Optimum mobility, contact properties, and open-circuit voltage of organic solar cells: A drift-diffusion simulation study

We investigate the role charge carrier mobility plays for loss mechanisms in organic bulk heterojunction solar cells. For this purpose, we perform drift-diffusion calculations for several recombination models and properties of the contacts. We show that in case of selective contacts, higher mobilities increase device efficiency, independent of injection barrier heights, energy level bending at the contacts, and the amount of background dark carriers in the device. Nonselective contacts provide a source of photocarrier loss at the “wrong” electrode. This is evident from a decrease of the open-circuit voltage ($V_{\mathrm{oc}}$) with an increased role of charge carrier diffusion, which originates from a higher mobility or from interface barriers reducing the built-in potential. In this case, $V_{\mathrm{oc}}$ furthermore depends on the device thickness. Considering the effect of different recombination models, a too high mobility of one charge carrier decreases $V_{\mathrm{oc}}$ significantly for Langevin recombination. That is why balanced mobilities are desirable for high efficiency in this case. In presence of recombination via CT states, $V_{\mathrm{oc}}$ is mainly governed by the dynamics of the charge transfer state. Based on these differentiations we show that the existence of an optimum mobility derived from simulation depends strongly on the assumptions made for contact and recombination properties and obtain a comprehensive picture how charge carrier mobility influences the parameters of organic solar cells.

Figure 1

A bulk heterojunction (BHJ) as effective medium between two metal contacts, determining the built-in potential $V_{\mathrm{bi}}$. Offsets between charge carrier transport levels and the metal work function (including potentially present dipoles) are injection barriers $\varphi$. The effective gap $E_{\mathrm{g}}^{\mathit{DA}}$ is the difference between the ionization potential of the donor (IP${}_{\mathrm{D}}$) and the electron affinity of the acceptor (EA${}_{\mathrm{A}}$).

Figure 2

Schematic visualization of the studied recombination processes: Left: direct recombination between electron $n$ and hole $p$ (bimolecular) with different models for the recombination constant $\beta$. Direct “recombination” into a CT state, with relaxation $k_{\mathrm{relax}}$ and dissociation rate $k_{\mathrm{diss}}$. Photogenerated excitons, entering the CT state, can recombine geminatly. That is why this recombination via CT-states is also called geminate recombination. Right: indirect, trap-assisted (SRH) recombination.

Figure 3

Efficiency $\eta$ as a function of electron and hole mobilities ${\mu }_n\text{and}{\mu }_p$ for different recombination models and contact properties: (a)–(c) selective contacts with (a) constant $\beta$ (qualitatively similar to Langevin recombination via a distinct CT state), (b) direct Langevin recombination (c) same as case (b), but with a modified $\beta \propto \min ({\mu }_n,{\mu }_p)$. (d) Same as case (a), but for nonselective contacts.

Figure 4

(a) $V_{\mathrm{oc}}$ and (b) FF as a function of the mobility ${\mu }_n={\mu }_p$ for different recombination models (symbols) and selective (solid lines) and nonselective contacts (dashed lines). In the case of a constant $\beta$, it is chosen to $7.23\times {10}^{-11}$ cm${}^3$s${}^{-1}$, which corresponds to ${\beta }_{\mathrm{L}}(\mu ={10}^{-4}{\text{cm}}^2/\text{Vs})$. The curves for selective and nonselective contacts coincide in case of Langevin recombination, which limits the solar-cell performance in both cases.

