Polarization-resolved terahertz third-harmonic generation in a single-crystal superconductor NbN: Dominance of the Higgs mode beyond the BCS approximation

Recent advances in time-domain terahertz (THz) spectroscopy have unveiled that resonantly enhanced strong THz third-harmonic generation (THG) mediated by the collective Higgs amplitude mode occurs in $s$-wave superconductors, where charge-density fluctuations (CDFs) have been shown to also contribute to the nonlinear third-order susceptibility. It has been theoretically proposed that the nonlinear responses of Higgs and CDF exhibit essentially different polarization dependences. Here we experimentally discriminate the two contributions by polarization-resolved intense THz transmission spectroscopy for a single-crystal NbN film. The result shows that the resonant THG in the transmitted light always appears in the polarization parallel to that of the incident light with no appreciable polarization-angle dependence relative to the crystal axis. When we compare this with the theoretical calculation here with the BCS approximation and the dynamical mean-field theory for a model of NbN constructed from first principles, the experimental result strongly indicates that the Higgs mode rather than the CDF dominates the THG resonance in NbN. A possible mechanism for this is the retardation effect in the phonon-mediated pairing interaction beyond BCS.

Figure 1

(a) Band structure of NbN obtained from a first-principles calculation with the weights of the band character of Nb 4d $e_g$, 4d $t_{2g}$, and N $2p$ displayed in green, orange, and blue, respectively. Red curves represent the bands in the effective three-orbital model. (b) Nb $4d_{xy}$ orbitals on the fcc lattice. The arrows represent the hoppings with amplitudes $t,t^{\prime }$, and $t^{\prime \prime }$. (c) Sets of two-dimensional square lattices made of $d_{xy},d_{yz}$, and $d_{zx}$ orbitals respectively on $xy,yz$, and $zx$ planes with a rotation of ${45}^{\circ }$ on each plane.

Figure 2

(a, b) The CDF contribution to the THG intensity spectra ${\left|\chi \left(\omega \right)\right|}^2$ with ${\mathit{e}}^I={\mathit{e}}^O={\mathit{e}}_{\theta }$ for $\theta =0^{\circ }$ (lower curve) and 45° (upper) calculated in the BCS approximation (a) and DMFT (b). (c, d) The polarization dependence of the CDF and Higgs-mode contributions to the THG intensity ${\left|\chi \left(\omega \right)\right|}^2$ at resonance $\left(2\omega =2\mathrm{\Delta }\right)$ with ${\mathit{e}}^I={\mathit{e}}^O={\mathit{e}}_{\theta }$ calculated in the BCS approximation (c) and DMFT (d). The intensity in each panel is normalized by the value at $\theta =0^{\circ }$. For the BCS approximation we take $\mathrm{\Delta }=2.7\mathrm{meV}=0.65\mathrm{THz}$ and $T=4\mathrm{K}$ and for DMFT we take the Holstein model with ${\omega }_0=0.5\mathrm{eV},g=3.0\mathrm{eV}$, and $T=0.05\mathrm{eV}$.

Figure 3

(a) A schematic experimental setup for THz transmission spectroscopy. WGP: wire-grid polarizer, BPF: bandpass filter. Inset shows the electric field polarization along the crystal axis on the sample surface. (b) Experimental result for the power spectra of the transmitted THz pulse with ${\theta }_2=22.5^{\circ }$. The black curve is obtained above $T_{\mathrm{c}}$. Red and blue curves show the data at $2\omega =2\mathrm{\Delta }\left(T\right)$ for polarizations parallel and perpendicular to the incident field, respectively. (c) Squared nonlinear susceptibility ${\left|\chi \right|}^2$, normalized at ${\theta }_2=0^{\circ }$, as a function of the incident polarization angle ${\theta }_2$.