Modes In Superconductors Research Articles

Optical response of Higgs mode in superconductors at clean limit: formulation through Eilenberger equation and Ginzburg–Landau Lagrangian

Both macroscopic Ginzburg–Landau Lagrangian and microscopic gauge-invariant kinetic equation suggest a finite Higgs-mode generation in the second-order optical response of superconductors at clean limit, whereas the previous derivations through the path-integral approach and Eilenberger equation within the Matsubara formalism failed to give such generation. The crucial treatment leading to this controversy lies at an artificial scheme that whether the external optical frequency is taken as continuous variable or bosonic Matsubara frequency to handle the gap dynamics within the Matsubara formalism. To resolve this issue, we derive the effective action of the superconducting gap near Tc in the presence of the vector potential through the path-integral approach, to fill in the long missing gap of the microscopic derivation of the Ginzburg–Landau Lagrangian in superconductors. It is shown that only by taking optical frequency as continuous variable within the Matsubara formalism, can one achieve the fundamental Ginzburg–Landau Lagrangian, and in particular, the finite Ginzburg–Landau kinetic term leads to a finite Higgs-mode generation at clean limit. To further eliminate the confusion of the Matsubara frequency through a separate framework, we apply the Eilenberger equation within the Keldysh formalism, which is irrelevant to the Matsubara space. By calculating the gap dynamics in the second-order response, it is analytically proved that the involved optical frequency is a continuous variable rather than bosonic Matsubara frequency, causing a finite Higgs-mode generation at clean limit.

Revealing Novel Aspects of Light-Matter Coupling by Terahertz Two-Dimensional Coherent Spectroscopy: The Case of the Amplitude Mode in Superconductors.

Recently developed terahertz (THz) two-dimensional coherent spectroscopy (2DCS) is a powerful technique to obtain materials information in a fashion qualitatively different from other spectroscopies. Here, we utilized THz 2DCS to investigate the THz nonlinear response of conventional superconductor NbN. Using broadband THz pulses as light sources, we observed a third-order nonlinear signal whose spectral components are peaked at twice the superconducting gap energy 2Δ. With narrow-band THz pulses, a THz nonlinear signal was identified at the driving frequency Ω and exhibited a resonant enhancement at temperature when Ω=2Δ. General theoretical considerations show that such a resonance can arise only from a disorder-activated paramagnetic coupling between the light and the electronic current. This proves that the nonlinear THz response can access processes distinct from the diamagnetic Raman-like density fluctuations, which are believed to dominate the nonlinear response at optical frequencies in metals. Our numerical simulations reveal that, even for a small amount of disorder, the Ω=2Δ resonance is dominated by the superconducting amplitude mode over the entire investigated disorder range. This is in contrast to other resonances, whose amplitude-mode contribution depends on disorder. Our findings demonstrate the unique ability of THz 2DCS to explore collective excitations inaccessible in other spectroscopies.

Optical response of Higgs mode in superconductors at clean limit

The phenomenological Ginzburg–Landau theory and the charge conservation directly lead to the finite Higgs-mode generation and vanishing charge-density fluctuation in the second-order optical response of superconductors at clean limit. Nevertheless, recent microscopic theoretical studies of the second-order optical response, apart from the one through the gauge-invariant kinetic equation (Yang and Wu, 2019), have derived a vanishing Higgs-mode generation but finite charge-density fluctuation at clean limit. We resolve this controversy by re-examining the previous derivations with the vector potential alone within the path-integral and Eilenberger-equation approaches, and show that both previous derivations contain flaws. After fixing these flaws, a finite Higgs-mode generation through the drive effect of vector potential is derived at clean limit, exactly recovering the previous result from the gauge-invariant kinetic equation as well as Ginzburg–Landau theory. By further extending the path-integral approach to include electromagnetic effects from the scalar potential and phase mode, in the second-order response, a finite contribution from the drive effect of scalar potential to the Higgs-mode generation at clean limit as well as the vanishing charge-density fluctuation are derived, also recovering the results from the gauge-invariant kinetic equation. Particularly, we show that the phase mode is excited in the second-order response, and exactly cancels the previously reported unphysical excitation of the charge-density fluctuation, guaranteeing the charge conservation.

