Resonance energy transfer and quantum dots

The mechanism of energy transfer between quantum dots is investigated theoretically. In order to incorporate explicit account of the selection rules for absorption of circularly polarized light, a quantum electrodynamical treatment of the electronic coupling is derived. The electronic coupling is mediated by the exchange of a virtual photon, which in the far zone limit acquires real character and is circularly polarized. A rotational average expression is also obtained. The conditions by which quantum information, in terms of exciton spin orientation (total angular momentum quantum number), can be exchanged or switched through resonance energy transfer are discussed. The spectral overlap factor is considered with explicit discussion of the roles of homogeneous and inhomogeneous line broadening. It is shown that the ensemble spectral overlap is determined by the inhomogeneous line broadening dictated by sample polydispersity.

Figure 1

The exciton fine structure that underlies the $1S_{3∕2}-1S_e$ exciton transition for CdSe QDs. Three states, drawn with dashed lines, are optically dark. Five are optically bright, one of these $\left({\Psi }_0^U\right)$ being linearly polarized. The other four states can be excited according to selection rules for circularly polarized light: the ${\Psi }_{+1}^L$ and ${\Psi }_{+1}^U$ states, which can be excited by right-circularly polarized light, and the ${\Psi }_{-1}^L$ and ${\Psi }_{-1}^U$ states, which can each be excited by left-circularly polarized light. We label each group of states enclosed in the dashed boxes ∣↑⟩ and ∣↓⟩, respectively.

Figure 2

(i) A pair of quantum dots A and B with center-to-center separation $R$. The donor is taken to be the QD labeled A. The QDs may be identical in size, otherwise typically A would be smaller than B. (ii) Coordinate system used to determine the orientation factors. We define an axis system for a quantum dot pair such that, for the donor plane normal to the crystal $\widehat{Z}$ axis, $\widehat{X}$ makes an angle $\theta$ with the donor-acceptor separation vector $\vec{R}$, and $\widehat{Y}$ makes an angle $\varphi$ with $\vec{R}$. Analogous angles and the axis system for the acceptor quantum dot are defined such that ${\widehat{X}}^{\prime }$ makes an angle ${\theta }^{\prime }$ with the acceptor-donor separation vector $\vec{R}$, and ${\widehat{Y}}^{\prime }$ makes an angle ${\varphi }^{\prime }$ with $\vec{R}$. In the diagram $\widehat{Y}$ and ${\widehat{Y}}^{\prime }$ emerge from the plane of the page.

Figure 3

Plots showing the variation of (a) ${\mid {\kappa }_{\uparrow \downarrow }\mid }^2$ and (b) ${\mid {\kappa }_{\uparrow \uparrow }\mid }^2$ as a function of the angle $\varphi$ defining a rotation of one quantum dot relative to another. Note that for the parallel case $\varphi =\pi ∕2$. The other angles are fixed: $\theta ={\theta }^{\prime }={\varphi }^{\prime }=\pi ∕2$.

Figure 4

The lower curve is a plot of experimental measurement of the absorption spectrum of CdSe colloidal quantum dots ($4.0\mathrm{nm}$ average diameter) in a polymer film at $4\mathrm{K}$. The upper two absorption curves (for the resonance energy transfer acceptor) are simulated using Eq. (29). The dashed curves represent two of the donor emission spectra considered in our calculations.

Figure 5

Plot of the ensemble averaged spectra overlap and the energy transfer rate calculated as a function of donor-acceptor energy gap, representing a variation in the average size of the donor quantum dots.