Dirac fermion time-Floquet crystal: Manipulating Dirac points

We demonstrate how to control the spectra and current flow of Dirac electrons in both a graphene sheet and a topological insulator (TI) by applying either two linearly polarized laser fields with frequencies $\omega$ and $2\omega$ or a monochromatic (one-frequency) laser field together with a spatially periodic static potential (graphene/TI superlattice). Using the Floquet theory and the resonance approximation, we show that a Dirac point in the electron spectrum can be split into several Dirac points whose relative location in momentum space can be efficiently manipulated by changing the characteristics of the laser fields. In addition, the laser-field-controlled Dirac fermion band structure—a Dirac fermion time-Floquet crystal—allows the manipulation of the electron currents in graphene and topological insulators. Furthermore, the generation of dc currents of desirable intensity in a chosen direction occurs when the biharmonic laser field is applied, which can provide a straightforward experimental test of the predicted phenomena.

Figure 1

(a) The two lowest zones of the quasienergy spectrum $\varepsilon (k_x,k_y)$ with application of the two-frequency laser field (2) with amplitudes $A_1=A_2=1$, time phase shift $\alpha =0$, and relative angle $\theta =\pi /10$. To highlight five Dirac points where $\varepsilon =0$, we plot $F=-ln\left[\varepsilon (k_x,k_y)\right]$ for the top energy zone in (b). (c) shows the evolution of the locations of the Dirac points when changing the relative angle $\theta$ of the electric fields, i.e., the angle between ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$; different colors code different angles from blue for $\theta =0$ to yellow for $\theta =2\pi$; all other parameters are the same as in (a). (d) is the same as (c) for the same set of parameters except for the weaker second-field amplitude $A_2=0.5$.

Figure 2

Distribution of the single-particle currents $j_x(k_x,k_y)$ (the left column) and $j_y(k_x,k_y)$ (the right column) for states with a fixed momentum $(k_x,k_y)$ when the harmonic field (2) with $A_1=A_2=1$ is applied. Four different cases are considered: (a), (b) $\theta =0$, $\alpha =0$; (c), (d) $\theta =\pi$, $\alpha =0$; (e), (f) $\theta =\pi /2$, $\alpha =0$; (g), (h) $\theta =0$, $\alpha =\pi /2$. Depending on the relative angle $\theta$ and the time shift $\alpha$ of the electric fields, one can expect different symmetry of the current distributions and a possible generation of a dc electric current (see discussion in the text). The colors run from light blue when $j_i=1$ to dark red when $j_i=-1$ and the thick dark curves are the points $(k_x,k_y)$ with $j_i=0$.

Figure 3

The mean currents ${\overline{j}}_{x,y}$ for a certain magnitude of the momentum $k$, but averaged with respect to $\mathbf{k}$ orientations (see text). (a) Red and black dots correspond to ${\overline{j}}_x\left(\alpha \right)$ and ${\overline{j}}_y\left(\alpha \right)$ current components, respectively, calculated for $A_1=A_2=1$, $k=0.25$, $\theta =\pi /4$, and $\alpha$ changing from 0 to $2\pi$; note that the nodes ${\overline{j}}_x\left(\alpha \right)=0$ and ${\overline{j}}_y\left(\alpha \right)=0$ coincide, giving $\theta =\pi /2+\pi n$ with integer $n$. (b) Red and black dots show the ${\overline{j}}_x\left(\theta \right)$ and ${\overline{j}}_y\left(\theta \right)$ dependence, calculated for $A_1=A_2=1$, $k=0.25$, $\alpha =0$; the nodes ${\overline{j}}_x\left(\theta \right)=0$ and ${\overline{j}}_y\left(\theta \right)=0$ are shifted by $\pi /2$ giving the possibility of having a nonzero $\alpha$ dependence of one current component and zero value of the other component at the points $\theta =\pi n/2$ with integer $n$. An example of such a case is shown in (c) where ${\overline{j}}_x\left(\alpha \right)\ne 0$ and ${\overline{j}}_y\left(\alpha \right)=0$ for $A_1=A_2=1$, $k=0.25$, $\theta =0$.

Figure 4

(a) The two lowest zones of the quasienergy spectrum $\varepsilon (k_x,k_y)$ for a graphene/TI superlattice [Eq. (6)] with $U_0=1$ and spatial period $L=2\pi /\mu =4\pi /3$, i.e., $\mu =1.5$, on application of a monochromatic laser field (7) along the $x$ axis ($\theta =0$) with amplitude $A_1=1$. To highlight the four Dirac points where $\varepsilon =0$, we plot $F=-ln\left[\varepsilon (k_x,k_y)\right]$ for the top energy zone in (b) for the same parameters as in (a) and in (c) for the same parameters as in (a) except for the laser field orientation ($\theta =\pi /2$).

Figure 5

(a) and (b) show the evolution of the locations of the Dirac points for a graphene/TI superlattice when the relative angle $\theta$ of the monochromatic electric laser field [Eq. (7)] and the electrostatic field $\nabla U$ is changed. The other parameters are $A_1=1$, $U_0=1$, $\mu =1.5$ for (a) and $\mu =0.5$ for (b); different colors code different angles from blue for $\theta =0$ to yellow for $\theta =2\pi$. (c)–(f) Distribution of the single-particle currents $j_x(k_x,k_y)$ (the left column) and $j_y(k_x,k_y)$ (the right column) for states with a fixed momentum $(k_x,k_y)$ when the monochromatic laser field (7) is applied to a graphene/TI superlattice for all parameters as in (a) while $\theta =0$ for (c), (d) and $\theta =\pi /2$ for (e), (f).

Figure 6

Schematic representation of the photon-shifted Dirac cones for the case of the first- (a) and the second- (b) order resonance approximations for monochromatic laser field $\mathbf{A}=(0,A_{1}cos\omega t)$. The black dots represent the lowest-energy zone. (c) The Dirac points calculated in the first- (red points) and the second- (blue points) order resonance approximations for $\omega =1,A_1=0.5$. The points for $k\approx \omega /2$ for the first- and second-order resonance approximations almost coincide. (d) 3D plot of the logarithm of the lowest-energy zone, $-ln\varepsilon (k_x,k_y)$, for the same parameters as in (c).

Figure 7

(a),(b) Contour plot of the lowest-energy band for a graphene superlattice with spatially periodic potential (6) driven by the monochromatic laser field $\mathbf{A}=(A_{1}cos\theta cos\omega t,A_{1}sin\theta cos\omega t)$, calculated using the spatial second-order resonance approximation (14) for the same parameters as the energy spectrum shown in Fig. 4 for $\theta =\pi /2$ (a) and $\theta =0$ (b). The blue and red small circles show the locations of Dirac points obtained within the spatial first-order and the spatial second-order resonance approximations.