High-harmonic generation in spin-orbit coupled systems

We study high-harmonic generation in two-dimensional electron systems with Rashba and Dresselhaus spin-orbit coupling and derive harmonic generation selection rules with the help of group theory. Based on the band structures of these minimal models and explicit simulations we reveal how the spin-orbit parameters control the cutoff energy in the high-harmonic spectrum. We also show that the magnetic field and polarization dependence of this spectrum provides information on the magnitude of the Rashba and Dresselhaus spin-orbit coupling parameters. The shape of the Fermi surface can be deduced at least qualitatively and if only one type of spin-orbit coupling is present, the coupling strength can be determined.

Figure 1

High-harmonic spectra for the charge current, $J_x$, and two components of the spin current, $J_{yx}$ and $J_{zy}$, of the Rashba model with $\alpha =2.5$, $\gamma =0$, $\mu =3.5$, and $B=0$. A linearly polarized light field with $E_0=0.2$, central frequency $\mathrm{\Omega }=0.3$, and polarization along the $x$ direction is used.

Figure 2

$\alpha$ dependence of the HHG spectrum for $\gamma =B=0$. A linearly polarized light field with $E_0=0.2$, central frequency $\mathrm{\Omega }=0.3$, and polarization along the $x$ direction is used. The inset shows the two branches of the band dispersion of Eq. (1) for $k_y=0$. The dashed lines indicate the values of the chemical potential. In the top panel, the chemical potential is fixed at $\mu =4$, while in the bottom panel, the filling of the system is kept constant.

Figure 3

HHG cutoffs for the charge current, $f_{cc}\left(\alpha \right)$, and the $J_{zy}$ spin current component, $f_{sc}\left(\alpha \right)$ when $\mu =4$ and $\gamma =B=0$. $2\alpha /\mathrm{\Omega }$ corresponds to the maximum band gap in units of $\mathrm{\Omega }$. The uncertainty in determining the cutoff values is estimated to be $\delta f\approx 2.0$.

Figure 4

$\alpha$ dependence of the high harmonic spectrum of the Rashba model with $\gamma =0.0$, $B=2.5$, and $\mu =4$. A linearly polarized light field with $E_0=0.2$, central frequency $\mathrm{\Omega }=0.3$, and polarization along the $x$ direction is used.

Figure 5

Dependence of the HHG spectra on the polarization direction. The top panel shows the total spectrum, whereas the bottom panel shows the contribution from the magnetization term. The inset shows the band structure along cuts parametrized by $k\langle 1,1\rangle$ (black) and $k\langle 1,-1\rangle$ (red), i.e., along the direction of the field polarization. $\alpha =\gamma =1.5$ and $B=1.8$. The field strength is set to $E_0=0.5$ and $\mu =2$.