Förster-type triplet-polaron quenching in disordered organic semiconductors

Triplet-polaron quenching (TPQ) is a major cause of the efficiency loss at large current densities in phosphorescent organic light-emitting diodes. The nature of the interaction process is presently not well understood. In this paper, we study TPQ due to Förster-type triplet-polaron interactions in energetically disordered organic semiconductors with a Gaussian polaron density of states. A continuum theory, which neglects the spatial inhomogeneity and energetic disorder, is from a kinetic Monte Carlo approach shown to correctly predict that the effective steady-state TPQ rate coefficient $k_{\text{TPQ,eff}}$ depends only sensitively on the polaron diffusion in a rather narrow range of diffusion coefficients. However, in this regime, significant discrepancies between the two approaches are found, in particular for realistic values of the TPQ-Förster radius, around 3 nm, and for systems with strong energetic disorder. Both approaches show that $k_{\text{TPQ,eff}}$ is not constant but can depend on the polaron density and the electric field. Various methods for deducing the TPQ mechanism from experiment are discussed, including an approach which utilizes the shape of the time-dependent photoluminescence after pulsed illumination.

Figure 1

(a) Shape of the depleted region in the infinite-time limit for materials with $R_{\mathrm{F}}=3\mathrm{nm}, R_0=1\mathrm{nm}$, and $\tau =1\mu \mathrm{s}$ for values of the hole diffusion coefficient in the range $D=1\times {10}^{-15}$–${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. (b) Time dependence of the shape $u(R,t)$ of the depleted zone for the case $D=1\times {10}^{-13}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. (c) Shape of the depleted zone at $t=\tau$ for various values of $D$.

Figure 2

Time dependence of the TPQ rate coefficient for materials with $R_{\mathrm{F}}=3\mathrm{nm}, R_0=1\mathrm{nm}$, and $\tau =1\mu \mathrm{s}$ for values of the hole diffusion coefficient in the range $D=1\times {10}^{-14}$–${10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. The symbols give the numerically exact results, and the curves show the approximation given by Eqs. (27) and (28). The dashed lines give the rate coefficients in the infinite-time limit, $k_{\mathrm{TPQ},\inf }$.

Figure 3

Dependence of the effective steady-state rate coefficient for Förster-type TPQ on the Förster radius, for materials with $\tau =1\mu \mathrm{s}$ and $R_0=1\mathrm{nm}$, and for values of the hole diffusion coefficient in the range $D=1\times {10}^{-12}$–${10}^{-7}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. The full curves give results obtained in the limit of a small polaron density. The dashed (dashed-dotted) curves give results for hole densities $n_{\text{h}}={10}^{24}\left({10}^{25}\right){\text{m}}^{-3}$, with $D={10}^{-10}{\text{m}}^2/\text{s}$. The upper and lower lines give the values in the strong-diffusion and no-diffusion limits, respectively.

Figure 4

Results of KMC simulations of Förster-type TPQ in materials with $a=1\mathrm{nm}$ and $c_{\mathrm{h}}={10}^{-3}$. (a) Dependence of $k_{\mathrm{TPQ},\mathrm{eff}}$ on the Förster radius for $\sigma /\left(k_{\mathrm{B}}T\right)=2$, 4, and 6, with $\tau ={\tau }_0=1.38\mu \mathrm{s}$. (b) Dependence of the ratio of the fraction of quenchings and the fraction of radiative decay processes on the Förster radius for materials with $\sigma /\left(k_{\mathrm{B}}T\right)=4$, with variable ratios $\tau /{\tau }_0$. The upper and lower full lines give the expected values in the strong-diffusion and no-diffusion limits, as obtained from Eq. (14) and as obtained from Eq. (12) with $n_{\mathrm{h}}=0$, respectively.

Figure 5

Results of KMC simulations of Förster-type TPQ in materials with $\tau =1.38\mu \mathrm{s}$ and $a=1\mathrm{nm}$, as a function of the hole concentration for various values of the disorder parameter, for (a) $R_{\mathrm{F}}=3\mathrm{nm}$ and (b) $R_{\mathrm{F}}=6\mathrm{nm}$. The symbols give the KMC simulation results. The dashed curves give the rate coefficients that follow from the continuum theory developed in Sec. 2b. The full curves show an empirical fit to the simulation data that is obtained using that theory with reduced values of the diffusion coefficient. For panel (a) the reduction factors for the cases $\sigma /\left(k_{\mathrm{B}}T\right)=2$, 3, 4, and 6 are 0.30, 0.26, 0.18, and 0.20, respectively. For panel (b), these factors are 0.8, 0.8, 0.6, and 0.6, respectively. The upper and lower full lines give the expected values in the strong-diffusion and no-diffusion limits.

Figure 6

Enhancement of the effective steady-state TPQ rate coefficient with respect to the value in the no-diffusion limit for systems with $R_{\mathrm{F}}=3\mathrm{nm}, a=1\mathrm{nm}$, and $\sigma /\left(k_{\mathrm{B}}T\right)=2$, 4, and 6 as a function of the hole diffusion length. The symbols, connected by curves that serve as a guide to the eye, give the KMC simulation results. The dashed curve gives the rate coefficient as expected from the continuum model developed in Sec. 2b. The uncertainty of the simulation points is smaller than or equal to the symbol size, unless indicated by error bars.

Figure 7

Dependence of the effective TPQ rate coefficient on the electric field $F$ for systems with $\sigma /\left(k_{\mathrm{B}}T\right)=4, c_{\mathrm{h}}={10}^{-3}, a=1\mathrm{nm}$, and $\tau =1.38\mu \mathrm{s}$ for various values of $R_{\mathrm{F}}$ as obtained from KMC simulations (symbols). The horizontal lines provide a reference showing that $k_{\text{TPQ,eff}}$ depends only for large values of $R_{\mathrm{F}}$ slightly on $F$. The data labeled NN give the results obtained from simulations assuming immediate nearest-neighbor-type TPQ; the curve is a guide to the eye.

Figure 8

Dependence of the $r$-ratio, defined in Eq. (32), on the hole diffusion length, as obtained from the continuum theory, for systems with $R_{\mathrm{F}}=3\mathrm{nm}$ (full curves) and 6 nm (dashed curves), $a=1\mathrm{nm}$ and for various values of the hole concentration. The three curves for $R_{\mathrm{F}}=3\mathrm{nm}$ correspond to the same hole concentrations as used for $R_{\mathrm{F}}=6\mathrm{nm}$, and show also an increase of $r$ with increasing $c_{\mathrm{h}}$. The symbols indicate for systems with $\sigma /\left(k_{\mathrm{B}}T\right)=2$ (spheres, cyan), 3 (triangles, magenta), 4 (diamonds, blue) and 6 (squares, red) the expected values for the case $\tau ={\tau }_0$ using the diffusion coefficients obtained from the KMC simulations. The full (open) symbols correspond to $R_{\mathrm{F}}=3\left(6\right)\mathrm{nm}$. The inset shows the actual time-dependent photoluminescence expected from the model (red curve) for a system with $R_{\mathrm{F}}=6\mathrm{nm}, \tau ={\tau }_0, c_{\mathrm{h}}={10}^{-3}, a=1\mathrm{nm}$ and no diffusion, and a reference line (dashed, black) that indicates the expected photoluminescence intensity for a system with the same total PL efficiency, in the strong-diffusion limit. At the $t_{1/2}$ times, the cumulative PL intensities have reached half the final value.