Leggett Modes Research Articles

Point-contact spectroscopy of Leggett modes in superconducting compounds with unconventional pairing

Proximity-coupled nanostructures of conventional superconductors (SCs) and half-metallic ferromagnets (hmFs) are promising candidates as materials with unconventional superconductivity. The interrelated superposition of spin singlet-triplet and frequency even-odd superconducting condensates characterizes the superconducting state in such heterostructures. In a multi-band SC, the collective modes associated with the excitations of the relative phase between superconducting bands without perturbation of the Cooper pairs symmetry (Leggett modes) are allowed. In this report, we present the results of experimental investigations via the point-contact transport measurements of the Leggett-like collective excitations in the superconducting state of the nanocomposite of s-wave two-band superconductor MgB2 and half-metallic ferromagnet (La,Sr)MnO3. Two types of point contacts (PCs) have been used: the nanocomposite-nonmagnetic metal PCs and the nanocomposite–hmF PCs. The conductance equidistant peaks against the background of the gap structure were observed in both types of high-quality point junctions. Their distinctive feature was their period: two times shorter for the nanocomposite–hmF contacts compared to the nonmagnetic metal PCs. We attribute these spin-selective conductance periodic peaks to the relative phase Leggett’s excitations between “parents” MgB2 even-frequency singlet condensates and proximity-induced triplet superconducting condensates. The data obtained on the hmF PCs also demonstrate the features that may indicate a dynamic coupling between even-frequency condensates and odd-frequency gapless superconducting condensates.

Role of Higgs and Leggett modes for the third harmonic response in noncentrosymmetric superconductors

The amplitude and phase of the third harmonic response in superconductors provide important insights into collective Higgs and Leggett modes, respectively, in the superconducting state. In particular, for twice the incident frequency equal to the binding energy (2ω=2Δ), one finds a resonance of the amplitude and a corresponding phase jump in the third harmonic response, respectively. We generalize these concepts to superconductors without an inversion symmetry, which can be effectively described by a two-band model with an order parameter consisting of spin-singlet (even parity) and spin-triplet (odd parity) components. In our work we use an effective action approach for the derivation of the nonlinear response and assign the underlying physical processes to their respective Feynman diagrams. We calculate the third harmonic signal exemplary for the noncentrosymmetric compound CePt3Si, showing that it contains contributions from three distinguishable sources, namely the Higgs mode, the Leggett mode, and quasiparticles (broken Cooper-pairs). Only in the clean limit diamagnetic Raman-like processes contribute to the third harmonic signal, whereas the quasiparticle contributions dominate the collective modes for all triplet-singlet ratios of the gap structure. In the dirty limit, we find a significant enhancement of the Higgs mode contributions to the third harmonic response, due to the inclusion of nonvanishing paramagnetic diagrams. Finally, we argue that the phase difference between the third harmonic and the fundamental signal might reveal a jump, where its size is dependent on the light-matter coupling. Published by the American Physical Society 2024

Spin-Selective Point-Contact Spectroscopy of Leggett Modes in Superconducting Proximity-Coupled MgB2/La0.67Sr0.33MnO3 Nanocomposites

Spin-Selective Point-Contact Spectroscopy of Leggett Modes in Superconducting Proximity-Coupled MgB2/La0.67Sr0.33MnO3 Nanocomposites

Leggett modes in a Dirac semimetal

Experiments have shown that several materials, including MgB2, iron-based superconductors and monolayer NbSe2, are multiband superconductors. Superconducting pairing in multiple bands can give rise to phenomena not available in a single band, including Leggett modes. A Leggett mode is the collective periodic oscillation of the relative phase between the phases of the superconducting condensates formed in the different bands. The experimental observation of Leggett modes is challenging because multiband superconductors are rare and because these modes describe charge fluctuations between bands and therefore are hard to probe directly. Also, the excitation energy of a Leggett mode is often larger than the superconducting gaps, and therefore they are strongly overdamped via relaxation processes into the quasiparticle continuum. Here, we show that Leggett modes and their frequency can be detected in a.c. driven superconducting quantum interference devices. We then use the results to analyse the measurements of such a quantum device, one based on a Dirac semimetal Cd3As2, in which superconductivity is induced by proximity to superconducting Al. These results show the theoretically predicted signatures of Leggett modes, and therefore we conclude that a Leggett mode is present in the two-band superconducting state of Cd3As2.

