Anomalous terahertz nonlinearity in disordered s-wave superconductor close to the superconductor-insulator transition

TL;DR

Disorder near the superconductor–insulator transition in NbN fundamentally reshapes terahertz nonlinear response and Higgs-mode visibility. By performing THG measurements on NbN films with varying $k_F l$ and comparing to disordered Au, the study reveals a weak normal-state THG above $T_c$ in the strongly disordered sample that remains under high magnetic fields, signaling disorder-driven electronic nonlinearity rather than superconducting fluctuations. Below $T_c$, THG is enhanced and dominated by the driven Higgs mode, while a multi-peak spectrum arises from interference between THG pathways associated with mesoscopic superconducting islands. The results connect disorder-tuned nonlinear THz response to mesoscopic inhomogeneity and are interpreted within the LO framework, highlighting how disorder fosters coupling between normal-state nonlinearity and the superconducting Higgs response near the SIT.

Abstract

Detection of the Higgs mode in superconductors using nonlinear terahertz spectroscopy is a key area of interest in condensed matter physics. We investigate the influence of disorder on the nonlinear terahertz response and the Higgs mode in NbN thin films with varying Ioffe-Regel parameters ($k_Fl$). In strongly disordered films near the superconductor-insulator transition (SIT), we observe an anomalous third-harmonic generation (THG) signal above $T_c$, which is absent in both cleaner superconducting and non-superconducting counterparts. The persistence of this normal-state THG signal in a high magnetic field excludes superconducting fluctuations as its origin. Below $T_c$, the THG intensity increases sharply, indicating a dominant contribution from the driven Higgs mode. The THG spectrum of the strongly disordered sample exhibits a broadened, multi-peak structure, which we attribute to quantum path interference between distinct channels involving unpaired electrons and Cooper pairs within emergent superconducting islands. Our findings not only demonstrate how disorder tunes the nonlinear terahertz response but also uncover a strong coupling between electrons responsible for normal-state THG and the superconducting Higgs mode below $T_c$ in strongly disordered samples.

Figures (13)

• Figure 1: THG response in NbN films with varying disorder levels. (a–d) Time-domain transmitted THz pulses for disordered NbN with $T_c$ values of (a) 15 K, (b) 13 K, (c) 5.9 K, and (d) 0 K under 0.42 THz driving. Insets show the corresponding Fourier transform spectra. A 1.26 THz bandpass filter was applied post-sample to attenuate the fundamental frequency. (e–g) Temperature-dependent THG intensity for samples with $T_c$ = (e) 15 K, (f) 13 K, and (g) 5.9 K. The estimated pump electric field strengths inside the samples (taking into account the Fabry–Pérot effect) are labeled in the figures, which are about one order smaller than the field strength before the sample due to the low transmission coefficient. The inset in (g) highlights the THG intensity above $T_c$. THG intensity is defined as the spectral integral of the $3\omega$ peak. Shaded regions represent error bounds (noise floors of frequency spectra). Error bars for $T_c$ = 13 K and 15 K are negligible relative to line widths. (h) Normalized THG intensity versus temperature. For the non-superconducting sample ($k_Fl < 1$), $T_c$ = 5.9 K is used for $T/T_c$ normalization.
• Figure 2: Magnetic field evolution of the linear and nonlinear THz response for the $T_c = 5.9$ K sample. (a) Real part, $\sigma_1(\omega)$, and (b) imaginary part, $\sigma_2(\omega)$, of optical conductivity at various magnetic fields, measured at 2 K. The magnetic field is applied perpendicular to the sample surface. The dashed black line represents the reference measurement at 6 K under zero magnetic field. (c) Time-domain waveforms of the THG signal measured at 2 K under different magnetic fields. The light gray curve at the top shows the fundamental waveform. Both THG and fundamental components are extracted from the transmitted waveforms (see SM) by digital filtering: THG waveforms are obtained using a high-pass filter with a cutoff frequency of 0.75 THz, while the fundamental waveform is obtained using a low-pass filter with the same cutoff frequency. (d) FFT of transmitted THz pulses measured at 2 K under different magnetic fields. A multi-peak feature appears in the superconducting state. (e) Temperature dependence of THG intensity at different magnetic field strengths. The inset shows a magnified view of the normal-state signal.
• Figure 3: THG signal of disordered Au films driven by 0.5 THz. (a) 1.2 THz high-pass filtered time-domain waveforms of transmitted THz pulses for different $k_Fl$ values measured at 5 K. (b) $k_Fl$ dependence of normalized THG intensity and $|\chi^{(3)}|$ measured at 5 K. $|\chi^{(3)}|$ is estimated by $I_{3\omega}/I_\omega^3$, where the intensity of the fundamental pulse, $I_\omega$, was measured at 5 K without a 3$\omega$ bandpass filter. (c) Heat map showing the temperature and $k_Fl$ dependence of THG intensity.
• Figure 4: Optical conductivity in the superconducting state. The normalized real part of the optical conductivity, $\sigma_1(\omega)/\sigma_1(N)$, is shown for (a) a sample with $k_Fl \sim 2.5$ and (b) a sample with $k_Fl \sim 5.5$, measured at 2 K. The blue dashed line and red solid line correspond to fits using the Mattis-Bardeen and Larkin-Ovchinnikov models, respectively. (c) Schematic of THG in a disordered sample, where superconducting islands are depicted as circles. The size and number of Cooper pairs represent the pairing amplitude, while their alignment indicates phase coherence.
• Figure S1: Resistivity of the four disordered NbN films.