Massless Dirac fermions in a laser field as a counterpart of graphene superlattices

We establish an analogy between spectra of Dirac fermions in laser fields and an electron spectrum of graphene superlattices formed by static one-dimensional periodic potentials. The general relations between a laser-controlled spectrum where electron momentum depends on the quasienergy and a superlattice miniband spectrum in graphene are derived. As an example we consider two spectra generated by a pulsed laser and by a steplike electrostatic potential. We also calculate the graphene excitation spectrum in continuous strong laser fields in the resonance approximation for linear and circular polarizations. Some physical phenomena, including an anomalous energy-gap anisotropy, related to the peculiar graphene energy spectrum in the strong electromagnetic field are discussed.

Figure 1

(a) The miniband electron structure for a graphene sample in the steplike spatial potential $U\left(x\right)=U_0\mathrm{sgn}\left[\sin (2\pi x/L)\right]$ with $U_0=1,L=1,k_y=1$. The gaps (${\delta }_1,{\delta }_2,{\delta }_3,...$) between different zones are clearly seen. (b) The reduced-zone momentum-quasienergy spectrum $k_x\left(\mathcal{E}\right)$ for graphene in the laser field $A=A_0\mathrm{sgn}\left[\sin (2\pi t/T)\right]$ with $A_0=1,T=1,k_y=1$. There are no gaps for momentum but not all values of quasienergy can be reached, resulting in gaps in the quasienergy ${\Delta }_1,{\Delta }_3,{\Delta }_3$ marked on the figure. The physical meaning of the energy gaps is readily seen in the extended-zone representation (c).

Figure 2

Left: (a) The crossing of electron-photon terms under the action of the laser field (see text). (b) Electron spectrum $\mathcal{E}\left(k_x\right)$ for graphene in a linearly polarized laser field with $\omega =1,A_0=1$ (all measured in units of $\pi /T$), and $k_y=0$: there are no gaps between the energy zones, in agreement with our spatial-temporal symmetry conclusions. (c) The same as (b) but for nonzero $y$ momentum $k_y=1$, resulting in the opening of gaps between the energy bands. (d) The same as (a) with zero $y$ momentum, but for the circular polarized laser field $\alpha =\pi /2$, also resulting in the opening of the gaps between zones. Right: Electron spectra $\mathcal{E}(k_x,k_y)$ for graphene in linearly polarized [(e) top and bottom energy branches, (f) two middle energy branches] and circularly polarized [(g) top and bottom energy branches, (h) two middle energy branches] laser fields with parameters $A_0=0.03,\hslash \omega =0.1$ (measured in units of $\hslash v_F\pi /a$ with $a$ being the atomic lattice constant). The two points for $k_y=0$ where the two middle energy branches touch one another are clearly seen in (f) for linearly polarized electromagnetic fields, while these branches are well separated for any $k_x,k_y$ for the circularly polarized electromagnetic fields ($\alpha =\pi /2$) as seen in (h).