlO3(001)

Novel physical phenomena arising from the complex interplay between the spin, charge, orbital, and lattice orders can emerge in perovskite-type oxide heterostructures. Here, we investigate the lattice and electronic structures on the interface of the YBa2Cu3O7/LaAlO3 heterostructure prepared by pulsed laser deposition with the combination of spherical aberration-corrected scanning transmission electron microscopy and density functional theory. Both reconstructed and normal interfaces are observed at sub-angstrom resolution, and these superconducting planes closest to the corresponding interface known as CuO2(a) and CuO2(c) planes, respectively. Due to the lattice reconstruction and charge density redistribution, CuO2(a) plane moves closer to the reconstructed interface with bond length of in-plane Cu and apical oxygen reduced. Consequently, the d3z2−r2 orbital loss electrons further results in two orbitals (dx2−y2 and d3z2−r2) crossing the Fermi level, showing two-band superconductivity behavior at the reconstructed interface. However, the CuO2(c) plane remains unaffected due to the protection of the charge reservoir layer at the normal interface and exhibits electronic structures similar to bulk cuprate superconductors, where only dx2−y2 orbitals contribute to the states at the Fermi level. These results suggest that the interfacial reconstruction might be a possible pathway to manipulate the electronic structures of the superconducting oxide heterostructures.

Checkerboard patterns of charge stripes in the two-gap superconductor ZrB12

Inhomogeneous superconductivity in the high quality single crystals of ZrB12 (Tc = 6 K) has been studied using the heat capacity and x-ray diffraction (XRD) data. Evidence of two-band superconductivity with two branches of upper critical field Hc2(Tc) is obtained in a magnetic field applied along the [110] axis of the crystal. On the contrary, at H //[100], the only dependence Hc2(Tc) is observed. This finding is supplemented with the checkerboard-type patterns of the charge stripes in ZrB12 deduced from the detailed analysis of XRD data. These patterns are compared to the structure of the charge stripes in the weakly bound superconductor LuB12, whose Tc is 15 times lower than that of ZrB12. Probable nature of the two-gap superconductivity in ZrB12 with strongly enhanced characteristics is discussed.

Collective Excitations in Two-Band Superconductors

We investigate the eigen oscillations of internal degrees of freedom (Higgs mode and Goldstone mode) of two-band superconductors using the extended time-dependent Ginzburg-Landau theory, formulated in a work Grigorishin (2021) \cite{grig2}, for the case of two coupled order parameters by both the internal proximity effect and the drag effect. It is demonstrated, that the Goldstone mode splits into two branches: common mode oscillations with the acoustic spectrum, which is absorbed by the gauge field, and anti-phase oscillations with the energy gap (mass) in the spectrum determined by the interband coupling, which can be associated with the Leggett mode. The Higgs mode splits into two branches also: a massive one, whose energy gap vanishes at the critical temperature $T_{c}$, another massive one, whose energy gap does not vanish at $T_{c}$. It is demonstrated, that the second branch of the Higgs mode is nonphysical, and it, together with the Leggett mode, can be removed by the special choice of coefficient at the "drag" term in the Lagrangian. In the same time, such a choice leaves only one coherence length, thereby prohibiting so-called type-1.5 superconductors. We analyze experimental data about the Josephson effect between two-band superconductors. In particular, it is demonstrated, that the resonant enhancement of the DC current through a Josephson junction at a resonant bias voltage $V_{\mathrm{res}}$, when the Josephson frequency matches the frequency of some internal oscillation mode in two-band superconductors (banks), can be explained with the coupling between AC Josephson current and Higgs oscillations. Thus, explanation of this effect does not need the Leggett mode.

Direct detection of odd-frequency superconductivity via time- and angle-resolved photoelectron fluctuation spectroscopy

We propose a measurement scheme to directly detect odd-frequency superconductivity via time- and angle-resolved photoelectron fluctuation spectroscopy. The scheme includes two consecutive, non-overlapping probe pulses applied to a superconducting sample. The photoemitted electrons are collected in a momentum-resolved fashion. Correlations between signals with opposite momenta are analyzed. Remarkably, these correlations are directly proportional to the absolute square of the time-ordered anomalous Green's function of the superconductor. This setup allows for the direct detection of the "hidden order parameter" of odd-frequency pairing. We illustrate this general scheme by analyzing the signal for the prototypical case of a two-band superconductor.

Probing the current-phase relation in Josephson point-contact junctions between hbox {Pb}_{0.6}In_{0.4} and hbox {Ba}_{0.6}K_{0.4}(FeAs)_2 superconductors

