Concentration dependence of Förster resonant energy transfer between donor and acceptor nanocrystal quantum dot layers: Effect of donor-donor interactions

The influences of donor and acceptor concentrations on Förster resonant energy transfer (FRET) in a separated donor-acceptor quantum dot bilayer structure have been investigated. Donor intra-ensemble energy transfer is shown to have an impact on the donor-acceptor FRET efficiency in the bilayer structure. At high donor concentrations the FRET distance dependence and the acceptor concentration dependence in the separated donor-acceptor layer structure agree well with theories developed for FRET between randomly distributed, homogeneous donor and acceptor ensembles. However, discrepancies between measurement and theory are found at low donor concentrations. A donor concentration study shows that the FRET efficiency decreases with increasing donor concentration even though a donor concentration-independent FRET efficiency is predicted by standard theory. The observed dependence of the FRET efficiency on the donor concentration can be explained within the FRET rate model, for a constant, donor concentration independent FRET rate, by taking into account the concentration dependent donor reference lifetime arising from intra-donor ensemble FRET. This shows that the decrease in the FRET efficiency with increasing donor concentration is not a signature of a change in the donor-acceptor FRET rate, but due to the competition of the donor-acceptor and donor-donor energy transfer for the higher energy donors. As the intra-donor ensemble FRET represents another decay mechanism, the donor quantum yield for the higher energy donors decreases with increasing donor quantum dot (QD) concentration, as can also be seen from the redshift of the donor emission spectrum. Using this concentration dependent donor quantum yield in the calculation of the Förster radius, the FRET theory for homogeneous donor and acceptor ensembles can be modified to include the effect of the donor intra-ensemble transfer and to correctly describe the trends and absolute values of the measured FRET efficiencies as a function of the donor and the acceptor concentrations. These results show that in QD systems where intra-donor ensemble FRET is as important as the radiative and nonradiative donor decay mechanisms, the FRET rate rather than the FRET efficiency more appropriately characterizes the donor-acceptor FRET. By fitting with the rate model, FRET rates as high as $(1.2\hspace{0.5em}{\mathrm{ns}}^{)}$ have been determined for the structures presented here.

Figure 1

Absorption (line) and photoluminescence (PL) spectra (symbols) for donor (solid, green line, and green squares) and acceptor (dashed, red line, and red circles) QD monolayers with concentrations of $c_{\mathrm{Don}}=1.1\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$ and $c_{\mathrm{Acc}}=0.28\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$, respectively. In the inset the donor PL decays for the blue (blue squares) and red (red circles) side of the QD ensemble emission spectrum as well as the unfiltered decay (black triangles) are shown.

Figure 2

Concentration dependence of the average lifetime of the PL decays on the blue side of the ensemble emission spectrum of the donor monolayers ${\tau }_{\mathrm{blue}}$ (filled squares). The line represents the fit of the lifetime data using a theory of FRET in two dimensions (Ref. 27). The intra-ensemble FRET efficiency and FRET rate, calculated from the fit of the measured lifetime data, are shown in the inset.

Figure 3

(a) Photoluminescence spectrum of a donor-acceptor bilayer structure with a donor-acceptor center-to-center separation of 3.6 nm (corresponding to a layer separation of 0.5 nm excluding the QD radii), $c_{\mathrm{Don}}=1.8\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$ and $c_{\mathrm{Acc}}=0.45\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$ (filled, black triangles). The reference spectra of donor (open, green squares) and acceptor (open, red circles) monolayers are also shown. (b) The donor emission decay (measured with a broad 500 nm filter) in the donor-acceptor bilayer structure (filled, black squares) and in the donor reference monolayer (open, green squares). In the inset the first 12 ns of the acceptor emission decay (using a broad 650 nm filter) are shown for the donor-acceptor bilayer structure (filled, black circles) and the acceptor reference monolayer (open, red circles).

Figure 4

Dependence of the measured FRET efficiency $E_{\mathrm{FRET}}$ on the acceptor concentration at high [$c_{\mathrm{Don}}=(3.1\pm 0.5)\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$, solid squares] and low donor concentrations [$c_{\mathrm{Don}}=(1.0\pm 0.2)\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$, open circles] for a donor-acceptor separation of 3.6 nm. The lines represent theoretically calculated trends for $R_0=3.1\hspace{0.5em}\mathrm{nm}$ (solid line), $R_0=3.9\hspace{0.5em}\mathrm{nm}$ (dotted line), and $R_0=4.3\hspace{0.5em}\mathrm{nm}$ (dashed line). In the inset the distance dependence at a donor concentration of $c_{\mathrm{Don}}=(3.9\pm 1.2)\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$ and an acceptor concentration of $c_{\mathrm{Acc}}=(0.85\pm 0.10)\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$ is shown, with the theoretical trend for $R_0=3.9\hspace{0.5em}\mathrm{nm}$ shown as solid line.

Figure 5

Measured FRET efficiency as a function of the donor QD concentration in the bilayer structure at low [$c_{\mathrm{Acc}}=(0.3\pm 0.2)\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$, solid squares] and high acceptor concentrations [$c_{\mathrm{Acc}}=(0.8\pm 0.3)\times {10}^{17}\hspace{0.5em}{\mathrm{m}}^{-2}$, open circles]. Also shown is the calculated FRET efficiency expected from the FRET theory for homogenous donor ensembles with a fixed ensemble quantum yield (Hom.-Theory, dotted line), the FRET efficiency using the modified Hom.-Theory which takes into account a concentration dependent donor quantum yield (dashed line) and the fit of the experimental data with the FRET rate equation model [Eq. (4a)] including a constant FRET rate $k_{\mathrm{FRET}}$ (solid line).