Topological resonance and single-optical-cycle valley polarization in gapped graphene

For gapped graphene, we predict that an intense ultrashort (single-oscillation) circularly polarized optical pulse can induce a large population of the conduction band and a large valley polarization. With an increase in the band gap, the magnitude of the valley polarization gradually increases from zero (for the native gapless graphene) to a value on the order of unity. The energy bandwidth of the electrons excited into the conduction band can be very large ($\gtrsim 10$ eV for a reasonable pulse amplitude of $\sim 0.5$ V/Å). These phenomena are due to the effect of topological resonance: The matching of the topological (geometric) phase and the dynamic phase. Gapped graphene with a tunable band gap can be used as a convenient generic model of two-dimensional semiconductors with honeycomb generic lattice structures and broken inversion symmetry, such as transition-metal dichalcogenides.

Figure 1

(a) Hexagonal lattice structure of graphene with sublattices A and B. (b) The first Brillouin zone of graphene with two valleys, $K$ and $K^{\prime }$. (c) Energy dispersion as a function of crystal momentum for gapped graphene with a band gap of 1 eV.

Figure 2

Residual CB population $N_{\mathrm{CB}}^{\left(\mathrm{res}\right)}\left(\mathbf{k}\right)$ for gapped graphene in the extended zone picture. The applied optical pulse has right circular polarization with an amplitude of $F_0=0.5\mathrm{V}{\AA }^{-1}$. Inset: Wave form of the pulse $\mathbf{F}\left(t\right)=\{F_x\left(t\right),F_y\left(t\right)\}$ as a function of time $t$. The white solid line shows the boundary of the first Brillouin zone with the $K,K^{\prime }$ points indicated. The band gap is (a) 0 eV, (b) 0.2 eV, (c) 0.8 eV, and (d) 1.6 eV, as indicated on the corresponding panels. The separatrix is shown by a solid black line.

Figure 3

Same as in Fig. 2, but for the pulse with left-hand circular polarization.

Figure 4

For a pulse with right-hand circular polarization and amplitude $F_0=0.5\mathrm{V}{\AA }^{-1}$, phases ${\varphi }_{\mathrm{cv}}^{\left(\mathrm{tot}\right)}(\mathbf{q},t)$, ${\varphi }_{\mathrm{cv}}^{\left(\mathrm{D}\right)}(\mathbf{q},t)$, ${\varphi }_{\mathrm{cv}}^{\left(\mathrm{B}\right)}(\mathbf{q},t)$, and ${\varphi }_{\mathrm{cv}}^{\left(\mathrm{T}\right)}(\mathbf{q},t)$ are shown for different initial wave vectors. (a) The initial wave vector $\mathbf{q}$ is inside the separatrix in the $K$ valley. (b) The initial wave vector $\mathbf{q}$ is outside of the separatrix in the $K$ valley. (c) The initial wave vector $\mathbf{q}$ is inside the separatrix in the $K^{\prime }$ valley. (d) The initial wave vector $\mathbf{q}$ is outside of the separatrix in the $K^{\prime }$ valley. The separatrix is shown in Fig. 2 by a solid black line.

Figure 5

Residual CB population $N_{\mathrm{CB}}^{\left(\mathrm{res}\right)}\left(\mathbf{k}\right)$ of gapped graphene with band gap ${\mathrm{\Delta }}_g=2$ eV is shown with color coding on the background of (a) the corresponding excitation energy and (b) the corresponding Berry curvature in the extended zone picture. The applied pulse is left handed with and amplitude of 0.5 $\mathrm{V}/\AA$

Figure 6

Valley polarization of gapped graphene as a function of the amplitude $F_0$ of the excitation right-handed pulse for different band gaps ${\mathrm{\Delta }}_g$: 0, 0.2, 0.4, 0.8, 1.2, 1.6, and 2 eV.