Phase signatures in the third-harmonic response of Higgs and coexisting modes in superconductors

Third-harmonic generation (THG) experiments on superconductors can be used to investigate collective excitations like the amplitude mode of the order parameter known as Higgs mode. These modes are visible due to resonances in the THG signal if the driving frequency matches the energy of the mode. In real materials, multiple modes can exist giving rise to additional THG contributions, such that it is difficult to unambiguously interpret the results. In this paper, we additionally analyze the phase of the THG signal for the first time, which contains microscopic details beyond classical resonances as well as signatures of couplings between modes that are difficult to observe in the amplitude alone. We investigate how the Higgs mode, impurities, or Coulomb interaction affects the phase response and consider exemplary two systems with additional modes. We argue that extracting this phase information could be valuable in future experiments.

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 $2f>2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\pi}}(T\ensuremath{\rightarrow}0)=1.03$ THz, the THG intensity increases monotonically with decreasing temperature. Moreover, for $2f<2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\pi}}(T\ensuremath{\rightarrow}0)$ 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.

1D topological systems for next-generation electronics

1D topological systems for next-generation electronics

Spin and charge currents driven by the Higgs mode in high-field superconductors

The Higgs mode in superconducting materials describes slowly decaying oscillations of the order parameter amplitude. We demonstrate that in superconductors with a built-in spin-splitting field the Higgs mode is strongly coupled to the spin degrees of freedom, allowing for the generation of time-dependent spin currents. Converting such spin currents to electric signals by spin-filtering elements provides a tool for the second-harmonic generation and the electrical detection of the Higgs mode generated by the external irradiation. The nonadiabatic spin torques generated by these spin currents allow for the magnetic detection of the Higgs mode by measuring the precession of the magnetic moment in the adjacent ferromagnet. We discuss also the reciprocal effect, which is the generation of the Higgs mode by the magnetic precession. Coupling the collective modes in superconductors to light and magnetic dynamics provides an opportunity for the study of superconducting optospintronics.Received 1 July 2019Revised 23 August 2020Accepted 25 August 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.033416Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFFLOOptical conductivitySpin currentSuperconducting phase transitionPhysical SystemsFerromagnetic superconductorsHiggs bosonsMultilayer thin filmsTunnel junctionsTechniquesMethods in superconductivityCondensed Matter & Materials Physics

Bardasis-Schrieffer polaritons in excitonic insulators

Bardasis-Schrieffer modes in superconductors are fluctuations in subdominant pairing channels, e.g., d-wave fluctuations in an s-wave superconductor. This Rapid Communication shows that these modes also generically occur in excitonic insulators. In s-wave excitonic insulators, a p-wave Bardasis-Schrieffer mode exists below the gap energy, is optically active and hybridizes strongly with photons to form Bardasis-Schrieffer polaritons, which are observable in both far-field and near-field optical experiments.

Current-assisted Raman activation of the Higgs mode in superconductors

The Higgs mode in superconductors is a scalar mode without electric or magnetic dipole moment. Thus, it is commonly believed that its excitation is restricted to a nonlinear two-photon Raman process. However, recent efforts have shown that a linear excitation in the presence of a supercurrent is possible, resulting in a new resonant enhancement at $\Omega=2\Delta$ with the driving light frequency $\Omega$ and the energy of the Higgs mode $2\Delta$. This is in contrast to the usual $2\Omega = 2\Delta$ resonance condition found in nonlinear third-harmonic generation experiments. In this communication, we show that such a linear excitation can still be described as an effective Raman two-photon process, with one photon at $\omega=2\Delta$ and one virtual photon at $\omega=0$ which represents the dc supercurrent. At the same time we demonstrate that a straightforward infrared activation with a single photon excitation is negligible. Moreover, we give a general context to our theory, providing an explanation for how the excitation of the Higgs mode in both THz quench and drive experiments can be understood within a conventional difference-frequency generation or sum-frequency generation process, respectively. In such a picture, the observed new resonance condition $\Omega = 2\Delta$ is just a special case. With the same approach, we further discuss another recent experiment, where we find a suppression of odd order higher harmonics in the presence of a dc supercurrent.

