Ultrafast Electron-Electron Scattering in Metallic Phase of $2\mathrm{H}\text{−}{\mathrm{NbSe}}_2$ Probed by High Harmonic Generation

Electron-electron scattering on the order of a few to tens of femtoseconds plays a crucial role in the ultrafast electron dynamics of conventional metals. When mid-infrared light is used for driving and the period of light field is comparable to the scattering time in metals, unique light-driven states and nonlinear optical responses associated with the scattering process are expected to occur. Here, we use high-harmonics spectroscopy to investigate the effect of electron-electron scattering on the electron dynamics in thin film $2\mathrm{H}\text{−}{\mathrm{NbSe}}_2$ driven by a mid-infrared field. We observed odd-order high harmonics up to 9th order as well as a broadband emission from hot electrons in the energy range from 1.5 to 4.0 eV. The electron-electron scattering time in 2H-${\mathrm{NbSe}}_2$ was estimated from the broadband emission to be almost the same as the period of the mid-infrared light field. A comparison between experimental results and a numerical calculation reveals that competition and cooperation between the driving and scattering enhances the nonperturbative behavior of high harmonics in metals, causing a highly nonequilibrium electronic state corresponding to several thousand Kelvin.

Figure 1

(a) Typical HHG spectra in $2\mathrm{H}\text{−}{\mathrm{NbSe}}_2$. Here, the MIR electric field inside the sample $E_{\mathrm{MIR}}$ is estimated to be $3.8\mathrm{MV}/\mathrm{cm}$. The blue line in the upper panel shows the emission polarized parallel to the MIR electric field. The orange shaded area shows the perpendicular component. The lower panel is the high harmonics spectrum obtained by subtracting the perpendicular component from the parallel component. Owing to the strong background signal by fundamental laser (1.55 eV), the spectrum between the 5th and 7th harmonics are noisy, and not shown for clarity. (b) Nonlinear emission intensity as a function of the transmission axis angle $\theta$ of the analyzer. Here, $\theta =0°$ corresponds to the MIR electric field direction. Red circles and blue rectangles show the intensities corresponding to the 5th and 7th harmonics energy, respectively. Owing to the much stronger signal of the 5th and 7th harmonic emissions than the broadband emission, the contribution from the broadband emission in their polarization state is negligible. Green triangles show the broadband emission integrated over the region between 2.5 and 4.0 eV. Note that the polarization state at the 9th harmonics energy is not shown due to the low signal-to-noise ratio and comparable to broadband emission intensity. (c) Broadband emission spectrum divided by the absorptivity of the sample with $E_{\mathrm{MIR}}=3.8\mathrm{MV}/\mathrm{cm}$ [24]. The dotted line represents the fitting result using the Planck distribution. The effective temperature estimated from the fitting is about 3800 K.

Figure 2

(a) Schematic time evolution of the carrier distribution. Left panel shows the temporal profile of MIR electric field and the corresponding vector potential. Middle and right panels show the electron distribution in a one-dimensional cosine band at each time without and with the scattering effect, respectively. (b) Total energy of electron distribution of $2\mathrm{H}\text{−}{\mathrm{NbSe}}_2$ after MIR-pulse excitation with $E_{\mathrm{MIR}}=3.8\mathrm{MV}/\mathrm{cm}$. The red line shows the total energy of the electron distribution that is asymptotic to the total energy before the excitation in the limit ${\gamma }_{\mathrm{eff}}^{-1}\to \infty$. Black dotted lines show the total energies of electrons for several temperatures.

Figure 3

Comparison of experimental and calculated results. (a) High harmonic spectra with $E_{\mathrm{MIR}}=3.8\mathrm{MV}/\mathrm{cm}$. Orange solid and black dashed lines show the experimental and numerical results (${\gamma }_{\mathrm{eff}}^{-1}=13\mathrm{fs}$), respectively. (b) Crystal orientation dependence of the 5th and 7th harmonics. Orange solid circles and blue open squares show the experimental results for the 5th and 7th harmonics, respectively. The brown dotted and dark-blue dashed lines show the corresponding numerical results ($E_{\mathrm{MIR}}=3.8\mathrm{MV}/\mathrm{cm}$ and ${\gamma }_{\mathrm{eff}}^{-1}=13\mathrm{fs}$). 0 degrees corresponds to the $\mathrm{\Gamma }-\mathrm{K}$ (zigzag) direction. For clarity, the intensity of the 7th harmonics is magnified. (c) Ratio of $n$ th order harmonic intensity to the $n$th power of MIR intensity ($I_n/I_{\mathrm{MIR}}^n$) as a function of MIR electric field. Symbols (circles and squares), solid, and dotted lines, respectively, show the experimental results, numerical results with and without scattering effect (${\gamma }_{\mathrm{eff}}^{-1}=13\mathrm{fs}$ and $=\infty$). The upper panel shows the 5th harmonic and lower panel shows the 7th. Here, all the data are normalized by the value at $E_{\mathrm{MIR}}=2.5\mathrm{MV}/\mathrm{cm}$ for clarity. (d) Ratio of the numerically simulated 5th harmonic intensity to the fifth power of MIR intensity ($I_5/I_{\mathrm{MIR}}^5$) as a function of MIR electric field at certain scattering times. The blue solid line shows the results without the scattering effect. Green dotted, yellow dashed, and red dashed-dotted lines show the results with an effective scattering time ${\gamma }_{\mathrm{eff}}^{-1}$ of 29, 17, and 13 fs, respectively. Here, all the data are normalized by the value at $E_{\mathrm{MIR}}=2.5\mathrm{MV}/\mathrm{cm}$. The shaded area shows the range of the MIR electric field in (c). In (b) and (c), the data corresponding to the 9th harmonics are not shown due to the low signal-to-noise ratio of the experiment.