High-order harmonic generation in imperfect crystals

High-order harmonic generation (HHG) in imperfect crystals, where the disorder is modeled by random shifts of the ionic positions, is studied using time-dependent density-functional theory. When irradiated by midinfrared laser pulses, the disorder-free system produces HHG spectra with two plateaus. Compared with the disorder-free system, disordered systems are found to emit suppressed harmonics in the first plateau region and enhanced harmonics in the second plateau region. The suppression of harmonics in the first plateau becomes less pronounced when decreasing the displacement of the nuclei, while the enhancement in the second plateau region is insensitive to the range of the ionic displacement. We have confirmed these findings for many different disordered sample systems and for different laser field strengths. The increase of the HHG signals in the second plateau region is proposed to stem from a change of the dynamics in the system, evidenced by the transition matrix elements between the field-free Kohn-Sham orbitals. In addition, a time-frequency profile of HHG spectra shows that the emission of harmonics is less regular in the time domain for a disordered system than for the disorder-free system.

Figure 1

(a) Comparison of the field-free KS potentials between the model system without disorder (solid black line) and a disordered system with $\delta =0.4$ [see Eq. (2)] (dotted red line). The inset shows a zoomed-in view of the potentials over 20 unit cells in the center of the model system. Also shown are band structures, which are obtained from the $k$-space distribution of the KS orbitals, for (b) the disorder-free system and (c) the disordered system considered in (a). The first Brillouin zone boundary is at $k=\pm \pi /a=\pm 0.7854$.

Figure 2

(a) HHG spectra of representative disordered model systems with $\delta =0.4$ [Eq. (2)]. (b) Same as (a) but with $\delta =0.2$. In both (a) and (b) the spectrum of the system without disorder (solid black curve) is shown for comparison. (c) HHG spectra obtained from calculations with the dynamic KS potential and the frozen KS potential (see the text) for a representative disordered system and the disorder-free system. The laser parameters are $A_0=0.3$, ${\omega }_0=0.015$, and $N_c=8$ [Eq. (7)].

Figure 3

HHG spectra as a function of vector-potential amplitude $A_0$ for (a) the system without disorder and (b) a disordered system [the sample system (a5) in Fig. 2]. The dashed lines indicate the first cutoff for the system without disorder. The remaining laser parameters are the same as in Fig. 2.

Figure 4

Norm of the transition matrix elements between field-free KS orbitals for (a) the system without disorder and (b) a disordered system with $\delta =0.4$ as also shown in Fig. 1. The horizontal axis and the vertical axis indicate the most probable momentum $k$ of the initial-state orbital and the KS orbital energy of the final-state orbital, respectively (see the text). The initial-state orbitals considered in this plot are in the first conduction band, while the corresponding final-state orbitals are in the first and the second conduction bands [see Figs. 1 and 1]. For a specific initial-state orbital, transitions to final-state orbitals with higher energies are shown.

Figure 5

Time-frequency profile of HHG spectra extracted by a Gabor transform for (a) the disorder-free system and (b) a disordered system [the sample system (a5) in Fig. 2]. The laser parameters are the same as in Fig. 2.