Figure 5

Temperature dependence of the open-circuit voltage $V_{\mathrm{oc}}$ for different recombination constants $\beta$ (x) (${\beta }_{\mathrm{c}}=7.23\times {10}^{-11}$ cm${}^3$s${}^{-1}$), CT state lifetimes $1/k_r$ ($E_{\mathrm{B}}=290$ meV) (o), and nonselective contacts (+) ($\mu =1$ cm${}^2$/Vs). Dashed lines are linear extrapolations of the 250–350 K region. The slope of $V_{\mathrm{oc}}\left(T\right)$ depends on the recombination constant $\beta$ and in case of recombination via CT states on the CT-state lifetime $1/k_r$. $V_{\mathrm{oc}}(T\to 0\text{K}$) approaches the effective energy gap $E_{\mathrm{g}}^{\mathit{DA}}$ and at recombination via CT states the CT-state energy $E_{\mathit{CT}}=E_{\mathrm{g}}^{\mathit{DA}}-E_{\mathrm{B}}$. In case of significant surface recombination, the extrapolation of $V_{\mathrm{oc}}$ to $T=0$ K does not result in the effective energy gap.

Figure 6

Electron ($n$) and hole ($p$) density and the recombination rate ($R$) within a device, where trap-assisted (SRH) recombination is dominating at $V_{\mathrm{oc}}$ of the points marked in Fig. 4 (0.49 V for $\mu ={10}^{-4}$ cm${}^2$/Vs, 0.44 V for $\mu ={10}^{-2}$ cm${}^2$/Vs). An increased recombination close to the contacts is found for a lower value of $\mu$.

Figure 7

(a) Charge carrier density and (b) current profiles at $V_{\mathrm{oc}}$ for selective (solid lines, $V_{\mathrm{oc}}$ = 0.62 V) and nonselective contacts (dashed lines, $V_{\mathrm{oc}}$ = 0.50 V). Here, $\beta =7.23\times {10}^{-11}$ cm${}^3$s${}^{-1}$ and $\mu =1$ cm${}^2$/Vs.

Figure 8

Thickness dependence of $V_{\mathrm{oc}}$ in the case of selective contacts ($\mu ={10}^{-4}$ cm${}^2$/Vs), nonselective contacts and $\mu =1$ cm${}^2$/Vs, and for recombination via CT states. In all cases G is constant and $\beta =7.23\times {10}^{-11}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$. The inset shows the logarithmic increase of $V_{\mathrm{oc}}$ with $d$ in case of nonselective contacts.

Figure 9

(a) $V_{\mathrm{oc}}$ and (b) FF as a function of the mobility ${\mu }_n={\mu }_p$ in the cases of an injection barrier of 0.3 eV. $V_{\mathrm{oc}}$ remains unchanged compared to Fig. 4 in the case of selective contacts (solid lines), whereas nonselective contacts (dashed lines) reduce $V_{\mathrm{oc}}$ and FF.

Figure 10

J-V curves for two different mobilities (${10}^{-2}$ and ${10}^{-5}{\text{cm}}^2$/Vs), Langevin recombination, and selective contacts. The injection barriers are 0.1 eV (solid lines, diamonds) and 0.3 eV (dashed, triangles). The inset compares the dependence of the FF on $\mu$ for the two barrier heights.

Figure 11

Electric field and charge carrier density profiles under the conditions marked in Fig. 10 (Langevin recombination, selective contacts, and varied $\varphi$): (a) and (b) $V=0.5$ V and $\mu ={10}^{-5}$ cm${}^2$/Vs, (c) and (d) $V=0.3$ V and $\mu ={10}^{-2}$ cm${}^2$/Vs.

Figure 12

$V_{\mathrm{oc}}$ as a function of the effective densities of states $N_{\mathrm{C}}=N_{\mathrm{V}}$ and the amount of potential drop $\Delta {\varepsilon }_{\mathrm{B}}$ due to bending of the energy levels at the contacts as depicted in the inset. $V_{\mathrm{oc}}$ is independent of $\varphi$ and does not directly correlate with $\Delta {\varepsilon }_{\mathrm{B}}$. The contacts are assumed to be selective and $\beta =7.23\times {10}^{-11}$ cm${}^3$s${}^{-1}$. The inset shows an energy-level diagram visualizing the bending of the energy levels $\Delta {\varepsilon }_{\mathrm{B}}/2$ close to each contact due to high charge carrier densities there.