In previous chapters, the energetic landscape for excitons has been treated as homogeneous. The assumption, however, is over simplistic when going forward and
inspecting the temperature dependence of LD. A more realistic picture accounts for
energetic disorder. This type of disorder has many origins. For instance, the local conformation of neighboring molecules may change the local dielectric environment. A change in the local dielectric environment then alters the energy of an exciton residing on that specific conjugation center. Over an entire film, many different site energies may then be realized, resulting in a continuous distribution. A common distribution function describing disorder in organic thin-films is the Gaussian distribution function (g[E]) written as:
Eq. 9.1 𝑔[𝐸] = 1
𝜎√2𝜋exp [− 𝐸 2𝜎2]
where σ is the disorder parameter, σ2 is the variance, and E is the energy. In combination with the molecular density of the film (ND), the density of states (ρ) is expressed as:
Eq. 9.2 𝜌[𝐸] = 𝑁𝐷𝑔[𝐸]
Importantly, the disorder being represented with this function is assumed to be static where the fluctuation in site energies is very long compared to the rate of energy transfer and diffusion.
Other options exist when selecting a distribution function [48,157]. If only the tail of a distribution needs to be considered, an exponential distribution function may be appropriate. If a large number of outliers need to be considered, a Lorentz distribution function may be appropriate. In this chapter, the Gaussian distribution function will be utilized.
Consider the generation of an exciton randomly within the density of states (DOS). At higher temperatures, excitons quickly relax through the DOS and establish a
because nearly all neighboring molecules are energetically accessible. Continuous hopping at equilibrium then takes place since there is sufficient thermal energy for an exciton to access a majority of neighboring sites. In this activated regime, the LD is
temperature dependent and sensitive to the degree of energetic disorder (σ) present in the system. At very low temperatures, relaxation through the DOS becomes frustrated since thermally-activated, upward hops become disallowed, reducing the number of energetically accessible, neighboring sites. The distance an exciton can travel is therefore limited by either trapping in the tail of the DOS or recombination owing to a finite τ [158]. Figure 9.1 schematically describes the activated and non-activated regimes by showing the reduction in relaxation pathways when thermal energy in the system is reduced. While both regimes exhibit sub-diffusive motion, especially at lower temperatures, an effective LD can be derived that is representative of exciton
transport [154,159]. These regimes have been used to qualitatively assess measurements of LD versus temperature; however direct quantitative fits have not been explored in
detail [160,161].
Figure 9.1 Schematic representation of the number of energetically and spatially accessible neighboring sites for an exciton near the tail in the density of states (DOS) in the temperature activated (a) and non-activated (b) regimes.
E ne rg y Distance Accessible in distance Accessible in energy
(a) Activated Regime
DOS E ne rg y Distance (b) Non-Activated Regime
9.1.1. Theoretical treatment for activated hopping
This section summarizes a previous derivation for the LD representative of
activated hopping described by Athanasopoulos et. al. [154]. In a similar manner to Miller-Abrahams hopping, the energy transfer rate Γ(E,E',r), here combined with a Förster-type mechanism, between a molecule or conjugation center with energy E to a target molecule or conjugation center of energy E' separated by distance d can be written as: Eq. 9.3 Γ[𝐸′, 𝐸, 𝑟] =1 𝜏( 𝑅0 𝑑) 6 {𝑒𝑥𝑝 [− 𝐸′−E 𝑘𝐵𝑇] , E ′− E > 0 1, E′− E < 0
where R0 is the Förster radius (Eq. 2.24), τ is the exciton lifetime, kB is the Boltzmann
constant, and T is the temperature. In this formalism, upward hops are weighted by a Boltzmann factor. Importantly, R0 is assumed to be temperature independent.
Accordingly, increases in τ do not affect LD as τ confers the time dependence to the
overall transfer rate. To simply the solution, the hopping rate can be rewritten as [154]: Eq. 9.4 Γ =1𝜏exp [−𝑢] where: Eq. 9.5 𝑢[𝐸, 𝐸′, 𝑟] = 6 ln [𝑟 𝑅0] + 𝜂[𝐸′−𝐸] 𝑘𝐵𝑇
Here, η is the Heaviside function. The mean-squared displacement is expressed as:
Eq. 9.6 〈𝑟2[𝐸]〉 = 4𝜋 ∫ 𝑟2𝑟2𝑑𝑟 ∫ 𝑔[𝐸′]𝑑𝐸′ 𝐸+𝑘𝑇(𝑢−6 ln[𝑟 𝑅0]] −∞ 𝑅0 exp[𝑢6] 0 4𝜋 ∫ 𝑟2𝑑𝑟 ∫𝐸+𝑘𝑇(𝑢−6 ln[ 𝑔[𝐸′]𝑑𝐸′ 𝑟 𝑅0]] −∞ 𝑅0 exp[𝑢6] 0 ≅ 𝑅02exp [〈𝑢[𝐸]〉3 ]
where 〈𝑢[𝐸]〉 is an average hopping parameter. The equilibrium diffusion coefficient is obtained by multiplying the mean-squared displacement with the hopping rate averaged over E. The LD results as:
Eq. 9.7 𝐿𝐷 = √𝐷𝜏 = 𝑅0(
∫−∞∞ 𝑔[𝐸] exp[−𝑘𝑇𝐸] exp[−23〈𝑢[𝐸]〉]𝑑𝐸 ∫−∞∞ 𝑔[𝐸] exp[−𝑘𝑇𝐸]𝑑𝐸 )
1 2
As a result, the LD in the activated regime will be sensitive to the size and shape of the
DOS. Increases to ND will increase LD whereas increases to σ will decrease LD.
