Interplay between spin-orbit interactions and a time-dependent electromagnetic field in monolayer graphene

We apply a circularly and linearly polarized terahertz field on a monolayer of graphene, taking into account spin-orbit interactions of the intrinsic and Rashba types. It turns out that the field can be used not only to induce a gap in the energy spectrum, but also to close an existing gap due to the different reaction of the spin components with circularly polarized light. Signatures of spin-orbit coupling in the density of states of the driven system can be observed even for energies where the static density of states is independent of spin-orbit interactions. Furthermore it is shown that the time evolution of the spin polarization and the orbital dynamics of an initial wave packet can be modulated by varying the ratio of the spin-orbit-coupling parameters. Assuming that the system acquires a quasistationary state, the optical conductivity of the irradiated sample is calculated. Our results confirm the multistep nature of the conductivity obtained recently, where the number of intermediate steps can be changed by adjusting the spin-orbit-coupling parameters and the orientation of the field.

Figure 1

Quasienergy spectrum under circularly polarized light (${\theta }_p={45}^{\circ }$) for various combinations of the SOC parameters: $({\lambda }_R/\Omega ,{\lambda }_I/\Omega )=$ (a) $(0,0)$, (b) $(0,0.1)$, (c) $(0.1,0)$, and (d) $(0.1,0.1)$. The field strength was set to $\alpha =0.3$.

Figure 2

Quasienergy spectrum under linearly polarized light (${\theta }_p=0^{\circ }$) for various combinations of the SOC parameters and momentum in plane orientations ($\tan {\varphi }_k=k_y/k_x$): $({\lambda }_R/\Omega ,{\lambda }_I/\Omega )=$ $(0,0)$ (left column), $(0,0.1)$ (second from left), $(0.1,0)$ (second from right), and $(0.1,0.1)$ (right). The field strength was set to $\alpha =0.3$.

Figure 3

Mean energies derived from Eq. (11) under circularly polarized light (${\theta }_p={45}^{\circ }$) for various combinations of the SOC parameters: $({\lambda }_R/\Omega ,{\lambda }_I/\Omega )=$ (a) $(0,0)$, (b) $(0,0.1)$, (c) $(0.1,0)$, and (d) $(0.1,0.1)$. The field strength was set to $\alpha =0.3$.

Figure 4

Mean energies derived from Eq. (11) under linearly polarized light (${\theta }_p=0^{\circ }$) for various combinations of the SOC parameters and momentum in plane orientations ($\tan {\varphi }_k=k_y/k_x$): $({\lambda }_R/\Omega ,{\lambda }_I/\Omega )=$ $(0,0)$ (left column), $(0,0.1)$ (second from left), $(0.1,0)$ (second from right), and $(0.1,0.1)$ (right). The field strength was set to $\alpha =0.3$.

Figure 5

Time-averaged density of states calculated from Eq. (13) under circularly polarized light (${\theta }_p={45}^{\circ }$) for various combinations of the SOC parameters: $({\lambda }_R/\Omega ,{\lambda }_I/\Omega )=$ (a) $(0,0)$, (b) $(0,0.1)$, (c) $(0.1,0)$, and (d) $(0.1,0.1)$. The field strength was set to $\alpha =0.3$.

Figure 6

Time-averaged density of states calculated from Eq. (13) under linearly polarized light (${\theta }_p=0^{\circ }$) for various combinations of the SOC parameters: $({\lambda }_R/\Omega ,{\lambda }_I/\Omega )=$ (a) $(0,0)$, (b) $(0,0.1)$, (c) $(0.1,0)$, and (d) $(0.1,0.1)$. The field strength was set to $\alpha =0.3$.

Figure 7

Time evolution of the $x$ and $z$ components of the spin polarization without electric field (top row), under a linearly polarized field (${\theta }_p=0^{\circ }$, $\alpha =0.5$) (middle row), and for circular polarization (${\theta }_p={45}^{\circ }$, $\alpha =0.5$) (bottom row), as a function of the Rashba coefficient. Parameters: ${\lambda }_I=0.25\Omega$, $k=0$.

Figure 8

Mean spin polarization as a function of the Rashba parameter (a) without electric field, (b) under a linearly polarized field (${\theta }_p=0^{\circ }$, $\alpha =0.5$), and (c) for a circular polarization (${\theta }_p={45}^{\circ }$, $\alpha =0.5$). The total simulation time is $\Omega t=10000$. Parameters: ${\lambda }_I=0.25\Omega$, $k=0$.

Figure 9

Orbital dynamics $\langle \mathbf{r}\left(t\right)\rangle$ calculated for a total simulation time of $\Omega t=1000$ for (a) circularly (${\theta }_p={45}^{\circ }$) and (b) linearly (${\theta }_p=0^{\circ }$) polarized light for various Rashba SOC coefficients: ${\lambda }_R=0$ (black line), $0.5$ (red), and $0.6$ (green). Parameters: ${\lambda }_I=0.5\Omega$, $\alpha =0.5$, $k=0$.

Figure 10

Optical conductivity under circularly polarized light (${\theta }_p={45}^{\circ }$) for various field strengths $\alpha =0$ (black dashed curves), $0.5$ [red lines in (a) and (b)], and $1.0$ [(c) and (d)], and SOC parameters in units of ${\sigma }_0=e^2/4$.

Figure 11

Optical conductivity under linearly polarized light (${\theta }_p=0^{\circ }$) for various field strengths $\alpha =0$ (black dashed curves), $0.5$ [red lines in (a) and (b)], and $1.0$ [(c) and (d)], and SOC parameters in units of ${\sigma }_0=e^2/4$.