Classification and characterization of nonequilibrium Higgs modes in unconventional superconductors

Lukas Schwarz and Dirk Manske

Understanding the complex properties of unconventional superconductors is important for designing and controlling new high-temperature superconducting materials. A trilateral collaboration of scientists from Stuttgart, Vancouver and Tokyo headed by Dirk Manske (MPI) and embedded in the Max Planck–UBC–UTokyo Center for Quantum Materials has proposed a classification scheme for excitations of the superconducting order parameter in nonequilibrium, which allows to gain information of the underlying symmetry properties of the superconducting condensate. Such a classification provides a new spectroscopic tool for future experiments to study the properties of novel superconductors.

In recent years it was realized that one can obtain information about ground state properties of a superconductor performing nonequilibrium pump-probe experiments. This is possible due to new ultrafast THz laser technology, which allows to perform controlled excitations of materials without destroying its coherent state. Hereby, an ultrashort THz laser pump pulse excites the superconductor (i.e. performs a quantum quench) and the nonequilibrium response is captured by a subsequent probe pulse after a time delay. Repeating the experiment with a variation of the time delay allows to scan the time-evolution of the excitation spectrum.

The symmetry properties of the superconducting wave function or order parameter, which correspond to an energy gap Δ in the excitation spectrum of a superconductor, control most of the spectroscopic properties as well as critical currents or critical magnetic fields and are therefore important to know for a given material. While conventional superconductors have an isotropic energy gap (s-wave symmetry), most unconventional superconductors have an anisotropic energy gap, like d-wave in the high-temperature cuprates. Hereby, the order parameter changes its sign across nodal lines at the diagonals, where the gap vanishes. In the nodal directions, excitations of Cooper pairs into quasiparticles are possible without energy costs. For most of the cuprates, the symmetry of the order parameter is known, however there are other new classes of unconventional superconductors, where there is still a debate about the symmetry of the superconducting wave function. So far, typical experiments to gain information about the gap symmetry include angle-resolved photoemission spectroscopy (ARPES) or experiments based on superconducting quantum interference devices (SQUIDs). Yet, both experiments have their difficulties to be conducted and one cannot obtain information about magnitude and phase of the order parameter simultaneously.

Free Energy 𝓕 of a superconductor as function of the order parameter Δ with the shape of a Mexian hat in equilibrium at t=t0. In a THz pump-probe experiment, the pump pulse acts as a quench, which shrinks the Mexican hat at t=t1. The order parameter, depicted as the black ball, is displaced in an out of equilibrium position, where it starts to oscillate with its characteristic frequency around the new minimum. This radial oscillation corresponds to the Higgs mode. The oscillation can be captured by a second probe pulse. The Fourier spectrum |Δ(ω)| of the Higgs oscillations |Δ(t)| shows peaks at the energies of the Higgs modes ωH. Depending on the gap symmetry, asymmetric oscillations of the condensate are possible, which can be classified according to group symmetry. These asymmetric oscillations for unconventional can lead to additional Higgs modes. — Free Energy 𝓕 of a superconductor as function of the order parameter Δ with the shape of a Mexian hat in equilibrium at t=t₀. In a THz pump-probe experiment, the pump pulse acts as a quench, which shrinks the Mexican hat at t=t₁. The order parameter, depicted as the black ball, is displaced in an out of equilibrium position, where it starts to oscillate with its characteristic frequency around the new minimum. This radial oscillation corresponds to the Higgs mode. The oscillation can be captured by a second probe pulse. The Fourier spectrum |Δ(ω)| of the Higgs oscillations |Δ(t)| shows peaks at the energies of the Higgs modes ω_H. Depending on the gap symmetry, asymmetric oscillations of the condensate are possible, which can be classified according to group symmetry. These asymmetric oscillations for unconventional can lead to additional Higgs modes.

© Max Planck Institute for Solid State Research

Free Energy 𝓕 of a superconductor as function of the order parameter Δ with the shape of a Mexian hat in equilibrium at t=t₀. In a THz pump-probe experiment, the pump pulse acts as a quench, which shrinks the Mexican hat at t=t₁. The order parameter, depicted as the black ball, is displaced in an out of equilibrium position, where it starts to oscillate with its characteristic frequency around the new minimum. This radial oscillation corresponds to the Higgs mode. The oscillation can be captured by a second probe pulse. The Fourier spectrum |Δ(ω)| of the Higgs oscillations |Δ(t)| shows peaks at the energies of the Higgs modes ω_H. Depending on the gap symmetry, asymmetric oscillations of the condensate are possible, which can be classified according to group symmetry. These asymmetric oscillations for unconventional can lead to additional Higgs modes.

© Max Planck Institute for Solid State Research

In superconductors, there exists a collective excitation of the order parameter, i.e. an oscillation of the energy gap with a characteristic frequency, which is called Higgs mode. It is an analogon to the Higgs boson in high energy physics from which it bears its name. This mode is typically difficult to excite as it does not couple to the usual linear light probes, but an excitation is possible within a nonequilibrium THz pump-probe experiment. It is known that the energy of the Higgs mode in s-wave superconductors is 2Δ, i.e. it corresponds to the value of the binding energy of Cooper pairs. For unconventional superconductors a much richer spectrum can occur.

In their work, the researchers have calculated the nonequilibrium Higgs oscillations for arbitrary gap symmetry. They have demonstrated that for anisotropic gap symmetry asymmetric oscillations of the superconducting condensate are possible. These lead to a characteristic spectrum, where multiple Higgs modes can occur. Experiments with a controlled excitation of these asymmetric oscillations and an observation of the resulting Higgs modes allow therefore, in principle, to deduce the underlying gap symmetry. Thus, the classification and characterization of the Higgs modes introduced in the work of the researches provides a new spectroscopic tool for experiments to gain information about the gap symmetry for new superconductors with unknown symmetry. This opens a new field in the area of superconductivity, called ″Higgs spectroscopy″.