Meaning of reaction orders in polymer:fullerene solar cells

Nongeminate recombination in polymer:fullerene solar cells is frequently characterized using transient optoelectronic measurements that allow the determination of recombination rates, charge carrier lifetimes, and average charge carrier concentrations as a function of voltage. These data are often interpreted in terms of an empirical reaction order defining how recombination depends on measured charge density. In polymer:fullerene solar cells, the empirical reaction orders are often considerably larger than 2, which had previously been explained in terms of the nonlinear relationship between mobile and trapped charge carriers in the presence of an exponential tail of localized states. Here, we show that experimentally determined reaction orders depend not only on the shape of the density of states but also on the spatial distribution of carriers. In particular, in solar cells with small depletion regions due to small active layer thicknesses or due to large unintentional background doping of the polymers, the reaction order can assume values that are much larger than the value expected from the shape of the density of states alone.

Figure 1

Schematic depiction of (a) the density of states and occupation in a system with exponential tails and (b) the recombination via tail states. The occupation of a tail with a slope $E_{\mathrm{ch}}>kT$ is always such that most carriers are trapped close to the quasi-Fermi levels (dashed lines). Thus, a recombination event requires one of the trapped carriers to become mobile and recombine with the other carrier trapped close to the interface at the respective quasi-Fermi level. CBT, conduction band tail; VBT, valence band tail.

Figure 2

Simulated dependence of (a) ideality factor $n_{\mathrm{id}}$ and (b) slope $m$ (defined by $n_{\mathrm{CE}}\sim \exp (q{V}_{\mathrm{oc}}/mkT)$), reaction order δ, and slope $\vartheta$ (defined by $\tau ={\tau }_0$ exp(−$q{V}_{\mathrm{oc}}$/$\vartheta kT$)) on tail slope $E_{\mathrm{ch}}$, assuming that recombination happens via conduction and valence band tail states (solid lines). The symbols represent the analytical approximations that hold for $E_{\mathrm{ch}}>kT$ and are defined in Eqs. (9) and (11)–(13).

Figure 3

Simulated dependence of (a) $n_{\mathrm{id}}$ and (b) $m$, δ, and $\vartheta$ on the valence band tail slope, assuming that the conduction band tail slope is fixed at $E_{\mathrm{chC}}=50$ meV (solid lines). Again, analytical approximations are shown that work well only in a limited range of characteristic energies. From the simulations, we conclude that the ideality factor $n_{\mathrm{id}}$ is mostly determined by the larger tail slope, while the reaction order δ is dominated by the smaller tail slope.

Figure 4

Simulated dependence of (a) $n_{\mathrm{id}}$ and (b) $m$, δ, and $\vartheta$ on the concentration of midgap states, assuming that recombination happens both via tails and via deep states (solid lines). The simulation shows a transition from the situation with tail recombination dominating to a situation that is dominated by recombination via deep states, mainly resulting in a change of $n_{\mathrm{id}}$ rather than $m$. High concentrations of deep states therefore lead to smaller reaction orders $\delta =m/n_{\mathrm{id}}$ and a high value of $\vartheta$. The thin dashed lines represent the analytical approximations assuming just symmetric tail states with slope $E_{\mathrm{chC}}=50$ meV.

Figure 5

Simulated dependence of (a) $n_{\mathrm{id}}$ and (b) $m$, δ, and $\vartheta$ on the mobility of both carriers, assuming that electrons can recombine at the anode and holes can recombine at the cathode (surface recombination). For higher mobilities, the relative importance of surface recombination increases, leading essentially to a smaller ideality factor $n_{\mathrm{id}}$. The thin dashed lines represent the analytical approximations assuming just symmetric tail states with slope $E_{\mathrm{chC}}=50$ meV.

Figure 6

(a) Band diagram, as well as (b) and (c) electron and hole concentration as a function of position for a 50-nm-thick active layer. The excess electron and hole concentrations are depicted for different light intensities, always at the open circuit, and the band diagram corresponds to the highest light intensity. The excess electron and hole concentrations are spatially inhomogeneous, which leads to a slow increase of the average carrier concentration with voltage because most change happens where the carrier concentrations are low, which contributes only slightly to the average carrier concentration.

Figure 7

(a) Band diagram, as well as (b) and (c) electron and hole concentration as a function of position for a 300-nm-thick active layer. The excess electron and hole concentrations are depicted for different light intensities, always at the open circuit, and the band diagram corresponds to the highest light intensity. In contrast to the case of a thin solar cell (Fig. 6), the carrier concentration is spatially homogeneous and is well described by analytical models based on one-dimensional approximations.

Figure 8

Simulated dependence of (a) $n_{\mathrm{id}}$ and (b) $m$, δ, and $\vartheta$ as a function of active layer thickness. The device is assumed to be dominated by recombination via deep states if thick. When the thickness is reduced, it becomes easier for the charge carriers to reach the opposite electrode and surface recombination becomes dominant, reducing the ideality factor. The slope $m$ and the reaction order increase strongly at low thicknesses because of the spatial inhomogeneity of the carrier concentration, as shown in Fig. 6.

Figure 9

(a) Band diagram, as well as (b) and (c) hole concentration as a function of position for a 150-nm-thick active layer with $p$-type doping. Panels (b) and (c) depicts the hole concentrations $p$, the concentration at 0 V in the dark $p_0$, and the excess concentration $\Delta p=p-p_0$. As for the case of a thin solar cell, the excess carrier concentration is spatially inhomogeneous, although the total concentration $p$ is rather homogeneous.

Figure 10

Simulated dependence of (a) $n_{\mathrm{id}}$ and (b) $m$, δ, and $\vartheta$ as a function of the concentration of effective acceptor doping. For a higher doping concentration, recombination depends only on the concentration of free minority carriers (here, electrons). The ideality factor therefore drops to 1, independent of the density of localized states. Because doping leads to a spatially inhomogeneous distribution of excess carriers, at high doping concentrations, the slope $m$ and therefore the reaction order δ increase drastically.