Effect of multiple conduction bands on high-harmonic emission from dielectrics

We find that, for sufficiently strong mid-IR fields, transitions between different conduction bands play an important role in the generation of high-order harmonics in a dielectric. The transitions make a significant contribution to the harmonic signal, and they can create a single effective band for the motion of an electron wave packet. We show how high harmonic spectra produced during the interaction of ultrashort laser pulses with periodic solids provide a spectroscopic tool for understanding the effective band structure that controls electron dynamics in these media.

Figure 1

The upper valence band and first three conduction bands used in the simulations. Electrons reaching the Bragg plane can stay in the same band, reflecting, shown here as the circle remaining black. Alternatively they can undergo a transition to the next CB, white circle.

Figure 2

(a) Shows the intraband current with only one CB; in this case interband transition to higher CBs cannot happen and so Bragg reflections are forced to occur, explaining the appearance of higher harmonic content. (b) Shows the intraband current for eight CBs.

Figure 3

The time-dependent conduction band population is plotted on a logarithmic scale in the extended zone scheme, so that the $n^{\text{th}}$ CB occupies crystal momenta $n-1<\left|k\right|/k_{\max }<n$. The high transition probability between CBs is easily seen.

Figure 4

The harmonic spectrum generated by the transient current is plotted for the single- (red, upper line) and many- CB (black) cases. The variation of the spectra with field strength is shown in the inset plot for the single CB case.

Figure 5

Time-frequency analysis, as described in the text, of the transient current for the single- (a) and many- (b) CB cases is plotted. One can immediately see that the emission for the single-CB case is occurring in bursts that are more defined than in the many-CB case.

Figure 6

The strength of the harmonic emission given by the transient current, when multiplied with a Gaussian window centered around 30 eV and FWHM of 5 eV, is plotted as a function of the time-dependent crystal momentum and time for the single- (a) and many- (b) CB cases. For the single-CB case we see the emission is occurring at the Bragg planes: $\left|k\right|=k_{\max }$. For the many-CB case a lot of the emission is at $k=0$, particularly at larger times.