Wannier-Bloch Approach to Localization in High-Harmonics Generation in Solids

Emission of high-order harmonics from solids provides a new avenue in attosecond science. On the one hand, it allows us to investigate fundamental processes of the nonlinear 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 phenomenon 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 in 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 precisely addresses 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.

Popular Summary

An intense laser pulse aimed at a gas, plasma, or solid can cause the target to emit light at a much higher frequency than that of the laser, extending into the extreme ultraviolet region of the electromagnetic spectrum. This “high harmonic generation,” or HHG, is important for creating pulses of light that last for just a billionth of a billionth of a second (an attosecond)—a useful probe of rapid processes such as the movement of electrons around an atom. A key feature of HHG in a gas is that ionized electrons recombine with their parent atoms; in a solid, the electron can recombine with any other atom in the crystal lattice. This “delocalization” is poorly understood, yet believed to be important for attosecond pulse generation and real-time imaging of the electronic wave function in the solid state. We have developed a mathematical model that pinpoints how this delocalization contributes to HHG emission in a solid.

Our analytic approach builds on a three-step model that is well known for accurately describing harmonic emission in a dilute gas. By using localized atomic sites in the valence band and a delocalized description in the conduction band, one can separate the contributions of neighboring lattice sites to each harmonic and hence determine delocalization in harmonic emission. These neighboring contributions vary significantly with harmonic frequency and band structure of a crystal. Interestingly, for crystals with narrower bands, the light emission shows significant localization, similar to what happens in a gas.

These results pave the way to controlling localized contributions to harmonic emission in a solid crystal, with important implications for the emerging field of atto-nanoscience, which explores physics on very small scales with unprecedented time resolution.

Figure 1

Scheme of the electron excitation-recombination process for the HHG from the periodic lattice crystal. The periodic lattice is represented by gray disks along the black dashed line. The Wannier wave function is depicted by the green area. At time $t^{\prime }$, a single-electron wave packet is excited from the $j^{\prime }$th atomic site in the valence band to the conduction band. This wave packet is denoted by blue area. Then, during $(t^{\prime },t)$, it is accelerated within the conduction band by the driving laser field. Finally, at time $t$, the electron recombination takes place on different $j$ atomic sites with the subsequent high-harmonic emissions, which are denoted by violet pulse oscillations. Note that the red arrows show excitation at $j^{\prime }=j_0$ and recombination at $j=j_0$ and $j_0-1$ atomic sites.

Figure 2

Harmonic spectra comparison of the Wannier-Bloch approach (black line) with the Bloch-Bloch approach (red line) is shown in panel (a). Decomposition of the harmonic contribution into different lengths of electron-recombination atomic sites $\mathrm{\Delta }j$ using the Wannier-Bloch method is depicted in panel (b). The calculations are carried out for laser polarization in the $\mathrm{\Gamma }-M$ direction of ZnO crystal. The laser parameters are as follows: carrier wavelength of ${\lambda }_0=3.25\mu \mathrm{m}$, laser intensity $I_0=3.15\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, total number of laser cycles $N=10$ periods, and ${\phi }_{\mathrm{CEP}}=0$.

Figure 3

Panels (a) and (b) depict the full harmonic spectra for two different laser time durations, $N=4$ and 10, respectively (thick black lines), calculated within the Wannier-Bloch approach. Furthermore, color regions show the harmonic contributions of the different relative electron-recombination atomic sites in the spatially periodic lattice structure $\mathrm{\Delta }j=0,1,\dots ,4$. The black vertical lines (read from left to right) point out the band-gap harmonic and the expected cutoff harmonic orders, respectively. The laser-pulse peak intensity is $I_0=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, the carrier wavelength ${\lambda }_0=3.0\mu \mathrm{m}$, and its CEP is ${\phi }_{\mathrm{CEP}}=0\mathrm{rad}$. The calculations are carried out for laser polarization in the $\mathrm{\Gamma }-\mathrm{A}$ direction.

Figure 4

(a) Harmonic spectra for different values of the carrier-laser wavelength ${\lambda }_0$. (b) Same as (a) but with energy units instead of harmonic order on the frequency axis. The grey line shows maximum emitted photon energy, which corresponds exactly to the maximum energy difference between the conduction and valence bands, $\mathrm{\Delta }{E}_{vc}=6\mathrm{eV}$. Laser intensity was set to $I_0=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, laser-pulse length to 10 laser periods, and ${\phi }_{\mathrm{CEP}}=0\mathrm{rad}$.

Figure 5

Harmonic spectra as a function of electric-field strength. The electric-field axis is in units of ${\mathcal{E}}_0=0.0038\mathrm{a}.\mathrm{u}.$, corresponding to the laser intensity $I_0=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$. The dashed magenta straight line denotes the estimated cutoff of the harmonic spectrum as a function of the electric-field strength. The carrier-laser wavelength was set to ${\lambda }_0=3.0\mu \mathrm{m}$, laser-pulse length to $N=10$ laser periods $T_0$, and ${\phi }_{\mathrm{CEP}}=0\mathrm{rad}$. The calculations are carried out for laser polarization in the $\mathrm{\Gamma }-\mathrm{A}$ direction of ZnO crystal.

Figure 6

Experimental and theoretical comparison: Linear scaling of the cutoff harmonic order with the laser electric-field strength as obtained in (a) the experiment [courtesy of Louis DiMauro from Fig. 1(b) of Ref. [5] and (b) theoretical calculations by means of the Wannier-Bloch approach.

Figure 7

(a) Evolution of the $a_j$ coefficients in time for $j=0,1,\dots ,5$ for band-structure and laser-pulse parameters as in Fig. 3. (b) Same as (a) but with the electric field turned off (${\mathcal{E}}_0=0$).

Figure 8

Electron trajectories on a 2D lattice as a function of the ellipticity $ϵ=E_y/E_x$ calculated by solving Newton’s equation of motion: (a) $ϵ=0.1$, (b) $ϵ=0.2$, (c) $ϵ=0.3$, and (d) $ϵ=0.5$.