General Rules for the Impact of Energetic Disorder and Mobility on Nongeminate Recombination in Phase-Separated Organic Solar Cells

State-of-the-art organic solar cells exhibit power conversion efficiencies of 18% and above. These devices benefit from the suppression of free charge recombination with regard to the Langevin limit of charge encounter in a homogeneous medium. It is recognized that the main cause of suppressed free charge recombination is the reformation and resplitting of charge-transfer (CT) states at the interface between donor and acceptor domains. Here, we use kinetic Monte Carlo simulations to understand the interplay between free charge motion and recombination in an energetically disordered phase-separated donor-acceptor blend. We identify conditions for encounter-dominated and resplitting-dominated recombination. In the former regime, recombination is proportional to mobility for all parameters tested and only slightly reduced with respect to the Langevin limit. In contrast, mobility is not the decisive parameter that determines the nongeminate recombination coefficient, $k_2$, in the latter case, where $k_2$ is a sole function of the morphology, CT and charge-separated (CS) energetics, and CT-state decay properties. Our simulations also show that free charge encounter in the phase-separated disordered blend is determined by the average mobility of all carriers, while CT reformation and resplitting involves mostly states near the transport energy. Therefore, charge encounter is more affected by increased disorder than the resplitting of the CT state. As a consequence, for a given mobility, larger energetic disorder, in combination with a higher hopping rate, is preferred. These findings have implications for the understanding of suppressed recombination in solar cells with nonfullerene acceptors, which are known to exhibit lower energetic disorder than that of fullerenes.

Figure 1

Scheme showing the states involved in free charge recombination. Here, $n_{\mathrm{CS}}$ and $n_{\mathrm{CT}}$ are the density of free carriers (charge-separated states) and charge-transfer states, respectively; ${\beta }_{\mathrm{enc}}$ is the coefficient for bimolecular free charge encounter forming the CT state; $k_d$ is the rate coefficient for CT dissociation into free charges; and $k_f$ is the rate coefficient for the decay of the CT state to the ground state. Also depicted are the two extremes. For $k_d\ll k_f$, resplitting of the CT state is very unlikely, and the recombination rate, R, is determined by the encounter rate, which is independent of the CT decay rate but proportional to the carrier-hopping rate and with this to the carrier mobility. In contrast, if $k_d\gg k_f$, CT states resplit at a much higher rate than they decay to the ground state. Because ${\beta }_{\mathrm{enc}}$ and $k_d$ depend linearly on the hopping rate, ${\nu }_0$, its effect on the recombination rate cancels out while, on the other hand, $R$ becomes strictly proportional to $k_f$.

Figure 2

KMC results of the dynamics of the total density of electron-hole pairs (solid lines), free electron-hole pairs (dotted lines), and CT states (dashed lines) for variable attempt-to-hop frequencies $({\nu }_0)$ (a) and the corresponding second-order NGR coefficient $(k_2)$ (b). Dotted arrows indicate the maximum CT population. Simulations are performed with the following parameters: ${\sigma }_{\mathrm{DOS},e}={\sigma }_{\mathrm{DOS},h}=90\mathrm{meV}$; $k_{\mathrm{CT}}$ (k_f) = 1 × 10⁸ s⁻¹; T = 300 K; intersite distance $(a_{\mathrm{NN}})=1.8\mathrm{nm}$; ${\nu }_{0,e}={\nu }_{0,h}=1\times {10}^8$, 10⁹, 10¹⁰, 10¹¹, and 10¹² s⁻¹ (purple to red).

Figure 3

Bimolecular recombination coefficient (solid lines) and charge carrier mobility (dashed lines) for variable attempt-to-hop frequencies at k_f = 1 × 10⁸ s⁻¹ (a) and 1 × 10⁷ s⁻¹ (b), plotted as a function of $t{\nu }_o$. In both graphs, the two y axes are aligned according to the encounter-dominated case with ${\gamma }_{\mathrm{enc}}=0.4$. Insets show the corresponding $k_2/{\nu }_0$ transients. KMC parameters are the same as those in Fig. 2.

Figure 4

Simulated transients of (a) mean energy of all charges, (b) mean energy of moving charges, (c) bipolar mobilities for holes (solid lines) and electrons (dashed lines), (d) energy of recombining CT states, (e) $k_2$ and (f) recombination reduction factor for variable attempt-to-hop frequencies plotted as a function of $t{\nu }_0$. KMC parameters are the same as those in Fig. 2. Blue vertical guideline marks the time when the mean energy crosses the transport energy $E_{\mathrm{tr}}=E\mp {\sigma }_{\mathrm{DOS}}^2/2k_{B}T$ (gray horizontal dashed guideline). Red horizontal guideline shows the equilibrium energy, $E_{\mathrm{\infty }}=E\mp {\sigma }_{\mathrm{DOS}}^2/k_{B}T$, where − indicates electrons and + indicates holes.

