Three-particle correlation from a Many-Body Perspective: Trions in a Carbon Nanotube

Trion states of three correlated particles (e.g., two electrons and one hole) are essential to understand the optical spectra of doped or gated nanostructures, like carbon nanotubes or transition-metal dichalcogenides. We develop a theoretical many-body description for such correlated states using an ab initio approach. It can be regarded as an extension of the widely used $GW$ method and Bethe-Salpeter equation, thus allowing for a direct comparison with excitons. We apply this method to a semiconducting (8,0) carbon nanotube, and find that the lowest optically active trions are redshifted by $\sim 130\mathrm{meV}$ compared to the excitons, confirming experimental findings for similar tubes. Moreover, our method provides detailed insights in the physical nature of trion states. In the prototypical carbon nanotube we find a variety of different excitations, discuss the spectra, energy compositions, and correlated wave functions.

Figure 1

Optical absorption and luminescence spectrum of an (8,0) carbon nanotube. (a) Optical absorption due to excitons (red dashed lines) and due to negatively charge trions (blue lines). The arrows at 1.09 (exciton) and at 1.07 eV (trion) indicate the onset of the spectrum, which is not visible because of zero optical dipole strength of the corresponding states. (b) Luminescence spectrum from the $E_{11}$ exciton and from the related optically allowed trions, assuming that the occupancy of the excited states relative to each other is given by a Boltzmann distribution at room temperature (see Supplemental Material [18]). This distribution highlights the lower-energy states near 1.4 eV and suppresses states above 1.5 eV. In all cases the electric-field vector of the light is along the tube axis. Note that the amplitudes of the exciton and trion spectrum cannot be compared with one another because the trion spectrum scales with the density of donated electrons. The trions have been calculated for $\mathbf{K}=0$, at which the optical dipoles are by far the strongest. We consider transitions to the conduction-band minimum. Including nonzero $\mathbf{K}$ yields very similar results. An artificial broadening of 0.02 eV is used.

Figure 2

Band structure of an (8,0) CNT, showing the composition of the lowest bright exciton (a) and trion (b). The black bell-shaped distributions on the highest valence band indicate the composition of the hole as a function of wave number ${\mathbf{k}}_v$ (resulting from the coefficients of the exciton and the trion, respectively). Similarly, the green bell-shaped distributions on the lowest conduction band and the $\mathrm{CBM}+3$ band indicate the contributions to the electrons (see text). The total momenta of the exciton and trion were chosen as zero.

Figure 3

Spatial electron distribution in the exciton (red) and trion (blue) as a function of position along the CNT. (a) Spatial distribution of the electron relative to the hole, which is kept at the center of the panel (black vertical line), for the lowest-energy exciton. (b) Distribution of one electron relative to the hole (again in the center) for the lowest-energy trion, after complete spatial averaging over the other electron. (c)–(f) Distribution of one electron of the lowest-energy trion. The hole and the other electron are kept in the center (black vertical line) and at positions (indicated by the green vertical line) of 15, 7, 4, and 0 lattice constants $a_0$ (63, 30, 17, and 0 Å) away from the hole. All distribution functions are given by $|\mathrm{\Phi }{|}^2$ of the wave functions of Eq. (1) and analog for excitons [18]. All positions are on a line parallel to the CNT, at a distance of 3.7 Å from its axis [i.e., on the outside of the CNT, 0.55 Å above the nuclei, see also yellow line in (g) and (f)]. (g)–(h) Three-dimensional view of the electron distribution, see (a) and (b).