Electron-hole recombination in disordered organic semiconductors: Validity of the Langevin formula

Accurate modeling of electron-hole recombination in organic light-emitting diodes (OLEDs) is essential for developing a complete description of their functioning. Traditionally, the recombination rate is described by the Langevin formula, with a proportionality factor equal to the sum of the electron and hole mobilities. In the disordered organic semiconductors used in OLEDs these mobilities have been shown to depend strongly on the carrier densities and on the electric field. Moreover, the energetic disorder leads to percolating pathways for the electron and hole currents, which may or may not be correlated. To answer the question whether the Langevin formula is still valid under such circumstances we perform Monte Carlo simulations of the recombination rate for Gaussian energetic disorder. We vary the disorder energy, the temperature, the densities, and mobility ratio of electrons and holes, the electric field, and the type of correlation between the electron and hole energies. We find that at zero electric field the Langevin formula is surprisingly well obeyed, provided that a change in the charge-carrier mobilities due to the presence of charge carriers of the opposite type is taken into account. Deviations from the Langevin formula at finite electric field are small at the field scale relevant for OLED modeling.

Figure 1

(Color online) The different types of correlation between on-site electron and hole energies considered in this work, reflected in the energies for the lowest-unoccupied molecular orbital and highest-unoccupied molecular orbital. Left/right: correlated/anticorrelated electron and hole energies.

Figure 2

(Color online) The two methods of calculating mobilities in this work. In the unipolar method we consider the presence of only one type of carrier and calculate its mobility. In the bipolar method we consider the presence of both types of carriers (open blue circles: electrons and solid red circles: holes) and calculate both their mobilities. In the bipolar method the mobilities of one carrier type are smaller than in the unipolar method because of the additional Coulomb interactions with the other carrier type. The figure indicates the typical situation in which almost all carriers are trapped in energetically deep-lying states in the Gaussian density of states, with only a few carriers that are mobile and contribute to the conduction.

Figure 3

(Color online) [(a), (c), and (e)]: Zero-field recombination rate $R$ relative to the Langevin recombination rate $R_{\text{Lan}}$ as a function of disorder energy $\sigma$, at temperature $T=300\text{K}$ and three different electron and hole densities $n_{\text{e}}$ and $n_{\text{h}}$. Red circles/blue triangles: correlated/anticorrelated electron and hole energies. Solid/dashed lines: Langevin recombination rate calculated with bipolar/unipolar mobilities. [(b), (d), and (f)]: Corresponding unipolar (black squares) and bipolar mobilities.

Figure 4

(Color online) [(a), (c), and (e)]: Zero-field recombination rate $R$ relative to the Langevin recombination rate $R_{\text{Lan}}$ as a function of relative hopping frequency ratio ${\nu }_{0,\text{e}}/{\nu }_{0,\text{h}}$ of electrons and holes, at temperature $T=300\text{K}$, electron and hole densities $n_{\text{e}}=n_{\text{h}}={10}^{-4}/a^3$, and three different disorder energies. [(b), (d), and (f)]: Corresponding electron and hole mobilities. Symbols and lines as in Fig. 3.

Figure 5

(Color online) [(a), (c), and (e)]: Zero-field recombination rate $R$ relative to the Langevin recombination rate $R_{\text{Lan}}$, calculated for correlated electron and hole energies, as a function of temperature $T$, at electron and hole densities $n_{\text{e}}=n_{\text{h}}={10}^{-4}/a^3$ and three different disorder energies. Red circles/black squares: Langevin recombination rate calculated with bipolar/unipolar mobilities. [(b), (d), and (f)]: Corresponding mobilities.

Figure 6

(Color online) (a): Zero-field recombination rate $R$ relative to the Langevin recombination rate $R_{\text{Lan}}$ as a function of disorder strength $\sigma$, at temperature $T=300\text{K}$, electron and hole densities $n_{\text{e}}=n_{\text{h}}={10}^{-4}/a^3$, and correlated electron and hole energies, calculated for different reintroduction procedures of electrons and holes. Red circles/green triangles: equilibrium/random reintroduction. Solid/dashed lines: Langevin recombination rate calculated with bipolar/unipolar mobilities. (b): Corresponding unipolar (black squares) and bipolar mobilities.

Figure 7

(Color online) [(a), (c), and (e)]: Recombination rate $R$ relative to the Langevin recombination rate $R_{\text{Lan},\text{bi}}$, calculated with bipolar mobilities and correlated electron and hole energies, as a function of electric field $F$, at temperature $T=300\text{K}$, at three different electron and hole densities, and three different disorder energies. [(b), (d), and (f)]: Corresponding bipolar mobilities.

Figure 8

Bars: distribution of the frequency of recombination events at sites as a function of the random part of the energy of the sites, at disorder energy $\sigma =100\text{meV}$, temperature $T=300\text{K}$, zero field, and densities $n_{\text{e}}=n_{\text{h}}={10}^{-4}/a^3$ for (a) correlated and (b) anticorrelated electron and hole energies. Lines: DOOS of electrons and holes and Gaussian DOS. One disorder configuration was used and the total number of recombination events was 67133 and 66817 for the correlated and anticorrelated case, respectively.