An improved method for the investigation of high-order harmonic generation from graphene*

Abstract

High-order harmonic generation (HHG) of bulk crystals in strong laser field is typically investigated with semiconductor Bloch equations (SBEs). However, in the length gauge, it suffers from the divergence for the crystals with a zero band gap, such as graphene, using both Bloch- and Houston-states expansion methods. Here, we present a method of solving the SBEs based on time-dependent Bloch basis, which is equivalent to semiconductor Bloch equations in the velocity gauge. Using this method, we investigate the HHG of a single-layer graphene. It is found that our results for population are in good agreement with the other results. For a initial condition py = 0, we find the electrons just move in single valence band or conduction band, which are in accord with classical results. Our simulations on the HHG dependence of polarization of driving laser pulse confirm that 5th, 7th, and 9th harmonic yields increase to the maximal value when laser ellipticity ε ≈ 0.3. What is more, similar to the case of atoms in the laser field, the total strength of 3rd harmonic decrease monotonically with the increase of ε. In addition, we simulate the dependence of HHG on crystallographic orientation with respect to the polarization direction of linear mid-infrared laser pulse, and the results reveal that for higher harmonics, their radiation along with the change of rotation angle θ reflects exactly the sixfold symmetry of graphene. Our method can be further used to investigate the behaviors of other materials having Dirac points (i.e., surface states of topological insulators) in the strong laser fields.

Multiple collisions in crystal high-order harmonic generation

“…[1,2,5] Since the experimental observation of nonperturbative HHG in crystal systems, [9] the solid-state HHG process has been studied extensively. [10][11][12][13][14] The conversion efficiency of the solid-state HHG is expected to be higher than that of the atomic system due to the high density of condensed materials. The solid-state HHG can be used to reconstruct the band structure of crystals in an all-optical manner, [15] to produce novel light sources such as extreme ultraviolet pulses, [11,16] and to probe the dynamics of electrons and holes in solids.…”

Crystal-Momentum-Resolved Contributions to Harmonics in Laser-Driven Graphene

“…We use the SBEs in velocity gauge to calculate the radiation from graphene driven by strong laser fields according to Refs. [9,10,27]. The unique electronic band structure of graphene is characterized by the gapless Dirac cones at the Dirac points in the Brillouin zone.…”

Broadband Terahertz Wave Generation from Monolayer Graphene Driven by Few-Cycle Laser Pulse

“…Our simulations are performed by solving the extended two-band semiconductor Bloch equations (SBEs) for monolayer graphene. [24] A detailed description of this method can be found in our recent work, [25] in which we proposed an efficient theoretical approach to solving the SBEs, which avoids the divergence of the dipole transition moment near the Dirac points in graphene. For a laser field propagating perpendicular to the graphene, we obtain current near Dirac points as follows: [25] 𝑗 (1) where 𝑣 F = 1 × 10 6 m/s denotes Fermi velocity, 𝜌 vv and 𝜌 cc represent valence population and conduction population, respectively, 𝜌 cv is the polarization, and 𝜃 𝑘 is the directional angle of wave number 𝑘.…”

“…[24] A detailed description of this method can be found in our recent work, [25] in which we proposed an efficient theoretical approach to solving the SBEs, which avoids the divergence of the dipole transition moment near the Dirac points in graphene. For a laser field propagating perpendicular to the graphene, we obtain current near Dirac points as follows: [25] 𝑗 (1) where 𝑣 F = 1 × 10 6 m/s denotes Fermi velocity, 𝜌 vv and 𝜌 cc represent valence population and conduction population, respectively, 𝜌 cv is the polarization, and 𝜃 𝑘 is the directional angle of wave number 𝑘. [25] Once we obtain the total integrated electric current, 𝐽 (𝑡), the THz radiation field can be calculated via a Fourier transform of the time derivative of the electron current:…”

“…For a laser field propagating perpendicular to the graphene, we obtain current near Dirac points as follows: [25] 𝑗 (1) where 𝑣 F = 1 × 10 6 m/s denotes Fermi velocity, 𝜌 vv and 𝜌 cc represent valence population and conduction population, respectively, 𝜌 cv is the polarization, and 𝜃 𝑘 is the directional angle of wave number 𝑘. [25] Once we obtain the total integrated electric current, 𝐽 (𝑡), the THz radiation field can be calculated via a Fourier transform of the time derivative of the electron current:…”