Nonlinear response of bilayer graphene at terahertz frequencies

A density-matrix formalism within the length gauge is developed for the purpose of calculating the nonlinear response of intrinsic bilayer graphene at terahertz frequencies. Employing a tight-binding model, we find that the interplay between the interband and intraband dynamics leads to strong harmonic generation at moderate field amplitudes. Specifically, we find that at low temperature (10 K), the reflected field of undoped suspended bilayer graphene exhibits a third harmonic amplitude that is 0.06% of the fundamental of the incident field, which corresponds to 30% of the fundamental in the reflected field for an incident 1-THz single-cycle pulse with a field amplitude of 1.5 kV/cm. More interestingly, we find that as the central frequency of the incident field is increased, the third harmonic amplitude also increases; reaching a maximum of 53% of the fundamental in the reflected field (0.11% of the fundamental in the incident field) for an incident frequency of 2 THz and amplitude of 2.5 kV/cm.

Figure 1

Schematic diagram of the crystal structure of BLG and the labeling conventions for atoms in the unit cell. $A$ sites are in white, while $B$ sites are in black.

Figure 2

Band structure of unbiased bilayer graphene. Lower energy bands touch at the Dirac point; higher energy bands are separated by $2t_{\perp }$.

Figure 3

Comparison of the real part of the full linear conductivity, calculated by computer simulation and numerical integration of the closed form expression. Fermi level is $60\text{meV}\left(0.2t_{\perp }\right)$, scattering time is $80\text{fs}$, and temperature is 50 K. Conductivity is measured in units of the universal conductivity of BLG, ${\sigma }_0=e^2/2\hslash$.

Figure 4

Comparison of the real part of the full linear conductivity, calculated by computer simulation and numerical integration of the closed form expression. Fermi level is $360\text{meV}\left(1.2t_{\perp }\right)$, scattering time is $80\text{fs}$, and temperature is 50 K. Conductivity is measured in units of the universal conductivity of BLG, ${\sigma }_0=e^2/2\hslash$.

Figure 5

Response of BLG to a incident field of 1 THz at $T=10$ K and for four different field amplitudes. (a) Intraband current density normalized to incident field amplitude. (b) Interband current density normalized to incident field amplitude.

Figure 6

Normalized electron density distribution in $k$ space (a) for the initial thermal distribution and (b) at time $t=1.75$ ps for a 0.5 kV/cm incident field. White lines indicate the position of the Dirac point.

Figure 7

Response of BLG to an incident field of 1 THz at $T=10$ K for four different field amplitudes. (a) The reflected field in the temporal domain, normalized to the amplitude of the incident field. Value of $E_r/E_0$ is multiplied by 100 for clarity. (b) The amplitude spectra of the reflected signal normalized to the peak at the fundamental frequency of 1 THz. (Inset) Amplitude spectra showing entire frequency range from 0 to 8 THz.

Figure 8

Response of BLG to a incident field of 1 THz at $T=100$ K for four different field amplitudes. (a) The reflected field in the temporal domain, normalized to the amplitude of the incident field. Value of $E_r/E_0$ is multiplied by 100 for clarity. (b) The amplitude spectra of the reflected signal normalized to the peak at the fundamental frequency of 1 THz.

Figure 9

Third harmonic amplitude spectra normalized to the peak of the fundamental central frequency, for three different incident frequencies: 1, 2, and 5 THz.