Classification of Lifshitz invariant in multiband superconductors: An application to Leggett modes in the linear response regime in Kagome lattice models

Multiband superconductors are sources of rich physics arising from multiple order parameters, which show unique collective dynamics including Leggett mode as relative phase oscillations. Previously, it has been pointed out that the Leggett mode can be optically excited in the linear response regime, as demonstrated in a one-dimensional model for multiband superconductors [T. Kamatani , ]. Here we identify the linear coupling term in the Ginzburg-Landau free energy to be the so-called Lifshitz invariant, which takes the form of d·(Ψi*∇Ψj−Ψj*∇Ψi), where d is a constant vector and Ψi and Ψj(i≠j) represent superconducting order parameters. We classify all pairs of irreducible representations of order parameters in the crystallographic point groups that allow for the existence of the Lifshitz invariant. We emphasize that the Lifshitz invariant can appear even in systems with inversion symmetry. The results are applied to a model of s-wave superconductors on a Kagome lattice with various bond orders, for which in some cases we confirm that the Leggett mode appears as a resonance peak in a linear optical conductivity spectrum based on microscopic calculations. We discuss a possible experimental observation of the Leggett mode by a linear optical response in multiband superconductors. Published by the American Physical Society 2024

Direct Observation of the Spin Exciton in Andreev Spectroscopy of Iron-Based Superconductors.

Quasiparticle excitations provide viable information on the physics of unconventional superconductors. Higgs and Leggett modes are some of the classic examples. Another important bosonic excitation is the spin exciton originating from the sign-changing superconducting gap structure. Here we report a direct observation of the temperature-dependent spin exciton in the Andreev spectra of iron-based superconductors. Combined with the other experimental evidence, our observation confirms the extended s-wave () order parameter symmetry and indirectly proves the spin-fluctuation mechanism of Cooper pairing.

Optical response of the Leggett mode in multiband superconductors in the linear response regime

We study optical responses of Leggett modes in multiband superconductors in the linear response regime. The Leggett mode is a collective mode unique to multiband superconductors that arises from relative phase fluctuations of superconducting orders for different bands. We use the Ginzburg-Landau (GL) description to study the collective modes in multiband systems. We find that multiband superconductors generally allow a linear coupling between the Leggett mode and external electric fields due to the presence of a cross term between different components of superconducting orders in the GL theory. The presence of a linear coupling for the Leggett mode is in sharp contrast with the absence of that for Higgs (amplitude) modes in single-band superconductors that only support nonlinear optical responses such as third harmonic generation (THG). We further confirm such a linear coupling in multiband superconductors by a more microscopic description based on a diagrammatic approach. We study the collective modes within the random phase approximation (RPA) and compute their contribution to the linear optical conductivity. These findings suggest a new route to observe the Leggett mode by optical absorption.

Leggett Modes Accompanying Crystallographic Phase Transitions

Higgs and Goldstone modes, well known in high energy physics, have been realized in a number of condensed matter physics contexts, including superconductivity and magnetism. The Goldstone-Higgs concept is also applicable to and gives rise to new insights on structural phase transitions. Here, we show that the Leggett mode, a collective mode observed in multi-band superconductors, also has an analog in crystallographic phase transitions. Such structural Leggett modes can occur in the phase channel as in the original work of Leggett, \href{https://doi.org/10.1143/PTP.36.901}{Prog.\ Theor.\ Phys.\ \textbf{36}, 901 (1966)}. That is, they are antiphase Goldstone modes (anti-phasons). In addition, a new collective mode can also occur in the amplitude channel, an out-of phase (antiphase) Higgs mode, that should be observable in multi-band superconductors as well. We illustrate the existence and properties of these structural Leggett modes using the example of the pyrochlore relaxor ferroelectric, Cd$_2$Nb$_2$O$_7$.