Field-free platform for Majorana-like zero mode in superconductors with a topological surface state

Superconducting materials exhibiting topological properties are emerging as an exciting platform to realize fundamentally new excitations from topological quantum states of matter. In this work, we explore the possibility of a field-free platform for generating Majorana zero energy excitations by depositing magnetic Fe impurities on the surface of candidate topological superconductors, LiFeAs and PbTaSe$_2$. We use scanning tunneling microscopy to probe localized states induced at the Fe adatoms on the atomic scale and at sub-Kelvin temperatures. We find that each Fe adatom generates a striking zero-energy bound state inside the superconducting gap, which do not split in magnetic fields up to 8T, underlining a nontrivial topological origin. Our findings point to magnetic Fe adatoms evaporated on bulk superconductors with topological surface states as a new platform for exploring Majorana zero modes and quantum information science under field-free conditions.

Higgs Mode in Superconductors

When the continuous symmetry of a physical system is spontaneously broken, two types of collective modes typically emerge: the amplitude and the phase modes of the order-parameter fluctuation. For superconductors, the amplitude mode is referred to most recently as the Higgs mode as it is a condensed-matter analog of a Higgs boson in particle physics. Higgs mode is a scalar excitation of the order parameter, distinct from charge or spin fluctuations, and thus does not couple to electromagnetic fields linearly. This is why the Higgs mode in superconductors has evaded experimental observations for over a half century after the initial theoretical prediction, except for a charge-density-wave coexisting system. With the advance of nonlinear and time-resolved terahertz spectroscopy techniques, however, it has become possible to study the Higgs mode through the nonlinear light–Higgs coupling. In this review, we overview recent progress in the study of the Higgs mode in superconductors.

Infrared Activation of the Higgs Mode by Supercurrent Injection in Superconducting NbN.

The Higgs mode in superconductors, i.e., the collective amplitude mode of the order parameter, does not associate with charge nor spin fluctuations, therefore it does not couple to the electromagnetic field in the linear response regime. Contrary to this common understanding, here, we demonstrate that if the dc supercurrent is introduced into the superconductor, the Higgs mode becomes infrared active and is directly observed in the linear optical conductivity measurement. We observed a sharp resonant peak at ω=2Δ in the optical conductivity spectrum of a thin-film NbN in the presence of dc supercurrent, showing a reasonable agreement with the recent theoretical prediction. The method as proven by this work opens a new pathway to study the Higgs mode in a wide variety of superconductors.

Theory of Green’s Functions of Nambu-Goldstone and Higgs Modes in Superconductors