Figure 5

(a) Scheme of CT formation, resplitting, and decay in a phase-separated disordered BHJ system. Charge encounter is determined by a time-dependent encounter rate, ${\beta }_{\mathrm{enc}}(t)={\gamma }_{\mathrm{enc}}{k}_L[\mu (t)]$, involving the entire density of occupied states. In contrast, for long enough delay after photoexcitation, dissociation rate $k_d$ is mainly determined by states around $E_{\mathrm{tr}}$. (b)–(d) KMC results of the reduction factor $\gamma$ of (nearly) equilibrated carriers for different hopping rates and disorders in phase-separated blends (taken at the times indicated in Fig. 13) for $k_f={10}^8{\mathrm{s}}^{-1}$ (orange circles) and $k_f={10}^7{\mathrm{s}}^{-1}$ (red circles). Solid lines show the best fit to data with Eq. (10). All simulations are performed at T = 300 K, except for green (T = 250 K) and blue (T = 200 K) dots in (c). (e) $k_d$ as a function of energetic disorder, as deduced from the fit of the reduction factor in  (b)–(d) for ${\nu }_0={10}^{11}{\mathrm{s}}^{-1}$ and $k_f={10}^8{\mathrm{s}}^{-1}$ (see Table 1 for parameters). Also shown is the analytical prediction of $k_d$ in the absence of disorder, according to Eq. (9), where $\mu$ is either the transient mobility when the carrier package crosses $E_{\mathrm{tr}}$ or after the carriers have equilibrated at $E_{\mathrm{\infty }}$ (dotted lines).

Figure 6

Simulated $k_2$ as a function of ${\mu }_{\mathrm{\infty }}$ for ${\sigma }_{\mathrm{DOS}}$ of 65 (triangles), 90 (squares), and 120 meV (circles) for (nearly) equilibrated charges taken at the times indicated in Fig. 13 (a) and the corresponding deduced reduction factor (b). Solid symbols indicate $k_f={10}^8{\mathrm{s}}^{-1}$, while open circles show the results for $k_f={10}^7{\mathrm{s}}^{-1}$ for the example of high energetic disorder. Dashed line is the prediction of the Langevin model; solid line guides the eye. Insets show experimental $k_2$ and $\gamma$ data as a function of ${\mu }_e+{\mu }_h$ of bulk-heterojunction solar cells taken from Ref. [33] (gray dots).

Figure 7

Transient mobility as a function of time (a)–(c) and reduced time $(t{\nu }_0)$ (d)–(f) for a phase-separated blend for variable energetic disorder and hopping frequencies under a voltage of 1 V. Universal mobility plot (g)–(i) with both mobility $(\mu /{\nu }_0)$ and time $(t{\nu }_0)$ modified by attempt-to-hop frequency ${\nu }_0$. Simulations are performed for ${\sigma }_{\mathrm{DOS},e}={\sigma }_{\mathrm{DOS},h}=65$ (a),(d),(g), 90 (b),(e),(h), and 120 meV (c),(f),(i); T = 300 K; a_NN= 1.8 nm; $k_{\mathrm{CT}}$ (k_f) = 1 × 10⁸ s⁻¹; ${\nu }_{0,e}={\nu }_{0,h}=1\times {10}^8$, 10⁹, 10¹⁰, 10¹¹, and 10¹² s⁻¹ (purple to red), respectively.

