Abstract
Emission of high-order harmonics from solids provides a new avenue in attosecond science. On one hand, it allows to investigate fundamental processes of the non-linear response of electrons driven by a strong laser pulse in a periodic crystal lattice. On the other hand, it opens new paths toward efficient attosecond pulse generation, novel imaging of electronic wave functions, and enhancement of high-order harmonic generation (HHG) intensity. A key feature of HHG in a solid (as compared to the well-understood phenomena of HHG in an atomic gas) is the delocalization of the process, whereby an electron ionized from one site in the periodic lattice may recombine with any other. Here, we develop an analytic model, based on the localized Wannier wave functions in the valence band and delocalized Bloch functions in the conduction band. This Wannier-Bloch approach assesses the contributions of individual lattice sites to the HHG process, and hence addresses precisely the question of localization of harmonic emission in solids. We apply this model to investigate HHG in a ZnO crystal for two different orientations, corresponding to wider and narrower valence and conduction bands, respectively. Interestingly, for narrower bands, the HHG process shows significant localization, similar to harmonic generation in atoms. For all cases, the delocalized contributions to HHG emission are highest near the band-gap energy. Our results pave the way to controlling localized contributions to HHG in a solid crystal, with hard to overestimate implications for the emerging area of atto-nanoscience.
Highlights
The techniques of attosecond science, traditionally applied to atoms and molecules in the gas phase [1], have been extended to the solid state [2,3,4,5,6,7,8,9,10,11,12]
Analogous to the three-step model for atoms [23], the high-order harmonic-generation (HHG) in a crystal solid via interband transitions can be described as a sequence of three stages [18]: (i) electron tunneling excitation from the valence band to the conduction one, (ii) electron acceleration in the conduction band, and (iii) electron-hole recombination, resulting in an emission of a high harmonic that is a multiple of the frequency of the driving laser
(i) the HHG cutoff shows a dependence on the maximum energy difference between the valence and conduction bands, as well as on the laser wavelength and peak intensity; (ii) for long laser pulses and few-cycle laser fields, the model depicts the full odd spectrum and a continuum spectrum, respectively; and (iii) we find a direct link between the emitted harmonic spectrum shape and the band structure
Summary
The techniques of attosecond science, traditionally applied to atoms and molecules in the gas phase [1], have been extended to the solid state [2,3,4,5,6,7,8,9,10,11,12]. The extent of spatial localization, measured experimentally by ellipticity dependence, is believed to be important for attosecond pulse generation and imaging of the electronic wave functions in the solid state [9,19]. In addition to adequately calculating the total HHG yield, the present Wannier-Bloch approach allows us to separate the contributions of individual lattice sites to each harmonic and determine the degree of localization of the HHG process as a function of experimental parameters. We find that this localization varies significantly both with the harmonic order and with the orientation of a crystal. Our results point to the possibility of controlling the spatial localization of the HHG process, with implications for HHG efficiency, imaging of attosecond electron dynamics in condensed matter, and for the emerging area of atto-nanoscience as a whole [22]
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