Crystal-momentum-resolved contributions to multiple plateaus of high-order harmonic generation from band-gap materials

We study the crystal-momentum-resolved contributions to the high-order harmonic generation (HHG) in band-gap materials, and identify the relevant initial crystal momenta for the first and higher plateaus of the HHG spectra. We do so by using a time-dependent density-functional theory model of one-dimensional linear chains. We introduce a self-consistent periodic treatment for the infinitely extended limit of the linear chain model, which provides a convenient way to simulate and discuss the HHG from a perfect crystal beyond the single-active-electron approximation. The multiplateau spectral feature is elucidated by a semiclassical $k$-space trajectory analysis with multiple conduction bands taken into account. In the considered laser-interaction regime, the multiple plateaus beyond the first cutoff are found to stem mainly from electrons with initial crystal momenta away from the $\mathrm{\Gamma }$ point ($k=0$), while electrons with initial crystal momenta located around the $\mathrm{\Gamma }$ point are responsible for the harmonics in the first plateau. We also show that similar findings can be obtained from calculations using a sufficiently large finite model, which proves to mimic the corresponding infinite periodic limit in terms of the band structures and the HHG spectra.

Figure 1

Band structures for the considered finite-system model with $N=200$, obtained from the $k$-space distribution of the KS orbitals. The markers are band structures for the corresponding infinite periodic system in the first Brillouin zone. The plot includes two valence bands (VB1 and VB2, red circles) and four conduction bands (from CB1 to CB4, green squares). While the free-space (FS) dispersion $k^2/2$ naturally appears for the finite system, this finite-box feature is absent for the infinite periodic system.

Figure 2

HHG spectra for a finite system of $N=200$ and the corresponding infinite periodic limit. The laser parameters are ${\omega }_0=0.0228, A_0=0.24$, and $n_{\text{cyc}}=15$ (see text). For the sake of comparison, the finite-system result has been scaled by $N^{-2}$ while the periodic-system result refers to the harmonics generated per atom in the infinitely extended model. The vertical arrows indicate the cutoffs of four plateaus.

Figure 3

The band-climbing processes illustrated by considering the central optical cycle (approximately treated as a sinusoidal form for simplicity). In the presence of the laser field, the instantaneous crystal momentum of an electron is $k_0+A\left(t\right)$, with $k_0$ the initial crystal momentum and $A\left(t\right)$ the vector potential (see text). The considered vector-potential amplitude $A_0$ fulfills $0.5\pi /a<A_0<\pi /a$, which applies to all the results presented in this paper. Tunneling excitation to an upper band is considered to occur at a band gap, e.g., at $k\left(t\right)=0$ ($\pm \pi /a$) for the VB2-CB1 (CB1-CB2) transition. Due to the form of $k\left(t\right)$, there are typically two moments for a band-gap transition within one optical cycle. Similar processes can happen among higher conduction bands with additional optical cycles, e.g., after being excited into CB2, the electron may undergo a CB2-CB3 transition when $k\left(t\right)$ reaches 0 again. Inset: The vector potential in one optical cycle, with the circles (squares) indicating the moments of the VB2-CB1 (CB1-CB2) transitions for the depicted paths.

Figure 4

Time-frequency profile of the periodic-system HHG spectrum in Fig. 2, extracted by a Gabor transform. The gray dots on top are obtained from a $k$-space semiclassical trajectory analysis (see text). The black dots are a subset of the trajectories with the real-space recollision constraint taken into account.

Figure 5

(a) Crystal-momentum-resolved profile, corresponding to the periodic-system HHG spectrum shown in Fig. 2 (see the caption of Fig. 2 for the laser parameters). The dashed lines indicate $k_0$ values $\pm A_0$ and $\pm (\pi /a-A_0)$, while the solid lines indicate $k_0$ values $\pm (A_0+{\delta }_k)$ and $\pm (\pi /a-A_0-{\delta }_k)$. The solid lines define two regions that contribute to the majority of interband harmonics: $R_{\text{I}}=\{k_0\left|\right|k_0|\le (\pi /a-A_0-{\delta }_k)\}$ and $R_{\text{II}}=\{k_0|(\pi /a-A_0-{\delta }_k)\le |k_0|\le (A_0+{\delta }_k)\}$, with ${\delta }_k\approx 0.016\pi$ accounting for tunneling transitions in the vicinity of the band gap (see text). $R_{\text{II}}$ captures all the harmonics beyond the first cutoff while $R_{\text{I}}$ roughly captures the first plateau. (b) and (c) HHG spectra obtained from the current calculated within the above-defined $k_0$ regions $R_{\text{I}}$ and $R_{\text{II}}$, respectively. The total spectrum is also shown for comparison, with the vertical arrows indicating the cutoffs of the four plateaus as in Fig. 2.

Figure 6

Similar as Fig. 5, but for a different set of laser parameters ${\omega }_0=0.01824$ and $A_0=0.3$.

Figure 7

(a) Mapping from the $k_0$ regions $R_{\text{I}}$ and $R_{\text{II}}$ defined in Fig. 5 to the VB2 orbital energy ranges $R_{\text{I}}^{\prime }$ and $R_{\text{II}}^{\prime }$, according to the band structure. (b) Orbital-index profile, corresponding to the finite-system HHG spectrum shown in Fig. 2 (without the scaling factor $N^{-2}$). The dashed and solid lines indicate the VB2 orbital energies corresponding to the $k_0$ values described in Fig. 5. For this finite system with $N=200$, the regions $R_{\text{I}}^{\prime }$ and $R_{\text{II}}^{\prime }$ are defined by indices from 331 to 400 and indices from 273 to 330, respectively. (c) and (d) HHG spectra obtained from the current calculated within the orbital-index ranges $R_{\text{I}}^{\prime }$ and $R_{\text{II}}^{\prime }$, respectively. The total spectrum is also shown for comparison, with the vertical arrows indicating the cutoffs of the four plateaus as in Fig. 2.