Visualization of two-dimensional transition dipole moment texture in momentum space using high-harmonic generation spectroscopy

Highly nonlinear optical phenomena can provide access to properties of electronic systems which are otherwise difficult to access through conventional linear optical spectroscopies. In particular, high-harmonic generation (HHG) in crystalline solids is strikingly different from that in atomic gases, and it enables us to access electronic properties such as the band structure, Berry curvature, and valence electron density. Here, we show that polarization-resolved HHG measurements with band-gap resonant excitation can be used to probe the transition dipole moment (TDM) texture in momentum space in two-dimensional semiconductors. TDM is directly related to the internal structure of the electronic system and governs the optical properties. We study HHG in black phosphorus, which offers a simple two-band system. We observed a unique crystal-orientation dependence of the HHG yields and polarizations. Resonant excitation of band edge enables us to reconstruct the TDM texture related to the interatomic bonding structure. Our results demonstrate the potential of high-harmonic spectroscopy for probing electronic wave functions in crystalline solids.

Figure 1

(a) Schematic diagram of interband-resonant HHG process in crystal momentum space. Orange solid arrows indicate driving electric field direction. The upper figure shows that electron-hole pairs are created at ${\mathit{k}}_{\mathrm{i}}$ (orange dashed line) and then accelerated in the direction of the driving field in their respective bands (red and blue solid lines), and they finally emit radiation at ${\mathit{k}}_{\mathrm{r}}$ (yellow green dashed line). ${\mathit{k}}_{\mathrm{AC}}$ and ${\mathit{k}}_{\mathrm{ZZ}}$ respectively denote the crystal momentum components along the armchair (AC) and zigzag (ZZ) directions of BP. The lower panel shows the same diagram projected on a two-dimensional plane. Here, θ(ϕ) is defined as the angle between the driving field (transmission axis of the polarizer for HHG emission) and the AC direction. The gray solid line represents the isoenergy line. (b) Crystal structure of puckered-layer black phosphorus from the upper side. A, B, C, and D indicate four sublattices. (c) Image of the black phosphorus flake used in the experiment. The horizontal direction corresponds to the AC direction. (d) Typical HHG spectrum from black phosphorus with MIR intensity $I_{\mathrm{MIR}}$ of $0.37\mathrm{TW}/\mathrm{c}{\mathrm{m}}^2$ at a focal point in air and MIR polarization $\theta ={35}^{\circ }$.

Figure 2

Crystal-orientation dependence of HHG yields for the (a) third, (b) fifth, (c) seventh, and (d) ninth harmonics.

Figure 3

(a,b) Polarization state of the (a) third and (b) fifth harmonics for MIR polarization $\theta =0^{\circ }$. (c,d) Polarization state of the (c) third and (d) fifth harmonics for MIR polarization $\theta ={40}^{\circ }$.

Figure 4

(a) Experimentally obtained in-plane TDM texture. The origin is set to be the $Z$ point. Red, orange, green, and blue solid lines show isoenergy lines corresponding to the emission energy of the third (0.8 eV), fifth (1.3 eV), seventh (1.8 eV), and ninth (2.3 eV) harmonics, respectively. The peak amplitude of the TDM vector on each isoenergy line is normalized for clarity. The gray shaded area indicates the region where we cannot obtain TDM due to the small signal to noise ratio. (b) Calculated TDM texture corresponding to (a) without correction. (c,d) Normalized crystal-orientation dependence of the third (c) and fifth (d) harmonic yields. (e,f) Normalized polarization states of (e) the third and (f) fifth harmonics for $\theta ={40}^{\circ }$. Solid lines in (c–f) indicate the ones calculated by using the corrected TDM texture in which the ZZ component is three times that of the tight-binding model (b). The corrected TDM is provided in the Supplemental Material [Fig. S14(c)] [27]. Open circles and squares in (c–f) indicate the corresponding experimentally obtained data [same as Figs. 2, 2, 3, and 3].