The Josephson effect in point contacts between an “ordinary” superconductor hbox {Pb}_{0.6}In_{0.4} (T_c approx 6.6 , {mathrm {K}}) and single crystals of the Fe-based superconductor Ba_{0.6}K_{0.4}(FeAs)_2 (T_c approx 38.5 , {mathrm {K}}), was investigated. In order to shed light on the order parameter symmetry of Ba_{0.6}K_{0.4}(FeAs)_2, the dependence of the Josephson supercurrent I_s on the temperature and on sin (dvarphi ) with d = 1, 2 was studied. The dependencies of the critical current on temperature I_c (T) and of the amplitudes of the first current steps of the current–voltage characteristic i_n^{exp} (sqrt{P})(n = 0, 1, 2) on the power of microwave radiation with frequency f = (1.5 div 8), {mathrm {GHz}} were measured. It is shown that the dependencies I_c (T) are close to the well-known Ambegaokar–Baratoff (AB) dependence for tunnel contacts between “ordinary” superconductors and to the dependence calculated by Burmistrova et al. (Phys Rev B 91, 214501 (2015)) for microshorts between an “ordinary” superconductor and a two-band superconductor with s pm order parameter symmetry at certain values of the transparency of boundaries and thickness of the transition layer. It is found that the dependencies i_n^{exp} (sqrt{P}) cannot be approximated within the resistively shunted model using the normalized microwave frequencies Omega = 2 pi f / (2eV_c / hbar ) with characteristic voltages V_c = I_c R_N, (R_N—normal resistance of the contact) found from the low-voltage parts of the current–voltage characteristics. The reasons for this failure are discussed and a method is proposed for accurately determining the value of Omega, which takes into account all the features of the point contact affecting the period of the dependence i_n^{exp} (sqrt{P}). An analysis of the I_c (T) and i_n^{exp} (sqrt{P}) dependencies shows that the superconducting current of the Josephson contacts under investigation is proportional to the sin of the phase difference varphi, I_s = I_c sin(varphi ). The implications of these results on the symmetry of the order parameter are also discussed.

Dynamics of fractional vortices in two-band superconductors

The entry of fractional vortices and their subsequent dynamics inside a two-band superconductor is explored based on the numerical solutions of time-dependent Ginzburg–Landau (TDGL) equations. We consider the case when superfluid electron condensates from two zones are characterized by quite different parameters, such as coherence lengths ξi, and London penetration depths λi, which in turn leads to the different critical magnetic fields Hc,iand fractional flux quanta ϕivalues for the superconducting state in these two zones. Numerical solutions of TDGL equations in increasing external magnetic field followed by mathematical modeling of magnetic flux penetration were performed for this case by finite element method. We have explored the time evolution for the fractional vortices penetration process and their subsequent dynamics inside the specimens for two geometries: the circular disk, and the circular disk with a triangular cutout. Obtained results indicate that magnetic flux penetrates inside the specimen in the form of fractional vortices when they can overcome the edge barrier, which may be different for these two vortex types. Therefore, in increasing external magnetic field first penetrate vortices with a lower barrier height (i.e., lower Hc,i) while the other type of fractional vortices start their penetration at higher external field value. Another mechanism for the formation of fractional vortices during their entrance in a two-band superconductor is related to the difference in their flux values and viscosity coefficients which determine the rate of vortex proliferation inside the sample. Within the specimen, fractional vortices move in order to arrange. Vortices of different types attract to each other and try to stick together thus forming composite vortices with the whole flux quantum value ϕ0 = h/2e.

Band-selective third-harmonic generation in superconducting MgB2 : Possible evidence for the Higgs amplitude mode in the dirty limit

We report on time-resolved linear and nonlinear terahertz spectroscopy of the two-band superconductor $\mathrm{Mg}{\mathrm{B}}_{2}$ with a superconducting transition temperature ${T}_{c}\ensuremath{\approx}36$ K. Third-harmonic generation (THG) is observed below ${T}_{c}$ by driving the system with intense narrow-band THz pulses. For the pump-pulse frequencies $f=0.3$, 0.4, and 0.5 THz, the temperature-dependent evolution of the THG signals exhibits a resonance maximum at the temperatures with the resonance conditions $2f=2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\pi}}(T)$ fulfilled, for the dirty-limit superconducting gap $2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\pi}}$. In contrast, for $f=0.6$ and 0.7 THz with $2f&gt;2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\pi}}(T\ensuremath{\rightarrow}0)=1.03$ THz, the THG intensity increases monotonically with decreasing temperature. Moreover, for $2f&lt;2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\pi}}(T\ensuremath{\rightarrow}0)$ the THG is found nearly isotropic with respect to the pump-pulse polarization. These results suggest a predominant contribution of the driven Higgs amplitude mode of the dirty-limit $\ensuremath{\pi}$-band superconducting gap, pointing to the importance of scattering for observation of the Higgs mode in superconductors.

Time-resolved optical conductivity and Higgs oscillations in two-band dirty superconductors

Recent studies have emphasized the importance of impurity scattering for the optical Higgs response of superconductors. In the dirty limit, an additional paramagnetic coupling of light to the superconducting condensate arises which drastically enhances excitation. So far, most work concentrated on the periodic driving with light, where the third-harmonic generation response of the Higgs mode was shown to be enhanced. In this work, we additionally calculate the time-resolved optical conductivity of single- and two-band superconductors in a two-pulse quench-probe setup, where we find good agreement with existing experimental results. We use the Mattis-Bardeen approach to incorporate impurity scattering and calculate explicitly the time-evolution of the system. Calculations are performed both in a diagrammatic picture derived from an effective action formalism and within a time-dependent density matrix formalism.

BCS-BEC crossover in a two-band superconductor with odd-parity hybridization

We study the crossover from the weak coupling Bardeen-Cooper-Schrieffer (BCS) state to Bose-Einstein condensation (BEC) at strong coupling in a two-band superconductor with orbitals of opposite parity coexisting at a common Fermi surface in the metallic state. In such systems hybridization can play a role similar to spin-orbit interaction in fermionic spinor gases, enhancing interband pairing and opening the possibility for driving the BCS-BEC crossover. Building on a mean-field analysis, we investigate the BCS-BEC crossover induced by a variation of the hybridization strength. We show that the relevant scale for the crossover depends strongly on the ratio of effective masses, with the crossover favored in systems with one dispersive and one flat band. Including the effect of thermal pair fluctuations in a one-loop approximation, we calculate the dependence of the critical temperature on the microscopic parameters in the strong coupling regime.