All-Optical Reconstruction of Crystal Band Structure

The band structure of matter determines its properties. In solids, it is typically mapped with angle-resolved photoemission spectroscopy, in which the momentum and the energy of incoherent electrons are independently measured. Sometimes, however, photoelectrons are difficult or impossible to detect. Here we demonstrate an all-optical technique to reconstruct momentum-dependent band gaps by exploiting the coherent motion of electron-hole pairs driven by intense midinfrared femtosecond laser pulses. Applying the method to experimental data for a semiconductor ZnO crystal, we identify the split-off valence band as making the greatest contribution to tunneling to the conduction band. Our new band structure measurement technique is intrinsically bulk sensitive, does not require a vacuum, and has high temporal resolution, making it suitable to study reactions at ambient conditions, matter under extreme pressures, and ultrafast transient modifications to band structures.

Figure 1

Comparison between photoemission and high harmonic generation. In photoemission experiments, an incoming photon (purple arrow) causes emission of an electron (black vertical arrow) with kinetic energy ($K_1$ and $K_2$) proportional to the initial binding energy ($E_1$ and $E_2$). In high harmonic generation, an electron first tunnels (marked by circled 1) to the conduction band. The electron-hole pair is then accelerated (circled 2) and recombines (circled 3) emitting a harmonic photon in the process. Electrons and holes are denoted by filled and empty blue circles, respectively. Two trajectories are identified by the wiggled solid black arrows, corresponding to different times of creation and recollision of the electron-hole pair.

Figure 2

Reconstruction of the bands. (a) High harmonic spectrum obtained from the target band structure as a function of delay between fundamental and second harmonic fields. The red line with dots is the optimum phase ${\mathrm{\Phi }}_{\text{osc}}$. (b) Target ${\mathrm{\Phi }}_{\text{osc}}$ (red dots with error bars) compared to those associated with the retrieved band structures (gray lines). (c) Target (red line) and retrieved (gray lines) momentum-dependent band gaps. The Brillouin zone extends up to half of the reciprocal lattice vector. The target band gap is $ϵ(k)={ϵ}_g+0.139$ $[1-\cos (ka)]+0.011$ [$1-\cos (2ka)$], and approximates that of a ZnO crystal as obtained from ab initio calculations. The crystal and laser parameters are defined in Table S2 of the Supplemental Material [19].

Figure 3

Experimental contribution of the split-off band. (a) Experimental spectrogram obtained from a ZnO crystal. The same setup of Ref. [15] has been used to collect this data. Experimental parameters are the same as in Fig. 2. Each harmonic order is independently normalized to facilitate comparison between orders. The odd harmonics saturate the spectrometer. The red line is the experimental ${\mathrm{\Phi }}_{\text{osc}}$. (b) The optimum phase extracted from the experiment (black dots with error bars), simulated for tunneling from the split-off valence band (green line), the heavy holes band (gold line) and the light holes band (red line). Contrary to Fig. 2, here we use an unmeasured offset phase to position the experimental data. This phase is, in principle, measurable in a more refined experiment. (c) Band structure for the same bands of part (b) with the same color coding. The blue band is the first conduction band. They are obtained from ab initio calculations. The arrows mark the recombination between electrons and holes in the split-off band for the 10th and 16th harmonic orders. The measured harmonic spectrum extends up to $\sim 25\%$ of the Brillouin zone.