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

Abstract

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.

Theory of coherent-oscillations generation in terahertz pump-probe spectroscopy: From phonons to electronic collective modes

“…While in a generic lattice model the Higgs is always excited by a light pulse [23][24][25][26][27][28] , its contribution to the differential probe field (49) is crucially determined by its Raman visibility. We then show that in general the experimental observation of THz-induced coherent oscillations at twice the gap value 5 or at twice the pump frequency 6 could be well reproduced by considering only density-like fluctuations, which certainly dominate the kernel K(ω) in the clean limit, as it has been widely discussed within the context of the third-harmonic generation 32,[34][35][36][37][38][39] . More recently, it has been pointed out 38,39 that in the presence of disorder the hierarchy of the two contributions can be reverted, making the Higgs contribution to the non-linear kernel K(ω) the dominant one.…”

“…(1) the differential field δE probe (t pp ) shows marked oscillations when entering in the SC state [4][5][6][7][8] . At the same time, below the SC critical temperature T c a marked THG has been observed 6,11,31 , which has been also ascribed to SC collective modes 6,8,[32][33][34][35][36][37][38][39] . For the case of conventional superconductors, like thin films of NbN, both pump-probe protocols and THG measurements show that the relevant frequency is twice the value of the SC gap ∆.…”

“…In Ref. 38 it has been argued that the Leggett mode gives a negligible contribution to the THG when the system is excited with a light pulse matching the larger gap value, 2Ω pump 2∆ 2 ω L . However, this result cannot be simply ascribed to the effect of disorder, since even in the clean case the frequency of the pump would be too high to excite the Leggett mode, and it will then always give negligible contribution to the non-linear kernel.…”

“…the Debye frequency for phonon-mediated superconductivity. For the Higgs contribution we have that χ H (ω) = χ 2 A 2 ∆ δ∆ 2 (ω), with δ∆ 2 (ω) the propagator for the Higgs mode and χ A 2 ∆ the Raman-like coupling between the gauge field and amplitude fluctuations δ∆, which explicitly reads 32,34,[37][38][39] :…”

“…A microscopic derivation of the Raman kernel for the amplitudon within a simplified electron-phonon model has ben recently provided in Ref.s 60 and 61. The model system is based on the electron-phonon Hamiltonian: (38) where ξ k = −2t(cos k x + cos k y ) − µ is the tight-binding dispersion on the square lattice, with t the hopping parameter and µ the chemical potential, such that at half filling (µ = 0) the band dispersion displays perfect nesting at Q = (π, π). As a consequence, the finite coupling g to a phonon mode at Q induces a CDW instability.…”