Harmonic generation of open-ended and capped carbon nanotubes investigated by time-dependent Hartree-Fock theory

The high-order harmonic generation (HHG) spectra of open-ended and capped single-walled carbon nanotubes (SWCNTs) have been studied theoretically. The time evolution of charge density is displayed. It is found that the harmonic yield in nanotubes is sensitive to the field polarization and that the selection rule established for HHG of infinitely extended SWNTs does not hold for all the finite-size tubes except for the armchair tubes. SWCNTs can be ideal candidates to generate coherent x-ray pulsed sources because of the high efficiency and intrinsic characterization of HHG.

Figure 1

(Color online) The HHG spectra of open-ended and capped (5,5) tubes in the LP and CP external fields. [(a) and (b)] HHG of ${\mathrm{C}}_{120}{\mathrm{H}}_{20}$, and (c) HHG of ${\mathrm{C}}_{120}$. In the simulation, we set ${\omega }_0=0.5\mathrm{eV}$ and $t_p=120\tau$ $(\tau =\frac{2\pi \hslash }{{\omega }_0})$ for ${\mathrm{C}}_{120}$ and ${\omega }_0=0.3\mathrm{eV}$ and $t_p=110\tau$ for ${\mathrm{C}}_{120}{\mathrm{H}}_{20}$. The laser intensity $I$ is marked in the panels. Red lines and green lines in panels (a), (b), and (c) are the HHG spectra in the CP laser field and the LP laser field, respectively. For comparison, we plot the calculated HHG spectra of ${\mathrm{C}}_{60}$ in (d).

Figure 2

(Color online) (a) The HHG spectra of capped (9,0) CNTs in the CP external field, ${\omega }_0=0.3\mathrm{eV}$ and $t_p=120\tau$, and (b) the HHG spectra of open-ended (9,0) BNNT in the CP external field, ${\omega }_0=1.0\mathrm{eV}$ and $t_p=120\tau$.

Figure 3

The HHG spectra of capped (9,0) ${\mathrm{C}}_{132}$ and H-terminated (6,4) ${\mathrm{C}}_{148}{\mathrm{H}}_{20}$ tubes in the CP external field, ${\omega }_0=0.1\mathrm{eV}$, $t_p=120\tau$, and $I=2.0\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 4

(Color online) The charge density of open-ended (5, 5) tube ${\mathrm{C}}_{120}{\mathrm{H}}_{20}$ interacting with a CP external field, ${\omega }_0=0.3\mathrm{eV}$ and $t_p=110\tau$. The electron density is plotted at a time period of $120\mathrm{fs}$ and the corresponding points in the external fields ($y$ and $x$ directions) are also displayed. The magnitude of charge density is expressed by three kinds of colors: blue, white, and red which represent negative, zero, and positive values, respectively. The heavier is the color, the larger is the electron density.

Figure 5

(Color online) The charge density of open-ended (5, 5) tube ${\mathrm{C}}_{120}{\mathrm{H}}_{20}$ interacting with a LP external field, ${\omega }_0=0.3\mathrm{eV}$ and $t_p=110\tau$. The electron density is plotted at a time period of $120\mathrm{fs}$; the point of corresponding external fields can be seen in Fig. 4.