High-harmonic generation in graphene: Interband response and the harmonic cutoff

We numerically examine the high-harmonic generation in graphene caused by an intense few-cycle terahertz laser field by solving the time-dependent Dirac equation. The observed spectra feature a complex interplay of interband and intraband electron dynamics. At high harmonic frequencies, we observe a plateau region with a cut-off frequency linearly proportional to the laser field. The linear dependence of the cutoff on the field resembles the behavior for electrons driven in bulk crystals but differs from the case of atoms or molecules. The unique features of the graphene band structure allow for a transparent decomposition of interband and intraband contributions and for an analytic estimate of the harmonic cut-off frequency in excellent agreement with the simulations.

Figure 1

(a) Calculated current (black curve) produced by electrons in graphene driven by a near-single-cycle 2-THz Gaussian envelope laser pulse of 40 kV/cm electric field strength (the black dashed curve shows its normalized vector potential). (b) Fourier transform of the vector potential in (a), (solid line) and after filtering out low-frequency components by a high-pass (HP) filter (dashed line).

Figure 2

(a)–(c) Integrated (top frame) and time-resolved (bottom frame) spectral power $P\left(E\right)$ of the far-field response of the graphene to a 2-THz, one-cycle Gaussian laser pulse with zero carrier-envelope phase and duration $t_p=250$ fs. We normalize the spectral power to the intensity of the first harmonic. The peak amplitudes of the laser field are (a) $F_0=20$ kV/cm, (b) $F_0=30$ kV/cm, and (c) $F_0=40$ kV/cm. We use the linear dispersion relation $E\left(t\right)=v_F\left|e\right|A\left(t\right)$ of Dirac fermions with $v_F={10}^6$ m/s (shown as black curve in bottom frame starting with $E\left(0\right)$ at the minimum energy $|E_{\min }|=v_F\left|e\right|A_0$). Solid vertical lines show the theoretically estimated harmonic cutoff $U_c^g$. Dashed vertical lines correspond to $0.5U_c^g$. (d) Theoretically estimated (red line, $v_F^0=0.78\times {10}^6$ m/s) and computed (black dots) harmonic cutoffs as a function of the laser field strength.

Figure 3

(a) Schematic illustration of high-energy harmonic generation in graphene and of the harmonic cutoff. The depicted initial state contributes to the harmonic emission near the cutoff. At $t_0\to -\infty$ the state has momentum $p_x\approx -\left|e\right|A_0$ ($p_y\ll p_x$) and energy $E_0\approx -v_F\left|e\right|A_0$. At the maximum of the vector potential (at time $t_1$) the momentum of the electron is close to $p_x\approx 0$ and its energy reaches the Dirac point (with $p_y=\mathrm{const}$), allowing for Landau-Zener tunneling between the two bands. The maximal energy distance between the two pieces of the split wave packet is reached at the following minimum of the vector potential (at time $t_2$) determining the cut-off energy of the harmonic radiation emitted upon electron-hole recombination. (b) Single-electron current (black curve) generated by a quantum trajectory $\psi \left(t\right)$ for an initial state (at $t\to -\infty$) with momentum $\vec{p}=(p_x,p_y)=(-0.8,0.05)\left|e\right|A_0$ and an energy close to $E_{\min }$ evaluated by solving the TDDE [Eqs. (1) and (6)]. The intraband current (for a state moving within only one band) evaluated using Eq. (13) and multiplied by –1 for better visibility is shown in red. Black-dashed curve represents the vector potential of the pulse. (c) Windowed Fourier transform of the quantum trajectory $\psi \left(t\right)$ depicted in (a). Black and red dashed curves follow the energy evolution of a state within a single (valence/conduction) band, i.e., $E_{\xi }\left(t\right)$ [Eq. (5)].

Figure 4

High-harmonic generation in graphene. The total nonlinear response of graphene is shown in black; red trace corresponds to the response of electrons near the Dirac point with energies $1.1E_{\min }<E<0$ contributing to the high-harmonic part of the spectra. The response of electrons further away from the Dirac point with $E<1.1E_{\min }$ (green) contribute only to low-order harmonics.

Figure 5

High-harmonic spectra of graphene subject (a) to a 2-THz Gaussian laser pulse with duration $t_p=250$ fs and carrier-envelope phase (CEP) of ${\varphi }_{\mathrm{CEP}}=\pi /2$; (b) to a 2-THz Gaussian laser pulse with duration $t_p=1240$ fs ($\mathrm{CEP}=0$); in each case for a peak field strength of $F_0=30$ kV/cm.

Figure 6

Comparison between (a) the TDDE and (b) the TDTB for the same Gaussian 2-THz laser pulse (${\varphi }_{\mathrm{CEP}}=0, t_p=250$ fs) with $F_0=30$ kV/cm calculated for identical Fermi velocity $v_F^0=0.78\times {10}^6$ m/s determined by our tight-binding parametrization. [Note the shift of the cutoff compared to that presented in Fig. 2.]

Figure 7

Cross sections through an ideal Dirac cone (black) and a deformed Dirac cone with trigonal warping (red) taken at different energies: $E=0.2$, 0.3, and 0.45 eV.

Figure 8

The spectral power $P\left(E\right)$ of the far-field response of graphene to a Gaussian 2-THz laser pulse with duration $t_p=250$ fs and field strength of 40 kV/cm calculated by TDDE with (red curve) and without (black curve) trigonal warping of the Dirac cone.

Figure 9

An example of single-electron current $\vec{j}\left(t\right)$ calculated by optical Bloch equations with different dephasing times: $T_d=\infty$ (black dashed curve), i.e., no dephasing; $T_d=500$ fs (red curve), i.e., $T_d$ is equal to the period of the pulse; and $T_d=80$ fs (green curve).

Figure 10

The spectral power $P\left(E\right)$ of the far-field response of graphene at $T=0$ to a Gaussian 2-THz laser pulse with duration $t_p=250$ fs and peak field strength of 40 kV/cm calculated by optical Bloch equations with different dephasing times: $T_2=\infty$ (black curve); $T_2=500$ fs (red curve); and $T_2=80$ fs (green curve).