Graphene in ultrafast and superstrong laser fields

For graphene interacting with a few-fs intense optical pulse, we predict unique and rich behavior dramatically different from three-dimensional solids. Quantum electron dynamics is shown to be coherent but highly nonadiabatic and effectively irreversible due to strong dephasing. Electron distribution in reciprocal space exhibits hot spots at the Dirac points and oscillations whose period is determined by nonlocality of electron response and whose number is proportional to the field amplitude. The optical pulse causes net charge transfer in the plane pf graphene in the direction of the instantaneous field maximum at relatively low fields and in the opposite direction at high fields. These phenomena promise ultrafast optoelectronic applications with petahertz bandwidth.

Figure 1

(a) Hexagonal lattice structure of 2D graphene. The graphene lattice consists of two inequivalent sublattices, which are labeled by “A” and “B.” The vectors ${\mathbf{a}}_1=a/2(\sqrt{3},1)$ and ${\mathbf{a}}_2=a/2(\sqrt{3},-1)$ are the direct lattice vectors of graphene. The nearest-neighbor coupling, which is characterized by the hopping integral $\gamma$, is also shown. (b) The first Brillouin zone of reciprocal lattice of graphene. Points $K$ and $K^{\prime }$ are two degenerate Dirac points, corresponding to two valleys of low energy spectrum of graphene. Blue line with arrows shows polarization of the time-dependent electric field of the pulse. The polarization is characterized by angle $\theta$.

Figure 2

Interband dipole matrix element $D_x$ is shown as a function of the wave vector $\mathbf{k}$. The red lines show the boundary of the first Brillouin zone. The dipole matrix element is singular near the Dirac points ($K$ and $K^{\prime }$ points).

Figure 3

Population of the CB. (a) The CB population, ${\mathcal{N}}_{CB}\left(t\right)$, and the corresponding electric field, $F\left(t\right)$, of the laser pulse are shown as function of time $t$. The polarization of the pulse is along axis $x$, i.e., $\theta =0$. (b) Series of the CB populations are plotted as functions of time for peak fields indicated on the graph. (c) The maximum and residual CB populations as functions of the peak electric field.

Figure 4

Conduction band population ${\mathcal{N}}_{CB}(\mathbf{k},t)={\left|{\beta }_{c\mathbf{k}}\left(t\right)\right|}^2$ (see SI for definition) as a function of the wave vector at different moments of time. Only the first Brillouin zone of the reciprocal space is shown. The peak electric field of the pulse is $F_0=1$ V/Å. Different colors correspond to different values of the conduction band population as shown in the figure.

Figure 5

Residual conduction band population ${\mathcal{N}}_{CB}^{\left(res\right)}\left(\mathbf{k}\right)$ as a function of wave vector $\mathbf{k}$ for different amplitudes $F_0$ of the optical pulse, as indicated. Only the first Brillouin zone is shown. The polarization of electric field is along axis $x$.

Figure 6

Time-dependent conduction band population and corresponding dipole matrix element $D_x$. The data are shown for a state with initial wave vector $\mathbf{q}$ of the reciprocal space. The conduction band population is calculated as $|{\beta }_{c\mathbf{q}}{\left(t\right)|}^2$ and the dipole matrix element is defined as $D_x({\mathbf{k}}_T(\mathbf{q},t)$. Two different initial wave vectors in panels (a) and (b) correspond to small and large residual conduction band populations, respectively. The inset in panel (a) illustrates schematically the electron dynamics in the reciprocal space: the electron is transferred along the path “1”$\to$“2”$\to$“3”$\to$“2”$\to$“1”. The polarization of the optical pulse is along axis $x$.

Figure 7

(a) Electric current density in graphene as a function of time for two amplitudes, $F_0=1.0$ V/Å and $F_0=2.0$ V/Å. (b) Transferred charge density through graphene monolayer as a function of $F_0$ for different levels of graphene doping (defined by electron Fermi energy $E_F$).