All-Optical Reconstruction of Crystal Band Structure
Authors: Giulio Vampa1, Thomas Hammond1, Nicolas Thiré2, Bruno Schmidt2, François Légaré2, Chris McDonald1, Thomas Brabec1, Dennis Klug3,
1Department of Physics, University of Ottawa, Ontario, Canada,
2INRS-EMT, Varennes, Québec, Canada,
3National Research Council of Canada, Ottawa, Ontario, Canada.
The electronic band structure of solids determines their key properties, for example electrical conductivity and optical absorption. The energy bands are conventionally measured with Angle Resolved Photoemission Spectroscopy (ARPES), in which the kinetic energy and the momentum of electrons ionized by highly energetic photons are simultaneously measured . The working principle is sketched in the left of Fig. 1. ARPES has been an incredibly successful method to probe conventional and less conventional materials, such as superconductors , graphene  and topological insulators .
However, electrons are not always accessible. Only high vacuum must oppose their propagation from the surface of the sample to the detector. This prevents measuring the band structure of the bulk, or of matter under extreme pressures, where the sample is enclosed in a diamond anvil cell, or in high magnetic fields, where electrons are deviated. Further, many surface chemical reactions that happen at ambient conditions, such as catalysis, are difficult to study with ARPES.
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 (K1 and K2) proportional to the initial binding energy (E1 and E2). 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.
In our recent letter , we report on an optical method to determine the band structure of solids. Because the technique does not detect electrons, it does not need high vacuum. The technique requires focusing a strong femtosecond laser pulse (1 fs = 10-15 s), with a peak field strength of ~0.1-1 V/Ang – comparable to the force binding electrons to their atoms – with a central wavelength in the mid-infrared spectral region (therefore < 1 eV photon energy) inside the bulk of the solid sample to generate several new frequencies at odd multiples of the mid-infrared one. The process, known as High Harmonic Generation , proceeds through three steps that are depicted in the right of Fig. 1. First, when the field is close to a maximum, electrons tunnel from the top of the valence to the bottom of the conduction band, creating an electron-hole pair with zero initial crystal momentum (circled “1” in Fig. 1). Then, the laser field accelerates the electron-hole pair to high crystal momentum (it can reach the edge of the Brillouin Zone!) - marked “2” in Fig. 1 – in a way that depends on the shape of their respective bands. Then, at a specific time in the laser cycle, the electron can recombine with the hole, emitting a high harmonic photon in the process with energy equal to the band gap at the momentum of recombination (marked “3” in Fig. 1).
The process is highly coherent: ionization, propagation and recombination are perfectly synchronized with the laser filed. This allows to link each harmonic photon energy to a specific trajectory of the electron-hole pair, which is ultimately determined by when it was created about the peak of the laser field. Therefore, knowledge of the trajectory is sufficient to determine the momentum at recollision, the only ingredient required to determined the band structure – together with the harmonic photon energy. The technique relies on a second laser pulse, superimposed to the first, at twice the mid-infrared frequency, to perturb – and therefore measure – the trajectory.
In a simulated experiment, we are able to fully reconstruct the momentum-dependent band gap of a target model solid, chosen to approximate the band structure of a ZnO crystal. The technique requires measuring many harmonic orders, a task at the moment beyond the capabilities of our experimental setup. In a real experiment we conducted, we are able to measure only 20% of the Brillouin Zone. Even with this limited experimental data, we determine that the split-off valence band on ZnO is the one mostly contributing to tunnelling of electrons to the conduction band.
In conclusion, rather than using weak incoherent fields of high energy photons – as in photoemission experiments – we use intense coherent light of low energy photons to create, accelerate and probe electron-hole pairs in solids. The information about the energy of the bands in which the electron and the hole move is imprinted on a high harmonic photon upon their recombination. Like other spectroscopy techniques based on high harmonic generation in gases , the method has intrinsic sub-laser cycle temporal resolution. This could be used to track band structure modifications following laser excitation .
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