Higher harmonic generation by massive carriers in buckled two-dimensional hexagonal nanostructures

Generation of high harmonics in novel two-dimensional (2D) nanostructures such as silicene, germanene, and stanene initiated by strong coherent electromagnetic radiation of arbitrary polarization, taking into account the spin-orbit coupling and the buckling of two Bravais lattices, is investigated. The buckled hexagonal lattice system is described by the four-band second-nearest-neighbor tight-binding model. The developed theory of the interaction of massive (nonzero effective mass) carriers with a strong driving wave field covers the full Brillouin zone of a 2D hexagonal nanostructure. The wave-matter interaction is taken in the length gauge that provides proper inclusion of inter- and intraband transitions with nonzero Berry curvatures. The closed set of differential equations for the single-particle density matrix of massive carriers is solved numerically. The obtained results show that novel 2D nanostructures can serve as an effective medium for the generation of even and odd high harmonics of arbitrary polarization. Moreover, for the nanostructures under consideration, the role of the band topology is significant at harmonic generation.

Figure 1

(a) Buckled hexagonal lattice: The atoms labeled $\mathrm{A}$ are all in the $xy$ plane ($z_A=0$), and all the $\mathrm{B}$ atoms are located behind the plane at $z_B=-\ell$. Hence, the sheet is composed of two atomic planes: one of $\mathrm{A}$ atoms and another of $\mathrm{B}$ atoms. The vectors ${\delta }_1,{\delta }_2$, and ${\delta }_3$ connect nearest-neighbor atoms. The dashed circle shows second-nearest-neighbor atoms of the same type. The vectors ${\mathbf{a}}_1={\delta }_1-{\delta }_3$ and ${\mathbf{a}}_2={\delta }_1-{\delta }_2$ are the two-dimensional basis vectors, and the shaded area is the unit cell with two atoms. (b) Reciprocal lattice of the triangular lattice. Its basis vectors are ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$. The reciprocal lattice unit cell is shown as a shaded rhombic area, with inequivalent $K_{+}$ and $K_{-}$ points. The $\mathrm{\Gamma }$ point of the Brillouin zone and the $M$ point are also shown.

Figure 2

Particle distribution function $N_c\left(\mathbf{k},t_f\right)$ (in arbitrary units) after the interaction as a function of scaled dimensionless momentum components ($k_x/k_b,k_y/k_b$) for (a) and (b) silicene and (c) and (d) germanene. The wave is assumed to be linearly polarized with an intensity of ${10}^{11}\mathrm{W}/{\mathrm{cm}}^2$. In (a) and (c) the wave is assumed to be polarized along the $y$ axis, while in (b) and (d) the wave is polarized along the $x$ axis. Multiphoton excitation with the trigonal warping effect for the photon energy $\hslash {\omega }_0=0.3\mathrm{eV}$ is shown.

Figure 3

The radiation spectrum via the logarithm of the normalized field strength ${\eta }_y\left(s\right)$ (in arbitrary units) for silicene, germanene, and stanene. The wave is assumed to be linearly polarized along the $y$ axis, and ${\chi }_{0y}=1.12$.

Figure 4

The emission rate at high harmonics via the normalized field strength ${\eta }_y\left(s\right)$ for silicene (${\chi }_{0y}=1.12$), germanene (${\chi }_{0y}=1.16$), and stanene (${\chi }_{0y}=1.36$). The wave is assumed to be linearly polarized along the $y$ axis. The emission rate in the perpendicular direction is zero, ${\eta }_x\left(s\right)=0$.

Figure 5

The emission rate at high harmonics for silicene for linearly polarized waves along the $x$ and $y$ axes.

Figure 6

The spectrogram of the high harmonic generation process via the windowed Fourier transform of the normalized field strength ${\eta }_y\left(t\right)$ for germanene. The wave is assumed to be linearly polarized along the $y$ axis. The color bar represents the emission rate (arbitrary units) multiplied by ${10}^6$.

Figure 7

The radiation spectrum via the logarithm of the normalized field strengths ${\eta }_x\left(s\right)$ and ${\eta }_y\left(s\right)$ for germanene at $\ell E_z=0.2\mathrm{eV}.$ The wave is assumed to be linearly polarized along the $y$ axis.

Figure 8

The even-harmonic emission rate for the normalized field strength ${\eta }_x\left(s\right)$ for silicene, germanene, and stanene. The wave is assumed to be linearly polarized along the $y$ axis.

Figure 9

The emission rate for harmonics polarized in both directions for germanene. The wave is assumed to be circularly polarized, and $E_z=0$.

Figure 10

The emission rate for harmonics polarized in both directions for germanene. The wave is assumed to be circularly polarized, and $\ell E_z=0.2\mathrm{eV}$.