Exciton diffusion coefficient

The theory of isotropic three-dimensional diffusion allows yss to be expressed by a product of the exciton diffusion coefficient Ds and the effective annihilation capture radius Rs [189],... 

As can be seen from eqn (14.8), the calculation of the tensor reduces to the calculation of two-particle correlation functions. The lack of sufficiently detailed data on the exciton band structure and the exciton-phonon coupling constants considerably complicates the accurate calculation of the two-particle correlation functions and the exciton diffusion coefficients. However, the temperature dependence of this coefficient differs significantly for coherent and incoherent excitons (see below). Therefore, studying the temperature dependence of diffusion has always been an important tool to analyze the character of the energy transfer in molecular crystals. In the remainder of this chapter, we will focus on the main characteristics of the diffusion constant and its temperature dependence... 

Here, th is the lifetime of the host excitons without traps. We furthermore make the simplifying assumption that every exciton is captured when it reaches a trap and that the excitation density is constant with time and remains homogeneous. If, in addition, the mean free path of the exciton is smaller than the capture radius R of the traps for excitons, and D is the exciton diffusion coefficient, then we obtain... 

The exciton migration within aggregates of cyanine dyes and the possibility of oxygen diffusion into the porous dye film result in a bulk generation of photocurrent [80]. Photoholes produced due to the oxidation of excitons by molecular oxygen diffuse to the back contact. The diffusion coefficient of charge carriers in dye layer (Dc) can be evaluated from the potential-step chronoamperometric measurements in the indifferent electrolyte. Considering dye film as a thin-layer cell, the current vs. time dependence can be described as follows [81] ... 

If an exciton hops about, starting from some initial site and ends up at another site at the time of its disappearance for one reason or another, the linear distance between these sites is called the diffusion length, lA. The diffusion length is related to the diffusion coefficient D through the exciton lifetime t ... 

Material Intermolecular distance3 c (nm) Exciton spin multiplicity Lifetime T (s) Diffusion length lA (nm) Diffusion coefficient D (cm2/s) Hopping time th (s) Total distance covered l (nm) l/l a... 

The relatively low values of yss and Ds in quasi-amorphous solids might be underlain by disorder (see Sec. 2.4.3) and/or a contribution of triplet excitons in quenching of fluorescent singlets (cf. Sec. 2.5.1.2). The diffusion coefficient of triplets is expected to be lower than of singlets since both energy donor and acceptor transitions are disallowed. A low value of yss has been found for the triplet-triplet annihilation rate constant from biexcitonic quenching... 

Mulder (1968) described photogeneration in anthracene by a surface-enhanced exciton model. Figures 27 and 28 show the photogeneration efficiency and singlet state absorption coefficient. According to Mulder, when the exciton diffusion length l and the absorption depth, //a, are much less than the thickness, the photogeneration efficiency is... 

If excitons are assumed to be particles that diffuse with a diffusion coefficient D in three-dimensional space and that annihilate one another if they come within the distance R of each other, the rate constant for exciton-exciton annihilation is given (23) by y = SuDR, If R is assumed to be 20 A, D is about O.OI cm /s. This value is surprisingly large, because the diffusion coefficient for singlet excitons in anthracene single crystals is thought to be (24) about 10 cm /s. The reported values of the exciton-exciton... 

Scheme 2. In this photophysical scheme it was proposed that M, and D interact by the generally accepted exciton diffusion mechanism. M was considered to be an isolated naphthalene chromophore wliich can transfer energy into M with a transfer rate characterized by the rate coefficient kt- Reverse transfer from M to M was considered unimportant for the following reason. Exciton diffusion is expected to be very efficient within sequences of naphthalene chrcmiophoies within the chain comprising the M sites. In view of the reduced lifetime of M relative to M and of the delocalised nature of the energy within extended chromc hore sequences which increases the effective sqiaiation of M and M, M to M enei transfer by Foster or Dexter mechanisms is dimini ed relative to the to Mf process.

Furthermore, the diffusion coefficients for triplet exciton migration extracted from this three-dimensional intramolecular model were nearly the same as those obtained using the conventional kinetic equation (i.e. eq 1). The hopping frequencies for triplet exciton migration in PVCA for these three models are summarized in Table 111(20). Neither the electron exchange mechanism(21) nor the Forster... 

The description of excitation motion outlined in the previous sections assumes completely incoherent nearest neighbor hopping. This was treated in detail because it is the case of widest applicability especially with the materials of interest discussed in the final section. However, it should be noted that in some cases excitons can move coherently over several lattice spacings before being scattered i). For this case the diffusion coefficient is expressed in terms of the group velocity of the exciton v and the time between scattering events r. 

Therefore the migration of excitons slows down when they reach the low-energy sites where they find fewer sites with lower energy in its neighborhood. Due to such dispersive migration, the exciton diffusion cannot be described using a constant diffusion coefficient, but a time-dependent one. 

We also mention Ref. (38), where EPR spectra of naphthalene doped by deuterated naphthalene were examined. In this work, using the EPR linewidth the frequency of transitions of triplet excitations from one molecule to another inside the naphthalene unit cell was estimated and from them the diffusion coefficient D for triplet excitons, obtaining the value D = 5 10 4 cm2/s. This value is in agreement with results obtained in other experiments. 

A useful measure for exciton migration is the diffusion length L = (Dto)1/2. Experimental data show that for Frenkel excitons in molecular crystals at room temperature the diffusion coefficient D 10 3 cm2/s and the lifetime of singlet excitons to 10 8s. This gives a typical diffusion length L 10 6cm (for anthracene crystals L 5 10 6cm). 

The motion of the exciton wavepacket causes the transport of energy. In order to find the appropriate energy diffusion coefficient we must estimate the mean free path and the mean free time of the wavepackets. This situation is quite similar to that of phonon heat conductivity (see, for example, (12)). 

In analogy to the mobility of electrons and holes in crystals, the diffusion coefficient for coherent excitons is determined by the relaxation time r. According to Frolich (13), we have... 

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