Valley-resolved interband excitation and emission in gapped graphene

We theoretically investigate the electronic dynamics in gapped graphene driven by a bicircular field. The results indicate that the residual population on the conduction band is obviously asymmetric in the K and ${\mathrm{K}}^{\prime }$ valleys. Through analyzing the electron trajectory, we find that the interband transitions are determined by the compensation of two phases: the dynamic phase and the full dipole phase (including the transition dipole phase and the Berry phase). When the rates of change of the dynamic phase and the full dipole phase nearby the ionization time have the inverse signs, the unidirectional interband transition from the valence band (VB) to the conduction band (CB) results in the constructive interference of the population. In contrast, when they have the same signs, the bidirectional transition (CB$\to$VB and VB$\to$CB) results in the destructive interference. The constructive interference and the destructive interference lead to the asymmetric distribution of the CB population. Moreover, we also investigate the harmonic emission and find that only the $3n+2$ harmonics are generated in the cutoff region. This phenomenon can be attributed to the asymmetric distribution of the CB population and the trefoil vector potential of the electric field. Due to the dependence on population distribution, the harmonics can be a promising optical way to detect the valley excitation of the gapped graphene.

Figure 1

(a) The honeycomb lattice of graphene with two sublattices, A and B. (b) The first Brillouin zone of graphene with two nonequivalent valleys, K and ${\mathrm{K}}^{\prime }$.

Figure 2

The normalized CB population after excitation with different band gaps: 0 eV (a), 1 eV (b), 2 eV (c), and 3 eV (d). The white dashed line shows the boundary of the first Brillouin zone with the K and ${\mathrm{K}}^{\prime }$ points indicated. The black solid lines in panel (d) show the boundary of the population.

Figure 3

(a) The residual CB population in the K and ${\mathrm{K}}^{\prime }$ valleys. A1, A2, B1, and B2 are initial points in $k$ space selected for the corresponding regions. (b) The phase of the $x$ component of transition dipole elements ${\mathbf{d}}_{cv}$. Arrows show the helicity of the phase. The phase of $d_y$ has the same helicity. (c) Difference of the Berry curvature between the VB and the CB. According to the Stokes theorem, the inverse helicity of the Berry phase around the K and ${\mathrm{K}}^{\prime }$ points is deduced (shown with arrows). (d) The transition dipole amplitude and the trajectories of A1, A2, B1, and B2 shown as the red or black dashed lines.

Figure 4

(a) The CB population (red line) and the modules of $\mathrm{\Omega }$ (purple line) normalized by the peak field strength along the electron trajectory of A1 and B2. (b) Total phase $S^{\prime }$ (blue line) and ${\mathbf{\nabla }}_t{S}^{\prime }$ (solid red line) with the minimum band gap indicated with a horizontal dashed red line. (c) Full dipole phase including the TDP and the BP.

Figure 5

(a) The CB population (red line) and the modules of $\mathrm{\Omega }$ (purple line) normalized by the peak field strength along the electron trajectory of A2 and B1. (b) Total phase $S^{\prime }$ (blue line) and ${\mathbf{\nabla }}_t{S}^{\prime }$ (solid red line) with the minimum band gap indicated with a horizontal dashed red line. (c) Full dipole phase including the TDP and the BP.

Figure 6

The residual CB population under the situations: (a) Both the TDP and the BP are discarded. (b) Only BP is discarded. (c) Only the TDP is discarded. (d) Both the TDP and the BP are included.

Figure 7

(a) The time-dependent CB population of A1 and B2 under the different situations, the same as those in Fig. 6. (b) The corresponding total phase.

Figure 8

(a) The harmonic spectra obtained from the different zones: the full Brillouin zone (green area), the ${\mathrm{K}}^{\prime }$ valley (blue line), and the K valley (orange line). (b) The harmonic spectra obtained from the selected zones shown in panel (c): the full Brillouin zone (green area), the outside zone of the ${\mathrm{K}}^{\prime }$ valley (blue line), and the outside zone of the K valley (orange line). (c) The selected zones between two concentric circles (i.e., inner black circles and outer white circles) in the K and ${\mathrm{K}}^{\prime }$ valleys. (d) A schematic diagram for harmonic emission. The contour lines describe the $k$-dependent band gap: $E_g=E_c-E_v$. The red lines represent the trajectories of the corresponding initial points (red points).

Figure 9

(a) The harmonic spectra obtained from the single point A (blue line) and the single point B (orange line). (b) The black dashed line shows the time-dependent band gap along the trajectory of point A, and the maximum band gap is marked with a white dashed line. The time-frequency spectrogram (color map) is for point A. The maximum values are normalized to 1 for clarity. A logarithmic color scale is used. (c) The same as panel (b), but for point B.

Figure 10

The relative-phase-resolved intensity of the harmonics in the cutoff region. A logarithmic color scale is used.