ABSTRACT

The first theoretical description of energy transfer from a donor to an acceptor molecule by dipole-dipole interaction was published by Theodor Förster in 1948 [24]. Therefore, this type of energy transfer is often referred to as Förster resonant energy transfer (FRET). Radiative energy transfer, corresponding to the emission of a photon by the donor and its re-absorption by the acceptor, and tunnelling, due to the spatial overlap of donor and acceptor wave-functions [25,26], are the other important energy transfer mechanisms aside from FRET. Photon re-absorption can be easily distinguished from the two other energy transfer mechanisms as

it does not cause a decrease of the donor photoluminescence lifetime [27,28]. Tunnelling also leads to a shortening of the donor lifetime, but distance and/or temperature dependent measurements can be carried out to differentiate between tunnelling and FRET [6,7,9,29]. Furthermore, tunnelling is a short-range energy transfer mechanism, usually only important on sub-nanometre scales [25,26], and can normally be excluded as the origin for energy transfer for donor-acceptor separations larger than 2 nm. 4.2.1 FRET in QD SystemsThe FRET equations discussed here have been obtained from FRET theories developed for molecules that can be represented as point dipoles. Due to the large size of the QDs, the inhomogeneous broadening of the QD ensembles as well as the different nature of the (non-)radiative processes in molecules and QDs, it has not been clear for a long time if these equations could also be applied directly to QD systems [8,30]. However, recent theoretical work on FRET between QDs suggests that it is valid to use the original FRET rate equations and the dipole approximation to describe FRET, and in particular the FRET rate, for spherical, direct-gap semiconductor nanocrystals [4,8,31-33]. Furthermore, some of the experimental work on FRET in QD systems presented in this chapter shows that these FRET equations can be used to describe the optical properties of QD FRET systems [5,6,34-37]. In the following, the most important theoretical equations describing Förster resonant energy transfer-represented by the energy transfer rate (kFRET), the energy transfer efficiency (EFRET) and the characteristic distance: the Förster radius (R0)—and its effect on the optical properties of the donor and acceptor species are summarized. 4.2.2 General ObservationsAs depicted schematically in Fig. 4.1a, a luminescent species “D” (the energy donor) can lose its excitation energy via a radiative channel with decay rate kr or non-radiative channels, giving rise to the decay rate knr. As shown in Eq. (4.1), these two processes determine the photoluminescence lifetime tD of the donor: D r nr1= +k kt . (4.1)

Figure 4.1 Schematic of donor decay paths. (a) Decay paths for an excited donor (D) via radiative and non-radiative channels with rates kr and knr respectively. (b) In the presence of a suitable acceptor (A) another decay channel via energy transfer, with an associated decay rate kFRET, can lead to the de-excitation of the donor.If a suitable acceptor “A” is present in proximity to the donor, as schematically presented in Fig. 4.1b, the excitation energy of the donor can also be transferred to the acceptor via energy transfer mediated by dipole-dipole interactions with a rate kFRET. Adding another possible decay channel increases the probability for non-radiative donor de-excitation and decreases the donor emission intensity as well as its photoluminescence lifetime tDA, as described in Eq. (4.2): DA –1r nr FRET D FRET1 1= =+ + +k k k kt t . (4.2)Energy transfer from a donor to an acceptor via FRET always results in a decrease of the donor lifetime and also has effects on the emission and photoluminescence lifetime of a luminescent acceptor. Typically, in a QD FRET system, both the donor and the acceptor are excited at the same time due to the broad absorption of the QDs. Energy transfer, in general, leads to an enhanced acceptor emission and the additional pumping via the donor, over timescales longer than the excitation pulse, and can give rise to a

prolonged acceptor emission. Furthermore, energy transfer can even lead to an increase in the overall acceptor emission intensity after the excitation pulse occurred, leading to a so-called rise-time in the time-trace of the acceptor photoluminescence [9,28,38].For broad donor and acceptor ensembles, arising mainly from the size distribution in the case for colloidal QDs, energy transfer from a sub-ensemble of donors to a particular sub-ensemble of acceptors might be more efficient than from others. Therefore, certain parts of the donor photoluminescence spectrum-corresponding to the emission from this sub-ensemble-might experience a stronger quenching than others; or a stronger enhancement in the case of the acceptors. This can lead to spectral changes in the donor and acceptor ensemble emission spectra, such as shifted peak emission wavelengths or narrowing/broadening of the emission features.One of the prerequisites for FRET, is the presence of resonant donor and acceptor states between which the energy can be transferred. For this reason the spectral overlap J, as given in Eq. (4.3), is an important parameter in FRET. It describes the overlap of the donor emission ID(l), using the area-normalized spectrum ∧ID(l) in Eq. (4.3), and the acceptor absorption spectra, represented by the wavelength dependent extinction coefficient eA(l). 4D A0 ˆ= ( ) ( )J I d l e l l l (4.3)Including eA(l) in units of M-1 . cm-1 and l in nm, J is calculated in nm4/M. cm with Eq. (4.3). Knowing the spectral overlap Jand the donor quantum yield QD, the Förster radius R0, the characteristic interaction distance of FRET, can be calculated with Eq. (4.4). The result is given in nm, if J is included in the calculation in nm4/M. cm. 1/62 D0 4= 0.211 QR Jn k     (4.4)The orientation factor k2 is typically 2/3 for randomly oriented donor and acceptor dipoles and n represents the refractive index of the surrounding medium. Typical values of R0 range between 1 and 10 nm [27,28].