Origin of reduced polaron recombination in organic semiconductor devices

We propose a model to explain the reduced bimolecular recombination rate found in state-of-the-art bulk heterojunction solar cells. When compared to the Langevin recombination, the experimentally observed rate is one to four orders of magnitude lower but gets closer to the Langevin case for low temperatures. Our model considers the organic solar cell as device with carrier-concentration gradients, which form due to the electrode/blend/electrode device configuration. The resulting electron concentration under working conditions of a solar cell is higher at the cathode than at the anode and vice versa for holes. Therefore, the spatially dependent bimolecular recombination rate, proportional to the local product of electron and hole concentrations, is much lower as compared to the calculation of the recombination rate based on the extracted and thus averaged charge-carrier concentrations. We consider also the temperature dependence of the recombination rate, which can be described with our model.

Figure 1

(Color online) Recombination mechanisms in low mobility materials, all being based on the Langevin recombination shown in (a). (a)–(c) consider local regions within a device, whereas (d) corresponds to the whole device of thickness $L$. $n\left(x\right)$ and $p\left(x\right)$ are the position-dependent electron and hole concentrations, respectively. The details are described in the text.

Figure 2

(Color online) Langevin recombination reduction factor $\zeta$ in dependence on temperature. Shown are the results for annealed devices of ours (diamonds) (Ref. 6) and of Juška et al. (Ref. 16) (circles). The models by Koster et al. (Ref. 15) (dashed) and Arkhipov et al. (Ref. 11) (dotted) are also included; they show a markedly different temperature dependence as compared to the experimental data.

Figure 3

(Color online) The simple model for the Langevin reduction factor $\zeta$, given by Eq. (10). It depends on the parameter $\alpha$, which represents the carrier-concentration gradients. The inset shows the carrier concentration (on a logarithmic scale) vs distance after Eqs. (7, 8). $p_p$ $\left(n_p\right)$ is the hole (electron) concentration at the hole injection electrode, the anode. Similarly, $n_n$ $\left(p_n\right)$ is the electron (hole) concentration at the electron injection electrode, the cathode.

Figure 4

(Color online) Simulated steady-state electron and hole concentrations in bulk heterojunction solar cells under short circuit (top graph), open circuit (middle), and at the built-in potential (bottom) at an illumination of 1 sun. In each graph, the concentration profiles for the balanced electron-hole mobilities $\mu ={10}^{-5}$ (short-dashed line), ${10}^{-8}$ (solid line), and ${10}^{-11}{\text{m}}^2/\text{V}\text{s}$ (long-dashed line) are shown. Holes have the highest concentration at the anode (left) and electrons at the cathode (right).

Figure 5

(Color online) (Top) Squared carrier concentrations, by global $\left(\overline{np}\right)$ or local multiplication $\left(\overline{n}\overline{p}\right)$ of electron and hole concentrations, respectively, in dependence on the charge-carrier mobility. The data shown were calculated for flatband conditions, as they are typically used in photo-CELIV measurements. (Middle) The recombination reduction factor in dependence on the charge-carrier mobility, $\zeta \left(\mu \right)$, for different injection barriers (${\Phi }_p$ at the anode and ${\Phi }_n$ at the cathode) of 0 eV (short-dashed line), 0.1 eV (solid line), and 0.2 eV (long-dashed line). $\zeta$ is lower for efficient extraction at high mobilities and for low injection barriers. (Bottom) The simulated bimolecular recombination rate vs the charge-carrier mobility, $R\left(\mu \right)$, after Eq. (1), as a result of the carrier-concentration gradients. Also shown is the recombination rate $R/\zeta$, which would be determined if the carrier-concentration gradients were neglected.

Figure 6

(Color online) Comparison of the temperature-dependent recombination reduction factor $\zeta$ for the experimental photo-CELIV data (as already shown in Fig. 2) with our macroscopic simulation. The difference of the absolute values of $\zeta$ in simulation and experiment, a factor of 1/200 for the data of Juška et al. (circles) compares to the calculated values with an injection barrier of 0.1 eV (black solid line). The data of Deibel et al. (diamonds) have similar shape with an additional static reduction factor of about 1/20 is needed to match the simulation. The details are described in the text.