Figure 8

Simulated transients for energetic disorder ${\sigma }_{\mathrm{DOS,}e}={\sigma }_{\mathrm{DOS},h}=90\mathrm{meV}$ of (a) mean energy of charges, (b) energy of moving charges, (c) bipolar mobilities for holes (solid lines) and electrons (dashed lines), (d) energy of recombining CT states (e) of k₂ and (f) of the reduction factor for variable attempt-to-hop frequencies plotted as a function of time (see Fig. 3 for the corresponding plot as a function of $t{\nu }_0$). KMC parameters are the same as those in Fig. 3. Star symbol marks the time when the mean energy crosses the effective transport energy, $E_{\mathrm{tr}}=E\mp {\sigma }_{\mathrm{DOS}}^2/2k_{B}T$ (gray horizontal dashed guideline), for different ${\nu }_0$, and orange horizontal guideline shows the equilibrium energy, $E_{\mathrm{\infty }}=E\mp {\sigma }_{\mathrm{DOS}}^2/k_{B}T$, where − corresponds to electrons and + to holes$.E_{G,\mathrm{tr}}-E_{\mathrm{bind}}=[E_{\mathrm{LUMO}}-({\sigma }_{\mathrm{DOS},e}^2/2k_{B}T)]-[E_{\mathrm{HOMO}}+({\sigma }_{\mathrm{DOS},h}^2/2k_{B}T)]-E_{\mathrm{bind}}$. Arrow indicates maximum CT population.

Figure 9

Simulated transients for energetic disorder ${\sigma }_{\mathrm{DOS},e}={\sigma }_{\mathrm{DOS},h}=65\mathrm{meV}$ of (a) mean energy of charges, (b) energy of moving charges, (c) bipolar mobilities for holes (solid lines) and electrons (dashed lines), (d) energy of recombining CT states (e) of k₂ and (f) of the reduction factor for variable attempt-to-hop frequencies plotted as a function of $t{\nu }_0$. KMC parameters as well as the meaning of all symbols and lines are the same as those in Fig. 3 except for the energetic disorder.

Figure 10

Simulated transients for energetic disorder ${\sigma }_{\mathrm{DOS},e}={\sigma }_{\mathrm{DOS},h}=120\mathrm{meV}$ of (a) mean energy of charges, (b) energy of moving charges, (c) bipolar mobilities for holes (solid lines) and electrons (dashed lines), (d) energy of recombining CT states (e) of k₂ and (f) of the reduction factor for variable attempt-to-hop frequencies plotted as a function of $t{\nu }_0$. KMC parameters as well as the meaning of all symbols and lines are the same as those in Fig. 3 except for the energetic disorder.

Figure 11

Dynamics of the energy of recombining CT states as a function of reduced time for variable energetic disorder and hopping frequencies. $E_{G,\mathrm{tr}}-E_{\mathrm{bind}}=[E_{\mathrm{LUMO}}-({\sigma }_{\mathrm{DOS,}e}^2/2k_{B}T)]-[E_{\mathrm{HOMO}}+({\sigma }_{\mathrm{DOS,}h}^2/2k_{B}T)]-E_{\mathrm{bind}}$. Except for energetic disorder, KMC parameters are the same as those in Fig. 2.

Figure 12

Dynamics of the energy of recombining CT states as a function of reduced time for variable energetic disorder and hopping frequencies for a homogenous blend (a) and a phase-separated blend with $k_f=1\times {10}^8{\mathrm{s}}^{-1}$ (b) and $k_f=1\times {10}^7{\mathrm{s}}^{-1}$ (c).

Figure 13

Reduction factor $\gamma =k_2/k_L$ as a function of reduced time for either a homogeneous blend with $k_f=1\times {10}^8{\mathrm{s}}^{-1}$ (a),(d),(g), or a phase-separated blend with $k_f=1\times {10}^8{\mathrm{s}}^{-1}$ (b),(e),(h), or $k_f=1\times {10}^7{\mathrm{s}}^{-1}$ (c),(f),(i). Gray dashed lines highlight regions of (nearly) constant $\gamma$, from which data in Figs. 5 and 6 are deduced at the times marked by solid vertical lines.

Figure 14

Simulated transients of k_L (dashed lines) and k₂ (solid lines) (a),(b) and reduction factor $\gamma$ (c),(d) as a function of reduced time for ${\nu }_0={10}^8{\mathrm{s}}^{-1}$ (a),(c) and ${\nu }_0={10}^{12}{\mathrm{s}}^{-1}$ (b),(d) and different temperatures. ${\sigma }_{\mathrm{DOS}}=90\mathrm{meV}$ and $k_f={10}^8{\mathrm{s}}^{-1}$ in all cases.

Figure 15

Simulated values of mobility (a), k₂ (b), and reduction factor $\gamma$ (c) for (nearly) equilibrated carriers, plotted as a function of hopping frequency for different energetic disorders, ${\sigma }_{\mathrm{DOS}}=\mathrm{65,\; 90,}\mathrm{and}120\mathrm{meV}$, with $k_f={10}^8{\mathrm{s}}^{-1}$ (solid lines) and $k_f={10}^7{\mathrm{s}}^{-1}$ (dashed lines).