Band-selective third-harmonic generation in superconducting ${\mathrm{MgB}}_2$: 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 $T_c\approx 36$ 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 $2f=2{\mathrm{\Delta }}_{\pi }\left(T\right)$ fulfilled, for the dirty-limit superconducting gap $2{\mathrm{\Delta }}_{\pi }$. In contrast, for $f=0.6$ and 0.7 THz with $2f>2{\mathrm{\Delta }}_{\pi }(T\to 0)=1.03$ THz, the THG intensity increases monotonically with decreasing temperature. Moreover, for $2f<2{\mathrm{\Delta }}_{\pi }(T\to 0)$ 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 $\pi$-band superconducting gap, pointing to the importance of scattering for observation of the Higgs mode in superconductors.

Figure 1

(a) Layered crystallographic structure of $\mathrm{Mg}{\mathrm{B}}_2$. THz electromagnetic waves propagated along the $c$ axis. (b) Transmitted THz electric field as a function of time delay, for $\mathrm{Mg}{\mathrm{B}}_2/\mathrm{Mg}\mathrm{O}$ at 4 and 36 K and for the bare MgO substrate at 4 K. The data for $\mathrm{Mg}{\mathrm{B}}_2/\mathrm{Mg}\mathrm{O}$ at 4 and 36 K are magnified by a factor of 30 for clarity. (c) Ratio of transmission for various temperatures with respect to 36 K. The arrows indicate the frequencies at which the ratio curves cross unity. (d) Real ${\sigma }_1$ and imaginary ${\sigma }_2$ parts of optical conductivity at 36 and 4 K. The arrow indicates the crossing point of the ${\sigma }_1$ and ${\sigma }_2$ curves at ${\tau }^{-1}=1.34$ THz for 36 K. (e) Ratio of ${\sigma }_1$ for various temperatures with respect to 36 K. The dashed line indicates the excitation gap $2{\mathrm{\Delta }}_{\pi }=1.03$ THz at 4 K.

Figure 2

(a) Electric field of the $f=0.5$ THz pump pulse in the time domain. (b) Spectrum of the generated radiation from the sample at 6 K. (c) Temperature dependence of superconducting gaps as determined by scanning tunneling microscopy (STM) for the $\sigma$ and $\pi$ bands (squares and circles) [31], and as estimated from Fig. 1 (spheres). Leggett mode vs temperature as observed in Raman spectroscopy (left triangles) [44, 45] and obtained from theoretical calculations (right triangles) [46]. The horizontal solid lines indicate $2f$ for $f=0.3$, 0.4, 0.5, 0.6, and 0.7 THz, respectively. (d) The corresponding normalized THG intensity $I_{3f}$ as a function of temperature, measured through two bandpass filters of $3f$. The vertical dashed lines indicate the peak positions in the temperature-dependent curves, where the corresponding resonance condition $2f=2{\mathrm{\Delta }}_{\pi }$ is fulfilled simultaneously.

Figure 3

(a) Schematic illustration of the setup for polarization-dependent THG measurements. The THz electromagnetic waves propagate along the sample $c$ axis. BPF-$f$ and BPF-3$f$ denote bandpass filters for the fundamental frequency and the THG, respectively. WGP's denote wire-grid polarizers. (b) Wave form of the $f=0.4$ THz pump pulse for the polarization-dependent measurements. (c) Normalized THG intensity as a function of the pump-pulse polarization, varied by tuning the angle ${\theta }_2$ of WGP2, which is obtained for $f=0.4$ and 0.5 THz at 20 and 4.6 K, respectively.