Atomistic theory of coherent spin transfer between molecularly bridged quantum dots

Time-resolved Faraday rotation experiments have demonstrated coherent transfer of electron spin between CdSe colloidal quantum dots coupled by conjugated molecules. We employ here a Green’s-function approach, using semiempirical tight binding to treat the nanocrystal Hamiltonian and extended Hückel theory to treat the linking molecule Hamiltonian, to obtain the coherent transfer probabilities from atomistic calculations, without the introduction of any new parameters. Calculations on 1,4-dithiolbenzene, 1,4-dithiolcyclohexane, and 1,6-dithiolhexane linked nanocrystals agree qualitatively with experiment and provide support for a previous transfer Hamiltonian model. We find a striking dependence on the transfer probabilities as a function of nanocrystal surface site attachment and linking molecule conformation. Additionally, we predict quantum interference effects in the coherent transfer probabilities for 2,7-dithiolnaphthalene and 2,6-dithiolnaphthalene linking molecules. We suggest possible experiments based on these results that would test the coherent, through-molecule transfer mechanism.

Figure 1

Linking molecules treated in this study: (a) 1,4-dithiolbenzene; (b) 1,4-dithiolcyclohexane; (c) 1,6-dithiolhexane; (d) 2,6-dithiolnaphthalene; (e) 2,7-dithiolnaphthalene.

Figure 2

$T\left(E\right)$ (top) and standard deviation of $T\left(E\right)$ (bottom) for 1,4-dithiolbenzene linked CdSe nanocrystals of 3.4 and 5.0 nm diameter. The MM2 optimized conformation of the molecule and attached surface Cd atom is shown as the inset cartoon. The black solid and black dashed lines indicate the average and median $T\left(E\right)$, respectively. The standard deviation of $T\left(E\right)$ results from variations due to the molecule-nanocrystal surface site attachments, as described in Sec. 3A1. The mean value of $T\left(E\right)$ in the conduction band region $(-0.7–0\mathrm{eV})$ oscillates between 0.06 and 0.13. Peaks in $T\left(E\right)$ in the band-gap region (0–2.4 eV) result from artificial surface states due to ligand removal in the calculation of $G\left(E\right)$.

Figure 3

$T\left(E\right)$ for the flat conformation of the 1,4-dithiolbenzene linker molecule, in which both the C-S and S-Cd bonds are collinear in the plane of the benzene ring. The upper figure shows mean $T\left(E\right)$ and the lower figure the standard deviation in $T\left(E\right)$ resulting from the variability in surface site attachment. As compared to the reference conformation in Fig. 2, the hole energy region transfer probabilities are reduced by half, and the electron energy region $T\left(E\right)$ is nearly zero.

Figure 4

Effect on $T\left(E\right)$ of in-plane bond angle changes from the flat conformation of 1,4-dithiolbenzene linked nanocrystals. Arrows from the surface Cd atom indicate the direction of the motion studied; each line in the plots indicates a different conformation along that direction.

Figure 5

Effect on $T\left(E\right)$ of out-of-plane bond angle changes from the flat conformation of 1,4-dithiolbenzene linked nanocrystals. Arrows from the surface Cd atom indicate the direction of the motion studied; each line in the three plots indicates a different conformation along that direction.

Figure 6

(a) Mean $T\left(E\right)$ for the MM2-optimized (“boat”) conformation of 1,4-dithiolcylohexane linked nanocrystals. Note the dramatic decrease in $T\left(E\right)$ over the entire energy range as compared with the conjugated linking molecule 1,4-dithiolbenzene, shown in Fig. 2. (b) Mean $T\left(E\right)$ for the MM2-optimized conformation of the linear 1,6-dithiolhexane linked nanocrystals. Note the expanded scale.

Figure 7

Effect of C-S and S-Cd bond rotations on the transmission probability, $T\left(E\right)$, for 1,4-dithiolcyclohexane. The results for each C-S and S-Cd rotation calculated as described in Sec. 3B are shown as individual lines in the figure. No significant changes in $T\left(E\right)$ are observed for any of these rotational motions.

Figure 8

Mean $T\left(E\right)$ and surface site standard deviation for 2,6-dithiolnaphthalene linked nanocrystals. In the ranges of $E=-1\text{–}0\mathrm{eV}$ and $E=2.0\text{–}3.5\mathrm{eV}$, $T\left(E\right)$ was found to be near zero for all three conformations. This results from destructive quantum interference of the electron paths, as discussed in detail in Sec. 4D.

Figure 9

Mean $T\left(E\right)$ and surface site standard deviation for 2,7-dithiolnaphthalene linked nanocrystals. In contrast to the result for 2,6-dithiolnaphthalene shown in Fig. 8, constructive quantum interference of the electron paths causes a nonzero $T\left(E\right)$ of between 0.02 and 0.09 in the region of $E=-1\text{–}0\mathrm{eV}$ that is evident for all three conformations.