Excitonic lifetimes and optical response of carbon nanotubes modulated by electrostatic gating

We investigate the excitonic optical properties of carbon nanotubes modulated by an electrostatic field applied in a direction transversal to the carbon nanotubes' axis. We find that excitation energies are redshifted while absorption peaks split due to symmetry breaking. Furthermore, from analysis of the electron-hole wave function calculated in the Wannier approximation, it is seen that the exciton wave function tends to polarize, with a separation of the electron and hole in the large-field limit. This has the effect of reducing the excitonic intrinsic radiative decay rates due to reduced electron-hole overlap. We compute thermalized effective decay rates for comparison with experiments, reflecting the interplay between exciton energy shifts and reduced intrinsic decay rates. Hence, these results suggest several possibilities for modulating the optical response of carbon nanotubes by the application of an electrostatic field in gatelike configurations.

Figure 1

Absolute square of exciton wave functions at $z=0$ for an $R=0.5$ tube with increasing applied field strength $F$. Note that the electrostatic field is reported in effective exciton units $R{y}^{*}/a_B^{*}$. The schematic illustrations indicate the electron (blue spheres) and hole (red spheres) configurations with largest wave function amplitude, and represent phases (i) to (iii) discussed in the text. Note that we draw multiple exciton complexes to indicate equivalent angular configurations; however, only two-particle interactions are taken into account (i.e., effects from trions, biexcitons, and the like are not included).

Figure 2

Excitonic lifetimes and energies calculated using the effective mass Wannier model for an $R=0.2$ tube. Lifetimes are normalized to the case of a vanishing electrostatic field, while energies are displayed in effective exciton Rydbergs and are measured relative to the unperturbed band gap energy. Top panels display the electron-hole wave functions at electrostatic fields indicated by the arrows. The axes of the respective top panels are identical to the ones in Fig. 1.

Figure 3

Imaginary part of the dielectric function for a (20,0) CNT with the exciting optical field aligned (a) parallel to the axial direction of the CNT and (b) perpendicular to this direction. The black lines indicate the response at a vanishing applied electrostatic field.

Figure 4

Logarithm of the imaginary part of the dielectric functions, radiative decay rates, and excitation energies of the ten lowest excitons found using the TB exciton model for a (a) (20,0) CNT and a (b) (7,0) CNT . We note that the top panels indicating the optical response display the same data as Figs. 3 and 3; however, here represented on a logarithmic scale to highlight the weak features. Please note that the line types (color/full/dashed) used in the middle and lower panels are identical, and indicate the respective exciton states.

Figure 5

Effective radiative decay rates for a (7,0) CNT and a (20,0) CNT as a function of temperature and electrostatic field strength.

Figure 6

Systematic study of the low-field response for various CNTs. Only semiconducting tubes are considered, hence all members of the $(3n,0)$ CNT family are excluded ($n$ indicating an integer). The lines represent a power law fit $\alpha =\beta r^3$, with $r$ indicating the CNT radius, while $\beta$ is given by 0.09 and 0.06 $\mathrm{eV}/\mathrm{V}{}^2\AA$ for the $(3n+1,0)$ and $(3n+2,0)$ families, respectively. The inset displays a selection of the fits used for extracting the polarizabilities (lines) and the shifts calculated using the full TB model (dots).