9.1.2. Theoretical treatment for non-activated hopping
This section summarizes a previous derivation for the LD representative of non-
activated exciton hopping described by Arkhipov et. al [162]. In this regime, excitons can only make downward hops in energy. The available part of the density of states (N[E]) is then:
Eq. 9.8 𝑁[𝐸] = ∫ 𝑔[𝐸𝐸 ′]𝑑𝐸′ −∞
A non-activated exciton energy distribution function can be derived as: Eq. 9.9 𝑓[𝐸, 𝑟, 𝑡] = 𝐴[𝑡]𝑟2𝜌[𝐸]𝑁[𝐸]exp [−4𝜋𝑟3 3 𝑁[𝐸] − 𝑡 𝜏( 𝑅0 𝑟) 6 ] where t is the time until the exciton decays and A[t] is a normalization constant:
Eq. 9.10 𝐴[𝑡] = exp [−𝑡𝜏] {∫ 𝑟2𝑑𝑟 ∫ 𝜌[𝐸]𝑁[𝐸] exp [−4𝜋𝑟3 3 𝑁[𝐸] − ∞ −∞ ∞ 0 𝑡 𝜏( 𝑅0 𝑟) 6 ] 𝑑𝐸}−1
The LD is calculated from the mean-squared displacement, 〈𝑟2〉, and the total
Eq. 9.11 〈𝑟2〉 =1 𝜏∫ 𝑑𝑡 𝜏 0 ∫ 𝑟2𝑑𝑟 ∞ 0 ∫ 𝑓[𝐸, 𝑟, 𝑡]𝑑𝐸 ∞ −∞
The n[t=0] can be determined by integrating the total jump rate ΓT from t=0 to infinity
where: Eq. 9.12 Γ𝑇[𝑡] =1𝜏∫ 𝑑𝑟0∞ ∫ (𝑅𝑟0) 6 𝑓[𝐸, 𝑟, 𝑡]𝑑𝐸 ∞ −∞ Eq. 9.13 𝐿𝐷 = √〈𝑟2〉𝑛[𝑡 = 0]
Importantly, the non-activated LD is not dependent on the degree of disorder (σ).
Increases to σ result in slower, yet longer, hops. Rather, it is dependent on the R0 and ND.
9.1.3. Extended Boltzmann approximation
In Eq. 9.3, the site energy and temperature dependence of the exciton transfer rate within the DOS is approximated by a Boltzmann distribution. The exact energy (wavelength) dependence, however, can be expressed via F0 as [33]:
Eq. 9.14 𝐹06 = 9𝜅2
128𝜋5𝑛4∫ 𝜆4𝐹𝐷[𝜆]𝜎𝐴[𝜆]𝑑𝜆
where κ2 is the dipole orientation factor, λ is the wavelength, FD is the area-normalized
photoluminescence spectrum, σA is the absorption cross-section, and n is the index of
refraction at the wavelength of maximum spectral overlap. Note that F0 is nearly
identical to R0 (Eq. 2.24) except the photoluminescence efficiency (ηPL) has been
removed. Assuming that energetic disorder leads to similar shifts in FD and σA, upward
hops become less favorable owing to a reduced spectral overlap integral—in agreement with Eq. 9.3. Downward hops will also become more favorable owing to an increase in the spectral overlap integral—behavior not captured in Eq. 9.3. Implementing this exact relationship is difficult, however, without precise knowledge of the purely
If the Boltzmann distribution is extended over the entire energy spectrum, the resulting exciton landscape correctly favors downhill exciton energy transfer and can be used to approximate Eq. 9.14. It should be noted that this approximation remains consistent with the notion of spectral diffusion where a time-dependent shift for the peak photoluminescence wavelength immediately after excitation characterizes the energetic evolution of excitons through the DOS [155,158,164,165].