Superfluid density and collective modes of fermion superfluid in dice lattice

The superfluid properties of attractive Hubbard model in dice lattice are investigated. It is found that three superfluid order parameters increase as the interaction increases. When the filling factor falls into the flat band, due to the infinite large density of states, the resultant superfluid order parameters are proportional to interaction strength, which is in striking contrast with the exponentially small counterparts in usual superfluid (or superconductor). When the interaction is weak, and the filling factor is near the bottom of the lowest band (or the top of highest band), the superfluid density is determined by the effective mass of the lowest (or highest) single-particle band. When the interaction is strong and filling factor is small, the superfluid density is inversely proportional to interaction strength, which is related to effective mass of tightly bound pairs. In the strong interaction limit and finite filling, the asymptotic behaviors of superfluid density can be captured by a parabolic function of filling factor. Furthermore, when the filling is in flat band, the superfluid density shows a logarithmic singularity as the interaction approaches zero. In addition, there exist three undamped collective modes for strong interactions. The lowest excitation is gapless phonon, which is characterized by the total density oscillations. The two others are gapped Leggett modes, which correspond relative density fluctuations between sublattices. The collective modes are also reflected in the two-particle spectral functions by sharp peaks. Furthermore, it is found that the two-particle spectral functions satisfy an exact sum-rule, which is directly related to the filling factor (or density of particle). The sum-rule of the spectral functions may be useful to distinguish between the hole-doped and particle-doped superfluid (superconductor) in experiments.

Frustration effect on escape rate in Josephson junctions between single-band and three-band superconductors in the macroscopic quantum tunneling regime

The escape rate of S → R switching (from superconducting S state to resistive R state) in Josephson junction formed by s-wave single-band (SB) and three-band (ThB) superconductors (SB/ThB junctions) in macroscopic quantum tunneling regime is investigated. We use the effective critical current approximation in SB/ThB junctions. It was shown that escape rate can exhibit qualitative features in the case of frustration effects in ThB superconductors. Inclusion of Leggett modes in ThB superconductors leads to enhancement of escape rate in three-channel Josephson junction in comparison with single-channel junctions.

Collective excitations of a BCS superfluid in the presence of two sublattices

We consider a generic Hamiltonian that is suitable for describing a uniform BCS superfluid on a lattice with a two-point basis, and study its collective excitations at zero temperature. For this purpose, we first derive an effective-Gaussian action for the pairing fluctuations, and then extract the low-energy dispersion relations for the in-phase Goldstone and out-of-phase Leggett modes along with the corresponding amplitude (i.e., the so-called Higgs) ones. We find that while the Goldstone mode is gapless at zero momentum and propagating in general, the Leggett mode becomes undamped only with sufficiently strong interactions. Furthermore, we show that, in addition to the conventional contribution that is controlled by the energy of the Bloch bands, the velocity of the Goldstone mode has a geometric contribution that is governed by the quantum metric tensor of the Bloch states. Our results suggest that the latter contribution dominates the velocity when the former one becomes negligible for a narrow- or a flat-band.

Escape rate in Josephson junctions between single-band and two-band superconductors

We theoretically study the escape rate of S → R switching (from superconducting S state to resistive R state) in Josephson junction (JJ) formed by s-wave single-band (SB) and two-band (TB) superconductors (SB/TB junctions). The escape rate is calculated in terms of effective critical current in thermal activation (TA) and macroscopic quantum tunneling (MQT) regimes. It was shown that escape rate can exhibit qualitative features in the case of s+--wave symmetry in TB superconductors, while s++-wave is identical to JJ between SB s-wave superconductors. Leggett modes in TB superconductors lead to enhancement of escape rate in two-channel JJ in comparison with single-channel junctions.

