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Chin. Phys. B, 2020, Vol. 29(10): 104206    DOI: 10.1088/1674-1056/abab76

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

Zhong Guan(管仲)1, Lu Liu(刘璐)2, Guo-Li Wang(王国利)1,†, Song-Feng Zhao(赵松峰)1, Zhi-Hong Jiao(焦志宏)1, and Xiao-Xin Zhou(周效信)1,
1 College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
2 Department of Physics, College of Science, National University of Defense Technology, Changsha 410073, China

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.

Keywords:  high-order harmonic generation      graphene      velocity gauge      divergence  
Received:  26 April 2020      Revised:  15 June 2020      Accepted manuscript online:  01 August 2020
PACS:  42.65.Ky (Frequency conversion; harmonic generation, including higher-order harmonic generation)  
  42.50.Hz (Strong-field excitation of optical transitions in quantum systems; multiphoton processes; dynamic Stark shift)  
  42.65.Re (Ultrafast processes; optical pulse generation and pulse compression)  
  72.20.Ht (High-field and nonlinear effects)  
Corresponding Authors:  Corresponding author. E-mail: Corresponding author. E-mail:   
About author: 
†Corresponding author. E-mail:
‡Corresponding author. E-mail:
* Project supported by the National Natural Science Foundation of China (Grant Nos. 11764038, 11864037, 11765018, and 11664035) and the Science Foundation of Northwest Normal University, China (Grant No. NWNU-LKQN-17-1).

Cite this article: 

Zhong Guan(管仲), Lu Liu(刘璐), Guo-Li Wang(王国利)†, Song-Feng Zhao(赵松峰), Zhi-Hong Jiao(焦志宏), and Xiao-Xin Zhou(周效信)‡ An improved method for the investigation of high-order harmonic generation from graphene 2020 Chin. Phys. B 29 104206

Fig. 1.  

The first Brillouin zone in graphene.

Fig. 2.  

Comparison of our calculated conduction band population ρcc(k,t) in the velocity gauge (right column) with those from the two-band model (left column) given in Ref. [58] for panels (a) and (b) F0 = 0.8 V/Å and panels (c) and (d) F0 = 2.25 V/Å.

Fig. 3.  

The same as Fig. 2, but for panels (a) and (b) t = 0.75 fs and panels (c) and (d) t = 2.25 fs with the same F0 = 1.0 V/Å.

Fig. 4.  

Comparison of the temporal evolution of the normalized single-electron current calculated by our method and those from TDDE for (a) py = 0, (b) py = 0.02A0, and (c) py = 0.05A0. In all the cases px/eA0 = −0.75. In Fig. 4(a) we also show the result with Houston basis.

Fig. 5.  

Left column: the vector potential of laser pulse and electron current; Right column: the time evolution of the wave packet (red and black lines are classical trajectories).

Fig. 6.  

Comparison of harmonic spectra of graphene generated by laser fields with different ellipticity.

Fig. 7.  

The dependence of intensity in two perpendicular directions (x and y) of harmonics 3rd, 5th, 7th, and 9th on the ellipticity of driving laser pulse.

Fig. 8.  

(a) Harmonic radiation with different rotation angle θ. (b) The interband polarization dcv (k) as a function of the crystal momentum k. (c) and (d) Harmonic spectra generated from inter-band polarization and intra-band current for θ of 0° and 20°, respectively.

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