Effect of Imbalanced Charge Transport on the Interplay of Surface and Bulk Recombination in Organic Solar Cells

Surface recombination has a major impact on the open-circuit voltage ($V_{\mathrm{OC}}$) of organic photovoltaics. Here, we study how this loss mechanism is influenced by imbalanced charge transport in the photoactive layer. As a model system, we use organic solar cells with a 2 orders-of-magnitude-higher electron than hole mobility. We find that small variations in the work function of the anode have a strong effect on the light-intensity dependence of $V_{\mathrm{OC}}$. Transient measurements and drift-diffusion simulations reveal that this is due to a change in the surface recombination rather than the bulk recombination. We use our numerical model to generalize these findings and determine the circumstances under which the effect of contacts is stronger or weaker compared to the idealized case of balanced charge transport. Finally, we derive analytical expressions for $V_{\mathrm{OC}}$ in the case that a pile-up of space charge is present due to highly imbalanced mobilities.

Figure 1

Current-voltage curves for ${\mathrm{Mo}\mathrm{O}}_x$ (squares) and PEDOT:PSS (circles) devices under simulated AM1.5G illumination ($100\mathrm{mW}{\mathrm{cm}}^{-2}$). The solid lines are the result of drift-diffusion simulations using the parameter set in Table 1. The only parameter varied is the injection barrier ${\phi }_{\mathrm{an}}$ at the anode. The inset shows a schematic energy-level diagram at $V=V_{\mathrm{OC}}$.

Figure 2

The light-intensity dependence of $V_{\mathrm{OC}}$ under AM1.5G illumination for (a) experimental and (b) simulated devices based on ${\mathrm{Mo}\mathrm{O}}_x$ (squares) and PEDOT:PSS (circles). The error bars in (a) are the standard deviation for five individual samples. The dotted lines indicate a scaling of $kT/q$ (“slope 1”) and $kT/2q$ (“slope 1/2”).

Figure 3

Simulated band diagrams and local carrier concentrations for (a,c) ${\phi }_{\mathrm{an}}=0$ and (b,d) ${\phi }_{\mathrm{an}}=250\mathrm{meV}$. The anode is positioned at $x=0$. The solid lines in panels (a) and (b) denote the transport bands and the dashed lines the quasi-Fermi-levels under AM1.5G illumination ($100\mathrm{mW}{\mathrm{cm}}^{-2}$). Panels (c) and (d) show the total electron and hole concentration for incident-light intensities of 1, 10, and $100\mathrm{mW}{\mathrm{cm}}^{-2}$. The result of the injection barrier is a gradient of $E_{F,p}$ at the anode, which reduces $V_{\mathrm{OC}}$ relative to its optimum value $V_{\mathrm{OC,max}}$ given by the quasi-Fermi-level splitting in the bulk.

Figure 4

The results of charge extraction and transient photovoltage measurements for ${\mathrm{Mo}\mathrm{O}}_x$ (squares) and PEDOT:PSS (circles) devices. (a) The extracted charge-carrier concentration versus the open-circuit voltage. (b) The recombination rate constant $\beta$ versus the carrier concentration.

Figure 5

(a) The quasi-Fermi-level $E_{F,p}$ close to the anode (located at $x=0$) for an ideal device with Ohmic contacts and balanced transport (dashed line) and for a device with an injection barrier and varied ${\mu }_n/{\mu }_p$ (solid lines). The arrows indicate the voltage loss due to the barrier height (1) and due to the imbalanced mobilities (2), respectively. Panels (b) and (c) show the corresponding hole concentration $p$ and hole conductivity ${\sigma }_p=q{\mu }_{p}p$.

Figure 6

The simulated $V_{\mathrm{OC}}$ for a varied hole mobility but a fixed electron mobility of ${\mu }_n={10}^{-4}{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ and different barrier heights ${\phi }_{\mathrm{an}}$ at the anode. Panel (a) shows the case of a constant $\beta$, while for (b) we assume reduced Langevin recombination with $\beta =\zeta {\beta }_L$. The vertical dashed lines indicate the case ${\mu }_n={\mu }_p$. The arrows in (a) exemplify the voltage loss $\mathrm{\Delta }{V}_{\mathrm{OC},1}$ for a barrier of 300 meV and $\mathrm{\Delta }{V}_{\mathrm{OC},2}$ for a mobility mismatch of ${\mu }_n/{\mu }_p=100$, respectively.

Figure 7

The simulated value of $V_{\mathrm{OC}}$ (solid lines) versus the light intensity for a device having a non-Ohmic anode (${\phi }_{\mathrm{an}}=0.3\mathrm{eV}$) at different ratios between ${\mu }_n$ and ${\mu }_p$. Equations (3) and (11) are indicated by the dashed and dotted lines, respectively. Our extended analytical expression, Eq. (11), shows striking agreement with the simulations for highly imbalanced mobilities. The dash-dotted line indicates Eq. (8), valid in the low-intensity regime, with $v_{d,n}$ approximated according to Ref. [12].