Model-potential method for high-order harmonic generation in monolayer graphene

We develop a model potential to simulate the effective potential of a single active electron in monolayer graphene by taking the electron energy band structure calculated by the density functional theory (DFT) as a reference. Based on the single-electron Schrödinger equation, the model potential is used to calculate not only the energy band structure but also the transition dipole moment, charge density, and other physical quantities of graphene. These quantities are compared with results from DFT and a good consistency is achieved. The simulation of laser-graphene interaction with the same laser parameters as Yoshikawa et al. [Science 356, 736 (2017)] is performed with the time-dependent Schrödinger equation. The obtained driving laser ellipticity scaling, the harmonic ellipticity, and the harmonic major-axis angle can well reproduce the experimental results. It is found that the carrier-envelope phase and chirp effects are capable of regulating high-order harmonic generation in monolayer graphene. The model potential method can be extended to field-free calculations and dynamic simulations in other materials as long as the corresponding model potential is constructed.

Figure 1

Band structure of monolayer graphene. (a) A sketch of graphene honeycomb lattice. Solid black circles represent carbon atoms. The rectangle area surrounded by the solid red line is the unit cell used through this article. (b) The high-symmetry path in part of the first Brillouin zone. The red triangle indicates the zero-gap Dirac point. The band structure has been calculated by (c) TISE and (d) DFT. Vertical dashed lines in (c) and (d) correspond to high-symmetry points presented in (b). The energy bands closest to zero are marked in red in (d).

Figure 2

Absolute values of transition dipoles in the high-symmetry path. Components of transition dipoles in (a) $x$ and (b) $y$ directions. Dipole amplitudes from TISE method (solid black lines) are compared with those from DFT calculations (red dashed lines). The longitudinal coordinates on the left-hand side in (a) and (b) correspond to TISE, while the right-hand-side ones in red correspond to DFT.

Figure 3

Single $k$-point charge distribution of monolayer graphene. Charge densities of two adjacent eigenstates around zero energy at $D$, the Dirac point marked in Fig. 1, are exhibited in (a) and (b) (below zero energy) and (c) and (d) (above zero energy). Carbon atoms are indicated by open black circles in TISE results [(a) and (c)]. They are omitted in the DFT outcomes [(b) and (d)] because they are in exactly the same place visually. But the coordinates of carbon atoms are different because the origin of the coordinates is shifted in the DFT calculation.

Figure 4

Laser ellipticity dependence of HHG in graphene and the generated harmonic ellipticity. (a) The pump pulse ellipticity dependence of the fifth high harmonic. (c) Under the linearly polarized driving pulse (${\varepsilon }_d=0$), the generated fifth harmonic ellipticity and its major axis orientation angle of the electric vector ellipse are shown. The same physical quantities are presented in (e) except that the ellipticity of the driving light is 0.3. Panels (b), (d), and (f) correspond to (a), (c), and (e), respectively, but for the seventh harmonic.

Figure 5

CEP and chirp effects of high-order harmonic generation in monolayer graphene. (a) Harmonic spectra in different CEPs. The shaded area of the spectrum denotes the harmonic emission when CEP is zero. The red line spectrum corresponds to the case when CEP is $0.5\pi$. (b) Spectra in different chirp parameters: upper red line, chirp = 0; middle cyan line, chirp = 0.5; middle yellow dot-dashed line, $\text{chirp}=-0.5$; lower green line, chirp = 1; and lower black dotted line, $\text{chirp}=-1$. Spectral and temporal emission profiles of HHG for three chirp parameters: (c) $\text{chirp}=-0.5$, (d) chirp = 0, and (e) chirp=0.5. Gray lines in (c–e) represent the electric fields with corresponding chirp parameters. $T_{laser}$ denotes the optical cycle.