Phonons and conduction in molecular quantum dots: Density functional calculations of Franck-Condon emission rates for bifullerenes in external fields

We report the calculation of various phonon overlaps and their corresponding phonon emission probabilities for the problem of an electron tunneling onto and off of the fullerene-dimer molecular quantum dots ${\mathrm{C}}_{72}$ and ${\mathrm{C}}_{140}$, both with and without the influence of an external field. We show that the stretch mode of the two balls of the dumbbell couples most strongly to the electronic transition and, in turn, that a field in the direction of the bond between the two fullerene balls is most effective at further increasing the phonon emission into the stretch mode. As the field is increased, phonon emission increases in probability with an accompanying decrease in probability of the dot remaining in the ground vibrational state. We also present a simple model to estimate the effect of molecular size on the phonon emission of composite dimer molecules and compare the results with the complete analysis of ${\mathrm{C}}_{72}$ and the experimentally tested ${\mathrm{C}}_{140}$. In our approach we do not assume that the Hessians of the molecule are identical for different charge states. Our treatment is hence a generalization of the traditional phonon overlap calculations for coupled electron-photon transitions in solids.

Figure 1

${\mathrm{C}}_{72}$ with $19\text{−}\mathrm{meV}$ stretch mode indicated. In this high-symmetry viewpoint, some atoms are obscured behind others.

Figure 2

(Color online) Wave functions ${\Psi }_0^{\left(1\right)}$ and ${\Psi }_0^{\left(2\right)}$ for harmonic potentials ${\Omega }_1^2$ and ${\Omega }_2^2$ in terms of a two-dimensional rescaled coordinate $({\mathbf{y}}_1,{\mathbf{y}}_2)$ separated by the rescaled length $\Delta =\sqrt{m}({\mathbf{r}}_2-{\mathbf{r}}_1)$. This and the succeeding figure (Fig. 3) are pictorial representations, with quadratic forms and displacements chosen to illustrate the variables used in the calculation.

Figure 3

(Color online) Overlap integrand corresponding to wave functions in Fig. 2, centered on $\tilde{\Delta }$ with quadratic form ${\overline{\Omega }}^2$.

Figure 4

${\mathrm{C}}_{140}$ with stretch mode shown schematically.

Figure 5

$g_{\alpha }$ for the ${\mathrm{C}}_{72}$ $0\to 1$ charge-state transition. The large peak is the stretch mode $\alpha =10$ at $19\mathrm{meV}$. Including the two-state emission lines would add an additional peak at $38\mathrm{meV}$ (twice the stretch mode) and an otherwise roughly continuous background (see Fig. 10).

Figure 6

$g_{\alpha }$ for the ${\mathrm{C}}_{140}$ $0\to 1$ charge-state transition. The large peak is the $\alpha =10$ stretch mode at $11\mathrm{meV}$. Again, we estimate that the only significant two-phonon line is at $22\mathrm{meV}$.

Figure 7

${\mid \Psi \mid }^2$ of the highest occupied molecular orbital level of ${\mathrm{C}}_{72}$ under an electric field of $4\times {10}^9\mathrm{V}∕\mathrm{m}$ along an intercage bond, showing the polarization of the electron density.

Figure 8

Field dependence of $g_{\alpha }$ for the ${\mathrm{C}}_{140}$ $11\text{−}\mathrm{meV}$ stretch mode from the DFT calculation. The solid line is the field dependence for our simple model calculation which is explained further in Sec. 8. Experimental values (triangles) are taken at zero field, but included in the plot in a vertical column for visibility.

Figure 9

Angle dependence of $g_{\alpha }$ for $\alpha$ giving the $11\text{−}\mathrm{meV}$ stretch mode for ${\mathrm{C}}_{140}$. Leftmost figure is $g$ plotted as function of the angle of electric field (an extremely high field magnitude of $5\times {10}^{12}\mathrm{V}∕\mathrm{m}$) for the $11\text{−}\mathrm{meV}$ stretch mode; the remaining figures to the right are for other modes and have been magnified considerably to show their shape. The vertical represents fields along the long axis of the molecule. At more reasonable fields these distributions would be added to a roughly isotropic background.

Figure 10

$I\text{−}V$ curve predicted for ${\mathrm{C}}_{72}$ for one-phonon processes (solid line) and up to two-phonon processes (approximate, dashed line), using the DFT STO-3G basis set. The arrow indicates the position of the two-phonon contribution from the stretch mode.