Power-Law Density of States in Organic Solar Cells Revealed by the Open-Circuit Voltage Dependence of the Ideality Factor

The density of states (DOS) is fundamentally important for understanding physical processes in organic disordered semiconductors, yet hard to determine experimentally. We evaluated the DOS by considering recombination via tail states and using the temperature and open-circuit voltage ($V_{\mathrm{oc}}$) dependence of the ideality factor. By performing Suns-$V_{\mathrm{oc}}$ measurements, we find that the energetic disorder increases deeper into the band gap, which is not expected for a Gaussian or exponential DOS. The linear dependence of the disorder on energy reveals the power-law DOS in organic solar cells.

Figure 1

(a) Suns-$V_{\mathrm{oc}}$ data (symbols) of a-Si:H solar cell fitted with Eq. (5) (solid lines). (b) Temperature-dependent ideality factor $n_{\mathrm{id}}$ extracted from the fits in comparison to Eqs. (3) and (4).

Figure 2

Temperature-dependent ideality factors $n_{\mathrm{id}}$ of (a) $\mathrm{P}3\mathrm{HT}:{\mathrm{PC}}_{61}\mathrm{BM}$, (b) PM6:Y6, and (c) $\mathrm{DCV}\text{−}\mathrm{V}\text{−}\mathrm{Fu}\text{−}\mathrm{Ind}\text{−}\mathrm{Fu}\text{−}\mathrm{V}:{\mathrm{C}}_{60}$ (symbols). Darker color corresponds to lower $V_{\mathrm{oc}}$, i.e., deeper subgap energy states. Dashed lines are the calculated $n_{\mathrm{id}}(E_U,T)$ according to Eq. (4) for $\mathrm{P}3\mathrm{HT}:{\mathrm{PC}}_{61}\mathrm{BM}$ and Eq. (3) for PM6:Y6 and $\mathrm{DCV}\text{−}\mathrm{V}\text{−}\mathrm{Fu}\text{−}\mathrm{Ind}\text{−}\mathrm{Fu}\text{−}\mathrm{V}:{\mathrm{C}}_{60}$. $E_U$ increases with the DOS depth for all three systems.

Figure 3

Urbach energies $E_U$ as a function of the quasi-Fermi level splitting, approximated by $V_{\mathrm{oc}}$. $E_U$ was calculated from $n_{\mathrm{id}}(T)$ according to Eq. (4) for (a) $\mathrm{P}3\mathrm{HT}:{\mathrm{PC}}_{61}\mathrm{BM}$ and Eq. (3) for (b) PM6:Y6 and (c) $\mathrm{DCV}\text{−}\mathrm{V}\text{−}\mathrm{Fu}\text{−}\mathrm{Ind}\text{−}\mathrm{Fu}\text{−}\mathrm{V}:{\mathrm{C}}_{60}$. $E_U$ is independent of $T$, while it depends linearly on $V_{\mathrm{oc}}$ within the highlighted areas, inferring that the DOS is described by a power law. Roughly constant $E_U$ outside of the highlighted area is an indication of the exponential DOS. Dashed lines correspond to $V_{\mathrm{oc}}$ at 1 sun illumination intensity at room temperature.