Leggett collective excitations in a two-band Fermi superfluid at finite temperatures

The Leggett collective excitations for a two-band Fermi gas with s-wave pairing and Josephson interband coupling in the BCS-BEC crossover at finite temperatures are investigated within the Gaussian pair fluctuation approach. Eigenfrequencies and damping factors for Leggett modes are determined in a nonperturbative way, using the analytic continuation of the fluctuation propagator through a branch cut in the complex frequency plane, as in Kurkjian et al (2019 Phys. Rev. Lett. 122 093403). The treatment is performed beyond the low-energy expansion, which is necessary when the collective excitation energy reaches the pair-breaking continuum edge. The results are applied in particular to cold atomic gases at the orbital Feshbach resonance and in a regime far from BEC, which can be relevant for future experiments.

Nonlinear optical response of collective modes in multiband superconductors assisted by nonmagnetic impurities

In multiband superconductors, multiple collective modes exist associated with the multiple order parameters. Oscillations of the amplitude and the relative phase of the order parameters are called Higgs and Leggett modes, respectively. Recently, it has been suggested that nonmagnetic impurity scattering would enhance nonlinear coupling between the Higgs mode and an electromagnetic wave with a frequency located in the superconducting gap region, while its effect on the Leggett mode is still unresolved. Here, we theoretically investigated the nonlinear optical response of multiband Bardeen-Cooper-Schrieffer-type superconductors in the presence of nonmagnetic impurities with a density matrix approach extending the Mattis-Bardeen model of linear response. We found that the drastic enhancement of nonlinear optical response due to the nonmagnetic impurity scattering occurs only for the Higgs modes and not for the Leggett mode. As a result, both the light-induced dynamics of the superconducting gaps and the resulting third-harmonic generation are dominated by the Higgs modes. We also examined the role of quasiparticle excitations to find that they give the subdominant contribution to the third-harmonic generation.

Non-London electrodynamics in a multiband London model: Anisotropy-induced nonlocalities and multiple magnetic field penetration lengths

The London model describes strongly type-2 superconductors as massive vector field theories, where the magnetic field decays exponentially at the length scale of the London penetration length. This also holds for isotropic multi-band extensions, where the presence of multiple bands merely renormalises the London penetration length. We show that, by contrast, the magnetic properties of anisotropic multi-band London models are not this simple, and the anisotropy leads to the inter-band phase differences becoming coupled to the magnetic field. This results in the magnetic field in such systems having N+1 penetration lengths, where N is the number of field components or bands. That is, in a given direction, the magnetic field decay is described by N+1 modes with different amplitudes and different decay length scales. For certain anisotropies we obtain magnetic modes with complex masses. That means that magnetic field decay is not described by a monotonic exponential increment set by a real penetration length but instead is oscillating. Some of the penetration lengths are shown to diverge away from the superconducting phase transition when the mass of the phase-difference mode vanishes. Finally the anisotropy-driven hybridization of the London mode with the Leggett modes can provide an effectively non-local magnetic response in the nominally local London model. Focusing on the two-component model, we discuss the magnetic field inversion that results from the effective non-locality, both near the surface of the superconductor and around vortices. In the regime where the magnetic field decay becomes non-monotonic, the multiband London superconductor is shown to form weakly-bound states of vortices.

Collective modes in a two-band superfluid of ultracold alkaline-earth-metal atoms close to an orbital Feshbach resonance

We discuss the collective modes in an alkaline-earth Fermi gas close to an orbital Feshbach resonance. Unlike the usual Feshbach resonance, the orbital Feshbach resonance in alkaline-earth atoms realizes a two-band superfluid system where the fermionic nature of both the open and the closed channel has to be taken into account. We show that apart from the usual Anderson-Bogoliubov mode which corresponds to the oscillation of total density, there also appears the long-sought Leggett mode corresponding to the oscillation of relative density between the two channels. The existence of the phonon and the Leggett modes and their evolution are discussed in detail. We show how these collective modes are reflected in the density response of the system.

