Excitonic effects in third-harmonic generation: The case of carbon nanotubes and nanoribbons

Linear and nonlinear optical properties of low-dimensional nanostructures have attracted great interest from the scientific community as tools to probe the strong confinement of electrons and for possible applications in optoelectronic devices. In particular it has been shown that the linear optical response of carbon nanotubes [F. Wang et al., Science 308, 838 (2005)] and graphene nanoribbons [Nat. Commun. 5 4253 (2014)] is dominated by bounded electron-hole pairs, excitons. The role of excitons in linear response has been widely studied, but still, little is known about their effect on nonlinear susceptibilities. Using a recently developed methodology [Phys. Rev. B 88, 235113 (2013)] based on well-established ab initio many-body perturbation theory approaches, we find that quasiparticle shifts and excitonic effects significantly modify the third-harmonic generation in carbon nanotubes and graphene nanoribbons. For both systems the net effect of many-body effects is to reduce the intensity of the main peak in the independent-particle spectrum and redistribute the spectral weight among several excitonic resonances.

Figure 1

Schematic description of the THG process: three photons of frequency $\omega$ are destroyed, and one photon of frequency $3\omega$ is created; that is, the system responds at frequency $3\omega$ to an applied field at frequency $\omega$.

Figure 2

Band structure and density of states for (a) and (b) the (10,0) CNT and (c) and (d) 9-AGNR. Notice that the band structures and densities of states are aligned in such a way that they have the same $y$ axis. The density of states is normalized in such a way that its integral on the occupied bands give the total number of electrons.

Figure 3

THG intensity $|{\chi }_{zzzz}^{\left(3\right)}\left(\omega \right)|$ and optical absorption in the longitudinal direction ($z$ axis) for the (10,0) CNT. (a) THG from the IP model (solid line) and the QP model (dotted line) for the Hamiltonian. (b) The imaginary part of the dielectric function at $\omega$ (solid line) and $\omega /3$ (dashed line) from the IP model for the Hamiltonian. (c) THG within the GW+BSE model (solid line) and QP model (dotted line) Hamiltonian. (d) The imaginary part of the dielectric function at $\omega$ (solid line) and $\omega /3$ (dashed line) within the GW+BSE model and the IP results at $\omega$ (black dotted line). The vertical violet line represents the KS fundamental gap, and the dashed line represents $1/3$ of the $GW$ fundamental gap.

Figure 4

THG response in GNRs and CNTs at the IP level. (a) THG for zigzag CNTs of increasing size. (b) THG for the $N=3p+1$ GNR family and (c) THG for the $N=3p$ family.

Figure 5

THG intensity $|{\chi }_{zzzz}^{\left(3\right)}\left(\omega \right)|$ and optical absorption in the longitudinal direction ($z$ axis) for 9-AGNR. (a) THG from the IP model (solid line) and the QP model (dotted line) for the Hamiltonian. (b) The imaginary part of the dielectric function at $\omega$ (solid line) and $\omega /3$ (dashed line) from the IP model for the Hamiltonian. (c) THG within the $GW$+BSE model (solid line) and QP model (dotted line) Hamiltonian. (d) The imaginary part of the dielectric function at $\omega$ (solid line) and $\omega /3$ (dashed line) at the same level of approximation. The vertical solid green (violet) line represents the $GW$ (Kohn-Sham) fundamental gap, and the green dashed line represents $1/3$ of the $GW$ fundamental gap.