We examine fundamental properties of Green’s functions of Nambu-Goldstone and Higgs modes in superconductors with multiple order parameters. Nambu-Goldstone and Higgs modes are determined once the symmetry of the system and that of the order parameters are specified. Multiple Nambu-Goldstone modes and Higgs modes exist when we have multiple order parameters. The Nambu-Goldstone Green function D(ω, q) has the form $1/(gN(0))^{2}\cdot (2{\Delta })^{2}/(\omega ^{2}-{{c}_{s}^{2}}\mathbf {q}^{2})$ with the coupling constant g and $c_{s}=v_{F}/\sqrt {3}$ for small ω and q, with a pole at ω = 0 and q = 0 indicating the existence of a massless mode. It is shown, based on the Ward-Takahashi identity, that the massless mode remains massless in the presence of intraband scattering due to non-magnetic and magnetic impurities. The pole of D(ω, q), however, disappears as ω increases as large as 2Δ: $\omega \sim 2{\Delta }$ . The Green function H(ω, q) of the Higgs mode is given by $H(\omega ,\mathbf {q})\propto (2{\Delta })^{2}/((2{\Delta })^{2}-\frac {1}{3}\omega ^{2} +\frac {1}{3}{c_{s}^{2}}\mathbf {q}^{2})$ for small ω and q, and H(ω, q) is proportional to $1/(gN(0))^{2}\cdot {\Delta }/\sqrt { (2{\Delta })^{2}+{c_{s}^{2}}\mathbf {q}^{2}-\omega ^{2}}$ for $\omega \sim 2{\Delta }$ with the cut in the interval − ω(q) ≤ z ≡ q0 ≤ ω(q) where $\omega (\mathbf {q})=\sqrt { (2{\Delta })^{2}+{c_{s}^{2}}\mathbf {q}^{2}}$ . The constant part of the action for the Higgs modes is important since it determines the coherence length of a superconductor. There is the case that it has a large eigenvalue, indicating that the large upper critical field Hc2 may be realized in a superconductor with multiple order parameters.

Hidden charge-conjugation, parity, and time-reversal symmetries and massive Goldstone (Higgs) modes in superconductors

A massive Goldstone (MG) mode, often referred to as a Higgs amplitude mode, is a collective excitation that arises in a system involving spontaneous breaking of a continuous symmetry, along with a gapless Nambu-Goldstone mode. It has been known in the previous studies that a pure amplitude MG mode emerges in superconductors if the dispersion of fermions exhibits the particle-hole (p-h) symmetry. However, clear understanding of the relation between the symmetry of the Hamiltonian and the MG modes has not been reached. Here we reveal the fundamental connection between the discrete symmetry of the Hamiltonian and the emergence of pure amplitude MG modes. To this end, we introduce nontrivial charge-conjugation ($\mathcal C$), parity ($\mathcal P$), and time-reversal ($\mathcal T$) operations that involve the swapping of pairs of wave vectors symmetrical with respect to the Fermi surface. The product of $\mathcal{CPT}$ (or its permutations) represents an exact symmetry analogous to the CPT theorem in the relativistic field theory. It is shown that a fermionic Hamiltonian with a p-h symmetric dispersion exhibits the discrete symmetries under $\mathcal C$, $\mathcal P$, $\mathcal T$, and $\mathcal{CPT}$. We find that in the superconducting ground state, $\mathcal T$ and $\mathcal P$ are spontaneously broken simultaneously with the U(1) symmetry. Moreover, we rigorously show that amplitude and phase fluctuations of the gap function are uncoupled due to the unbroken $\mathcal C$. In the normal phase, the MG and NG modes become degenerate, and they have opposite parity under $\mathcal T$. Therefore, we conclude that the lifting of the degeneracy in the superconducting phase and the resulting emergence of the pure amplitude MG mode can be identified as a consequence of the the spontaneous breaking of $\mathcal T$ symmetry but not of $\mathcal P$ or U(1).

Two-photon absorption and amplitude mode in conventional superconductors with paramagnetic impurities

The two-photon absorption (TPA) spectrum of BCS superconductors is theoretically investigated. We discuss a possibility of observing the amplitude mode (one of collective modes in superconductors) in this spectrum. An absorption edge by this mode appears at the same frequency as that by quasiparticle excitations in the TPA spectrum of superconductors with time-reversal symmetry, and it will be difficult to distinguish the amplitude mode from quasiparticle excitations. This property originates from the fact that the magnitude of superconducting order parameter is one half of the energy gap in the linear absorption spectrum. In contrast to this one half of the energy gap is smaller than the magnitude of order parameter in superconductors with paramagnetic impurities. Though the amplitude mode is not well-defined owing to the broken time-reversal symmetry, a numerical calculation shows that the peak by this mode persists in the TPA spectrum. This peak appears at the frequency different from that of the absorption edge by quasiparticle excitations, and it is possible to specify the amplitude mode in the TPA spectrum.