Theory of light-induced resonances with collective Higgs and Leggett modes in multiband superconductors

We theoretically investigate coherent optical excitations of collective modes in two-band BCS superconductors, which accommodate two Higgs modes and one Leggett mode corresponding, respectively, to the amplitude and relative-phase oscillations of the superconducting order parameters associated with the two bands. We find, based on a mean-field analysis, that each collective mode can be resonantly excited through a nonlinear light-matter coupling when the doubled frequency of the driving field coincides with the frequency of the corresponding mode. Among the two Higgs modes, the higher-energy one exhibits a sharp resonance with light, while the lower-energy mode has a broadened resonance width. The Leggett mode is found to be resonantly induced by a homogeneous ac electric field because the leading nonlinear effect generates a potential offset between the two bands that couples to the relative phase of the order parameters. The resonance for the Leggett mode becomes sharper with increasing temperature. All of these light-induced collective modes along with density fluctuations contribute to the third-harmonic generation. We also predict an experimental possibility of optical detection of the Leggett mode.

Leggett modes and multiband superconductivity inSr2RuO4

Sr$_2$RuO$_4$ is a prototypical multi-band superconductor with three bands crossing the Fermi level. These bands exhibit distinct dimensional characteristics, with one quasi-2D $\gamma$-band and two quasi-1D $\alpha$- and $\beta$-bands. This leads to the expectation that the superconductivity on the $\gamma$-band may be only weakly Josephson-coupled to that on the other two bands. Based on an explicit microscopic weak coupling calculation appropriate for Sr$_2$RuO$_4$, we study the collective Leggett modes associated with the relative phase oscillations between the bands and show that a relatively soft Leggett mode exists due to the comparatively weaker inter-band Josephson coupling. These calculations also provide insight into why the superconducting gap magnitudes may be comparable on all three bands, despite the noticeable differences between the $\gamma$ and $\alpha, \beta$ bands. The analyses can be readily applied to other multi-band superconductors.

Coupling of Higgs and Leggett modes in non-equilibrium superconductors.

In equilibrium systems amplitude and phase collective modes are decoupled, as they are mutually orthogonal excitations. The direct detection of these Higgs and Leggett collective modes by linear-response measurements is not possible, because they do not couple directly to the electromagnetic field. In this work, using numerical exact simulations we show for the case of two-gap superconductors, that optical pump–probe experiments excite both Higgs and Leggett modes out of equilibrium. We find that this non-adiabatic excitation process introduces a strong interaction between the collective modes, which is absent in equilibrium. Moreover, we propose a type of pump–probe experiment, which allows to probe and coherently control the Higgs and Leggett modes, and thus the order parameter directly. These findings go beyond two-band superconductors and apply to general collective modes in quantum materials.

Giant phonon anomaly associated with superconducting fluctuations in the pseudogap phase of cuprates

The pseudogap in underdoped cuprates leads to significant changes in the electronic structure, and was later found to be accompanied by anomalous fluctuations of superconductivity and certain lattice phonons. Here we propose that the Fermi surface breakup due to the pseudogap, leads to a breakup of the pairing order into two weakly coupled sub-band amplitudes, and a concomitant low energy Leggett mode due to phase fluctuations between them. This increases the temperature range of superconducting fluctuations containing an overdamped Leggett mode. In this range inter-sub-band phonons show strong damping due to resonant scattering into an intermediate state with a pair of overdamped Leggett modes. In the ordered state, the Leggett mode develops a finite energy, changing the anomalous phonon damping into an anomaly in the dispersion. This proposal explains the intrinsic connection between the anomalous pseudogap phase, enhanced superconducting fluctuations and giant anomalies in the phonon spectra.