High harmonic generation in undoped graphene: Interplay of inter- and intraband dynamics

We develop a density-matrix formalism in the length gauge to calculate the nonlinear response of intrinsic monolayer graphene at terahertz frequencies. Employing a tight-binding model, we find that the interplay of the interband and intraband dynamics leads to strong harmonic generation at moderate field amplitudes. In particular, we find that at low temperature, the reflected field of undoped suspended graphene exhibits a third harmonic amplitude that is 32% of the fundamental for an incident field of 100 V/cm. Moreover, we find that up to the seventh harmonic and beyond are generated.

Figure 1

(a) Schematic band structure of undoped $({\mu }_F=0)$ graphene near the Dirac point demonstrating interband and the intraband dynamics, (b) temporal plot of the incident pulse employed in the simulations, and (c) the corresponding amplitude spectrum of the pulse.

Figure 2

(a) Intraband current in the first case when the $\xi$'s are set to zero, (b) interband current in the second case when all the gradients in Eqs. (6) and (7) are eliminated, and (c) intraband and (d) interband current for the full calculation. In all plots the current is calculated for the four different incident electric fields with peak $E_i$ of 1, 50, 100, and 200 V/cm. All currents have been normalized to the corresponding $E_i$.

Figure 3

Electron density distribution in $k$ space at (a) the initial conditions and (b) at a time $t=2.75$ ps for a 100 V/cm incident field. The white lines identify the position of the Dirac point.

Figure 4

(a) Reflected electric field normalized to the corresponding amplitude of the incident electric field for four different incident fields $E_i$; (b) amplitude spectra of the reflected signal normalized to the peak at the fundamental frequency of 1 THz. The inset shows the dependence of the third harmonic amplitude as a function of the Fermi level for $E_i=200$ V/cm.

Figure 5

Amplitude of the third, fifth, and seventh harmonic normalized to the peak in the reflected field spectrum at the fundamental frequency at scattering time of 50 fs as a function of the incident field strength with the black curve shows a cubic fit.

Figure 6

Two possible mixing processes for ${\chi }^3$ susceptibility: (a) direct ${\chi }^3$ process and (b) indirect ${\chi }^3$ process as a result of ${\chi }^5$ process.

Figure 7

Third harmonic amplitude of the reflected signal normalized to the peak in the reflected field spectrum at the fundamental frequency vs. the incident field strength with scattering time constants of 10, 25, 50, and 100 fs.