Contribution of Floquet-Bloch states to high-order harmonic generation in solids

We identify the contribution of Floquet-Bloch states to the high-order harmonic generation (HHG) in solids by numerically solving the time-dependent Schrödinger equation for both a one-dimensional and a two-dimensional model. Results from the single $k$ point and the full Brillouin zone are compared to each other and the symmetry- breaking effect is discussed. We show that the observed phenomena can be explained under the framework of the Floquet-Bloch theory and the strong-field approximation, respectively. Our results indicate that the total yield of the harmonic radiation increases nonmonotonically with the intensity of the driving pulse. After a rough consideration of the focusing volume effects, we find that the yield of the harmonics shows a steplike structure, which is similar to several recent experimental observations. The present work can contribute to a better understanding of the channel-closing effect in the HHG of solids and may provide a way to detect the Floquet-Bloch bands in a laser field.

Figure 1

(a) Left panel: the first three energy bands of the one-dimensional model. Right panel: the corresponding quasienergies as a function of the amplitude of the vector potential. The solid lines and dashed lines represent states with different parities. (b) The yield of the 11th-order harmonic induced by a trapezoidal (blue solid line) and cosine square (yellow dashed line) envelope at various $A_0$. The full HHG spectrum is, respectively, shown in (c) for the trapezoidal envelope and (d) for the cosine square envelope. The black dashed line and white dashed line are ${\mathrm{\Omega }}_{\text{C1}}\left(A_0\right)-{\mathrm{\Omega }}_{\text{V2}}\left(A_0\right)$ and $E_{\text{C1}}\left(A_0\right)-E_{\text{V2}}\left(A_0\right)$, respectively.

Figure 2

The harmonic spectra calculated by SFA at different peak vector potential $A_0$ of laser pulses with a trapezoidal or cosine square envelope. (a1)–(a4) The amplitude of the conduction band for the laser pulse: (a1) a trapezoidal pulse with $A_0=0.52\pi /a$, (a2) a trapezoidal pulse with $A_0=0.50\pi /a$, (a3) a cosine square pulse with $A_0=0.52\pi /a$, and (a4) a cosine square pulse with $A_0=0.50\pi /a$. In these four subfigures, the red circle centered at the origin has the same absolute size to indicate the scale and the small arrows represent the tunneling amplitude in each saddle point. (b1)–(b4) The corresponding HHG spectra, calculated by TDSE (red dashed line) and by SFA (blue solid line). (c1),(c2) An overview of the HHG spectra calculated by SFA as a function of $A_0$ for (c1) a trapezoidal pulse and (c2) a cosine square pulse.

Figure 3

The full HHG spectrum integrated over the full BZ calculated by TDSE for (a) the trapezoidal envelope and (b) the cosine square envelope. (c) The yield of 11th-order harmonics as a function of peak vector potential. Other parameters are the same as those in Fig. 1. The SFA results (not shown here) agree with the TDSE calculation quantitatively in the plateau region. The inset in (c) shows the momentum-space trajectory for electrons starting from $k_0>0$ (blue line) and from $-k_0$. The red points represent the saddle points.

Figure 4

(a) The Brillouin zone of the 2D model and high-symmetric points in the reciprocal space. The four lowest energy bands are shown along the laser polarization, (b) from $(0,\pi /a)$ to $X(2\pi /a,\pi /a)$ and (c) from (0,0) to $(2\pi /a,0)$. The corresponding dipoles between the valence and the two lowest conduction bands are shown in (d) and (e), respectively.

Figure 5

The HHG spectrum for the 2D model integrated over the full BZ and a single line along the $M-X$ direction from the $(0,0)$ point are shown in (a) and (b), respectively. An eight-cycle trapezoidal laser pulse is used, with wavelength $\lambda =1200\mathrm{nm}$ (${\omega }_0\approx 0.038\mathrm{a}.\mathrm{u}.$), linearly polarized along the $M-X$ direction. The black dashed line indicates the enhanced positions of the channel closing, which are almost at the same values of the vector potential in (a) and (b).

Figure 6

The yield of the first platform (7–15 order) from the 2D model case with all $k$ points from 2D momentum space. The blue dashed line is the averaged result after considering the laser-focusing volume effects for the trapezoidal envelope.