Short-distance breakdown of the Higgs mechanism and the robustness of the BCS theory for charged superconductors

Through the Higgs mechanism, the long-range Coulomb interaction eliminates the low-energy Goldstone phase mode in superconductors and transfers spectral weight all the way up to the plasma frequency. Here we show that the Higgs mechanism breaks down for length scales shorter than the superconducting coherence length while it stays intact, even at high energies, in the long-wavelength limit. This effect is a consequence of the composite nature of the Higgs field of superconductivity and the broken Lorentz invariance in a solid. Most importantly, the breakdown of the Higgs mechanism inside the superconducting coherence volume is crucial to ensure the stability of the BCS mean-field theory in the weak-coupling limit. We also show that changes in the gap equation due to plasmon-induced fluctuations can lead to significant corrections to the mean-field theory and reveal that changes in the density-fluctuation spectrum of a superconductor are not limited to the vicinity of the gap.

Tracing the Electronic Pairing Glue in Unconventional Superconductors via Inelastic Scanning Tunneling Spectroscopy.

Scanning tunneling microscopy has been shown to be a powerful experimental probe to detect electronic excitations and further allows us to deduce fingerprints of bosonic collective modes in superconductors. Here, we demonstrate that the inclusion of inelastic tunnel events is crucial for the interpretation of tunneling spectra of unconventional superconductors and allows us to directly probe electronic and bosonic excitations via scanning tunneling microscopy. We apply the formalism to the iron based superconductor LiFeAs. With the inclusion of inelastic contributions, we find strong evidence for a nonconventional pairing mechanism, likely via magnetic excitations.

Majorana Zero Modes Protected by a Hopf Invariant in Topologically Trivial Superconductors.

Majorana zero modes are usually attributed to topological superconductors. We study a class of two-dimensional topologically trivial superconductors without chiral edge modes, which nevertheless host robust Majorana zero modes in topological defects. The construction of this minimal single-band model is facilitated by the Hopf map and the Hopf invariant. This work will stimulate investigations of Majorana zero modes in superconductors in the topologically trivial regime.

Amplitude Higgs Mode and Admittance in Superconductors with a Moving Condensate

We consider the amplitude (Higgs) mode in a superconductor with a condensate flow (supercurrent). We demonstrate that, in this case, the amplitude mode corresponding to oscillations δ|Δ|_{Ω}exp(iΩt) of the superconducting gap is excited by an external ac electric field E_{Ω}exp(iΩt) already in the first order in |E_{Ω}|, so that δ|Δ|_{Ω}∝(v_{0}E_{Ω}), where v_{0} is the velocity of the condensate. The frequency dependence δ|Δ|_{Ω} has a resonance shape with a maximum at Ω=2Δ. In contrast to the standard situation without the condensate flow, the oscillations of the amplitude δ|Δ(t)| contribute to the admittance Y_{Ω}. We provide a formula for admittance of a superconductor with a supercurrent. The predicted effect opens new ways of experimental investigation of the amplitude mode in superconductors and materials with superconductivity competing with other states.

Damping of the collective amplitude mode in superconductors with strong electron-phonon coupling

We study the effect of strong electron-phonon interactions on the damping of the Higgs amplitude mode in superconductors by means of non-equilibrium dynamical mean-field simulations of the Holstein model. In contrast to the BCS dynamics, we find that the damping of the Higgs mode strongly depends on the temperature, becoming faster as the systen approaches the transition temperature. The damping at low temperatures is well described by a power-law, while near the transition temperature the damping shows exponential-like behavior. We explain this crossover by a temperature-dependent quasiparticle lifetime caused by the strong electron- phonon coupling, which smears the superconducting gap edge and makes the relaxation of the Higgs mode into quasiparticles more efficient at elevated temperatures. We also reveal that the phonon dynamics can soften the Higgs mode, which results in a slower damping.