Effect of exciton diffusion on the triplet-triplet annihilation rate in organic semiconductor host-guest systems

We study the contribution of triplet exciton diffusion to the efficiency loss resulting from Förster-type triplet-triplet annihilation (TTA) in organic phosphorescent semiconductor host-guest systems, using kinetic Monte Carlo (KMC) simulations. Our study focusses on diffusion due to Förster-type guest-guest transfer, but includes also a comparison with simulation results for the case of Dexter-type guest-guest transfer. The simulations are carried out for a wide range of Förster radii, and for guest concentrations up to 100 mol%, with the purpose to support analyses of time-resolved photoluminescence experiments probing TTA. We find that the relative contribution of diffusion to the TTA-induced efficiency loss may be deduced quite accurately from a quantitative experimental measure for the shape of the time-dependent photoluminescence intensity, the so-called $r$ ratio. For small guest concentrations and Förster radii that are most relevant to organic light-emitting diodes (OLEDs), the diffusion contribution is in general quite small. Under these weak-diffusion conditions, the absolute diffusion contribution to the TTA-induced efficiency loss can be understood quantitatively using a capture radius formalism. The effective guest-guest diffusion coefficient that follows from the TTA simulations, using the capture radius formalism, agrees well with the diffusion coefficient that follows from direct KMC diffusion simulations. The simulations reveal that the diffusion coefficient is strongly affected by the randomness of the distribution of guest molecule locations.

Figure 1

KMC simulation results of Förster-type TTA in the absence of diffusion. (a) Dependence of the total PL efficiency, ${\eta }_{\mathrm{PL}}$, on $R_{\mathrm{F},\mathrm{TT}}$, for two values of $R_{\max }$ (symbols), for $T_0=1\times {10}^{24}{\mathrm{m}}^{-3}$. The curves give the results for $b=$ 0.65 (blue) and 0.72 (red). (b) $T_0$ dependence of $k_{\mathrm{TT},1}$ and $k_{\mathrm{TT},2}$ (dashed and full curves) as obtained for various values of $R_{\mathrm{F},\mathrm{TT}}$ from Eq. (5) with $b$ = 0.65, and KMC simulation data for $T_0=1\times {10}^{24}{\mathrm{m}}^{-3}$. (c). Dependence of the $r$ ratio on $R_{\mathrm{F},\mathrm{TT}}$ for $T_0=1\times {10}^{24}{\mathrm{m}}^{-3}$, as obtained from KMC simulations (see also Table S1 in Ref. [19]) and from Eq. (5) with $b=$ 0.65.

Figure 2

KMC simulation results of the rate coefficient $k_{\mathrm{TT},2}$ describing the PL efficiency loss due to Förster-type TTA combined with Förster-type diffusion. [(a) and (b)] Dependence on the guest concentration and $R_{\mathrm{F},\mathrm{diff}}$, for $R_{\mathrm{F},\mathrm{TT}}=$ 3.5 nm and $R_{\mathrm{F},\mathrm{TT}}=$ 5 nm, respectively. (c) Dependence on $R_{\mathrm{F},\mathrm{TT}}$ for $R_{\mathrm{F},\mathrm{diff}}=$ 3.5 nm, for (from bottom to top) the following conditions: no diffusion, $c_{\mathrm{g}}=$ 1, 2, 4, 8, 16, 32, 64, and (only for $R_{\mathrm{F},\mathrm{TT}}=$ 3.5, 4, and 4.5 nm) 100 mol%. The dashed strong-diffusion line gives $k_{\mathrm{TT},\mathrm{sd}}$ (see Sec. 2) and the dashed no-diffusion line gives $k_{\mathrm{TT},2,\mathrm{nd}}$ as obtained from Eq. (7) with (for simplicity) $p_1=4{\pi }^2/3$ and $p_2=0$.

Figure 3

Dependence of the rate coefficient $k_{\mathrm{TT},2}$ on the guest concentration for various values of $R_{\mathrm{F},\mathrm{TT}}$ and for $R_{\mathrm{F},\mathrm{diff}}=$ 2 (a), 2.5 (b), 3 (c) and 3.5 nm (d), from KMC simulations for $T_0={10}^{24}{\mathrm{m}}^{-3}$ and $\tau =$ 1 $\mu \mathrm{s}$. The lines give a linear interpolation between successive data points.

Figure 4

Sensitivity of $k_{\mathrm{TT},2}$ to the guest concentration as a function of $R_{\mathrm{F},\mathrm{TT}}$, obtained from KMC simulations for various values of $R_{\mathrm{F},\mathrm{diff}}$ (symbols). The dashed lines give a fit using Eq. (8). The sensitivity vanishes for $R_{\mathrm{F},\mathrm{TT}}=R_0=1.8$ nm.

Figure 5

Dependence of the $r$ ratio on the guest concentration, for various values of $R_{\mathrm{F},\mathrm{diff}}$ and for $R_{\mathrm{TT},2}=$ 3 (a) and 5 nm (b). The full curves (and their dashed extrapolation) provide a guide to the eye. The long-dashed (short-dashed) horizontal lines give the $r$ ratio in the absence of diffusion, obtained from simulations with $R_{\max }=2R_{\mathrm{TT},2}\left(4R_{\mathrm{TT},2}\right)$. The numerical $2\sigma$ accuracy is approximately $\pm 0.1$.

Figure 6

Relative enhancement of the $r$ ratio with respect to the value in the strong-diffusion, normalized to the enhancement for the no-diffusion limit, $(r-1)/(r_{\mathrm{nd}}-1)$, as a function of the enhancement of $k_{\mathrm{TT},2}$ due to diffusion, $k_{\mathrm{TT},2}/k_{\mathrm{TT},2,\mathrm{nd}}$, as obtained from KMC simulations for all systems included in this study. The dashed curve gives an empirical fit [see Eq. (11)].

Figure 7

Relative emission profiles (symmetrized) as obtained from KMC simulations within which absorption at random guest sites in the $x=0$ plane and subsequent Förster-type exciton diffusion with $R_{\mathrm{F},\mathrm{diff}}$ = 1.5 nm takes place, for host-guest systems with guest concentrations of 1, 15, and 50 mol%.

Figure 8

KMC simulation results of the exciton diffusion coefficient in host-guest systems, resulting from Förster-type transfer with $\tau =1\mu \mathrm{s}$. (a) Guest concentration dependence for various values of the Förster radius. The dashed lines indicate the result expected from Eq. (A2). (b) Dependence on the Förster radius, for various values of the guest concentration. The full red curves are a guide-to-the-eye.