Optical response of graphene under intense terahertz fields

Optical responses of graphene in the presence of intense circularly and linearly polarized terahertz fields are investigated based on the Floquet theory. We examine the energy spectrum and density of states. It is found that gaps open in the quasienergy spectrum due to the single-photon and multiphoton resonances. These quasienergy gaps are pronounced at small momentum, but decrease dramatically with the increase of momentum, and finally tend to be closed when the momentum is large enough. Due to the contribution from the states at large momentum, the gaps in the density of states are effectively closed, in contrast to the prediction in the previous work by Oka and Aoki [Phys. Rev. B 79, 081406(R) (2009)]. We also investigate the optical conductivity for different field strengths and Fermi energies, and show the main features of the dynamical Franz-Keldysh effect in graphene. It is discovered that the optical conductivity exhibits a multi-steplike structure due to the sideband-modulated optical transition. It is also shown that dips appear at frequencies being the integer numbers of the applied terahertz field frequency in the case of low Fermi energy, originating from the quasienergy gaps at small momentums. Moreover, under a circularly polarized terahertz field, we predict peaks in the middle of the “steps” and peaks induced by the contribution from the states around zero momentum in the optical conductivity.

Figure 1

Circularly polarized THz field with $\beta =0.4$. (a) Quasienergies ${\xi }_{\mathbf{k}\eta }^n$ against the normalized momentum. The color coding represents the weight $W_{\mathbf{k}\eta }^n$ of the corresponding sideband. The blue crosses represent the selected quasienergies of the quasielectron states. The arrows illustrate the optical transitions between the sidebands at $\mathbf{k}=(0.9\Omega /v_{\mathrm{F}},0)$ for $N=1$ with $N$ defined in Eq. (31). (b) Quasienergy ${\varepsilon }_{\mathbf{k}+}$ (red solid curve) and mean energy ${\stackrel{̲}{\varepsilon }}_{\mathbf{k}+}$ (blue dashed curve) of the quasielectron state as well as the field-free electron energy (green dotted curve) against the normalized momentum. In the inset of (b), we plot the quasienergy around the gap at $k=0.5\Omega /v_{\mathrm{F}}$ and ${\varepsilon }_{\mathbf{k}\eta }=0.5\Omega$ [the region labeled by the box in (a)] as well as the corresponding mean energy as a function of momentum. The solid (dashed) curves with and without crosses represent the quasienergies (mean energies) of the quasielectron ($\eta =+$) and quasihole ($\eta =-$) states, respectively. The black chain line indicates the momentum of the crossover point $k_0$. Note the scale of mean energy is on the right-hand side of the frame.

Figure 2

Circularly polarized THz field with $\beta =2.3$. (a) Quasienergies ${\xi }_{\mathbf{k}\eta }^n$ against the normalized momentum. The color coding represents the weight $W_{\mathbf{k}\eta }^n$ of the corresponding sideband. The blue crosses represent the selected quasienergies of the quasielectron states. (b) Quasienergy ${\varepsilon }_{\mathbf{k}+}$ (red solid curve) and mean energy ${\stackrel{̲}{\varepsilon }}_{\mathbf{k}+}$ (blue dashed curve) of the quasielectron state as well as the field-free electron energy (green dotted curve) versus the normalized momentum. The thin black lines mark the mean energies being $1.5\Omega$, $1.8\Omega$, $2.3\Omega$, and $2.7\Omega$, which are used in Fig. 6c. Note the scale of mean energy is on the right-hand side of the frame.

Figure 3

Linearly polarized THz field with $\beta =2.3$. Quasienergies ${\xi }_{\mathbf{k}\eta }^n$ against the normalized momentum for different polar angles ${\theta }_{\mathbf{k}}$. The color coding represents the weight $W_{\mathbf{k}\eta }^n$ of the corresponding sideband. $k_{\stackrel{\wedge }{\mathbf{n}}}$ in (c) stands for $\mathbf{k}·\stackrel{\wedge }{\mathbf{n}}$, with $\stackrel{\wedge }{\mathbf{n}}$ being the unit vector at an angle $\pi /6$ with respect to the $x$ axis.

Figure 4

(a) Time-averaged DOS versus $\omega$ under circularly (red solid curve) and linearly (blue dashed curve) polarized THz fields for $\beta =2.3$. The region close to $\omega ~0$ is enlarged in the inset. (b) Time-averaged DOS versus $\omega$ under a circularly polarized THz field for $\beta =2.3$ with all relevant states (red solid curve), limited in the momentum regimes $v_{\mathrm{F}}k<2\hspace{0.5em}\Omega$ (blue dashed curve) and $v_{\mathrm{F}}k<0.5\hspace{0.5em}\Omega$ (green dotted curve). The yellow chain curve represents the DOS from Ref. 21.

Figure 5

Circularly polarized THz field. (a) Time-dependent optical conductivity as a function of the optical frequency and time with $\beta =2.3$. (b) Time-averaged optical conductivity as a function of the optical frequency with different field strengths. $E_{\mathrm{F}}=6\Omega$ in both figures.

Figure 6

Circularly polarized THz field. (a) Time-dependent optical conductivity as a function of the optical frequency and time with $\beta =2.3$ and $E_{\mathrm{F}}=2.7\Omega$. (b) Time-averaged optical conductivity as a function of the optical frequency with different field strengths; $E_{\mathrm{F}}=2.7\Omega$. (c) Time-averaged optical conductivity as a function of the optical frequency with $\beta =2.3$ for different Fermi energies.

Figure 7

Linearly polarized THz field with $\beta =2.3$. (a) Time-averaged optical conductivity as a function of the optical frequency for different Fermi energies. (b) Time-averaged optical conductivities versus the optical frequency from the states in different polar-angle regions $[{\theta }_{\mathbf{k}}^m-\pi /128,\hspace{0.5em}{\theta }_{\mathbf{k}}^m+\pi /128)$ as well as the one with all polar angles integrated; $E_{\mathrm{F}}=2.5\Omega$. (c) Mean energies of the quasielectron states with different polar angles ${\theta }_{\mathbf{k}}$ against the normalized momentum. The thin black line indicates the mean energy being $2.5\Omega$.

Figure 8

Time-averaged optical conductivities from the mean-energy-determined distribution (solid curves) and the projected distribution (dashed curves) are plotted as a function of the optical frequency under a circularly polarized THz field with $\beta =2.3$ for different Fermi energies.

Figure 9

Quasienergy gap ${\Delta }_m$ versus the corresponding momentum from the exact calculation (dots) and the approximate formula (B24) (squares) under circularly polarized THz fields with $\beta =0.4$ (solid curves) and $2.3$ (dashed curves). The triangles represent the results from the exact calculation without the intraband term ${\mathrm{H̃}}_{\mathrm{THz}}^{\mathrm{intra}}$. The indices $m$ of the gaps at finite momentum from the exact calculation for $\beta =2.3$ are labeled in the figure. The other gap indices can be read from the $m$ axis on the upper frame. Note that gaps only appear at the momentums where the dots (squares, triangles) indicate. The curves are only plotted as a guide for the eyes.

Figure 10

Time-averaged optical conductivities from the exact calculation (solid curves) and the the approximate formula (C5) (dashed curves) are plotted as a function of the optical frequency under a circularly polarized THz field with $\beta =2.3$ for different Fermi energies.