Signatures of Multiband Effects in High-Harmonic Generation in Monolayer ${\mathrm{MoS}}_2$

High-harmonic generation (HHG) in solids has been touted as a way to probe ultrafast dynamics and crystal symmetries in condensed matter systems. Here, we investigate the polarization properties of high-order harmonics generated in monolayer ${\mathrm{MoS}}_2$, as a function of crystal orientation relative to the mid-infrared laser field polarization. At several different laser wavelengths we experimentally observe a prominent angular shift of the parallel-polarized odd harmonics for energies above approximately 3.5 eV, and our calculations indicate that this shift originates in subtle differences in the recombination dipole strengths involving multiple conduction bands. This observation is material specific and is in addition to the angular dependence imposed by the dynamical symmetry properties of the crystal interacting with the laser field, and may pave the way for probing the vectorial character of multiband recombination dipoles.

Figure 1

Experimental HHG spectra measured in the parallel (left column) and perpendicular configuration (right column) for ${\mathrm{MoS}}_2$ irradiated by intense MIR pulses of central wavelength $3.4\mu \mathrm{m}$ (top row), $3.7\mu \mathrm{m}$ (middle row), and $4.4\mu \mathrm{m}$ (bottom row). Yields are plotted in linear scale and arbitrary units. In the insets, the two polarizer and analyzer configurations are sketched, with $\alpha$ being the angle between the crystal mirror plane and the laser polarization (red arrow), and the polarizer direction indicated by the green arrow. The magenta bullets mark the nodes predicted by a DS analysis, and the dashed ellipses mark the angular shifts of the parallel odd harmonics.

Figure 2

Experimental and simulation $\alpha$-dependent HHG yields with a $3.7\mu \mathrm{m}$ driver, for the (a) parallel and (b) perpendicular configurations. The yields are normalized to the maximum.

Figure 3

Simulated harmonic yields as a function of $\alpha$, for the parallel-polarized (a) H9 and (b) H13. The full (black) curves are the full calculations, the dashed (red) curves are for calculations with ${\tilde{\mathbf{p}}}_{13}^{\mathbf{k}}=\mathbf{0}$, and the dotted (blue) curves are with ${\tilde{\mathbf{p}}}_{12}^{\mathbf{k}}=\mathbf{0}$. The insets show sketches of the dominating pathways for generating H9 and H13 (see text).

Figure 4

Modulus squares of the recombination dipoles along (a) $\widehat{\mathbf{x}}$ ($\alpha =30°$) and (b) $\widehat{\mathbf{y}}$ ($\alpha =0°$) directions in a.u. The gray hexagon traces the first Brillouin zone. Panels (c) and (d) plot the same as (a) and (b), but along paths in reciprocal space marked by the yellow lines. Panel (e) and (f) show the band gaps along these paths, with the gray (black) curve between CB1 (CB2) and the VB. The areas marked by gray indicate regions accessible by electron-hole pairs created at the $K$ point, with maximum excursions $\pm A_0$, and the orange dots in (c) and (d) highlight the maximum values in these regions.