Ideal diode equation for organic heterojunctions. I. Derivation and application

The current-voltage characteristics of organic heterojunctions (HJs) are often modeled using the generalized Shockley equation derived for inorganic diodes. However, since this description does not rigorously apply to organic semiconductor donor-acceptor (D-A) HJs, the extracted parameters lack a clear physical meaning. Here, we derive the current density-voltage $\left(J\text{−}V\right)$ characteristic specifically for D-A HJ solar cells and show that it predicts the general dependence of dark current, open-circuit voltage $\left(V_{oc}\right)$, and short-circuit current $\left(J_{sc}\right)$ on temperature and light intensity as well as the maximum $V_{oc}$ for a given D-A material pair. We propose that trap-limited recombination due to disorder at the D-A interface leads to the introduction of two temperature-dependent ideality factors and show that this describes the dark current of copper phthalocyanine/${\text{C}}_{60}$ and boron subphthalocyanine/${\text{C}}_{60}$ cells at low temperature, where fits to the generalized Shockley equation break down. We identify the polaron pair recombination rate as a key factor that determines the $J\text{−}V$ characteristics in the dark and under illumination and provide direct measurements of this process in our companion paper II [N. C. Giebink, B. E. Lassiter, G. P. Wiederrecht, M. R. Wasielewski, and S. R. Forrest, Phys. Rev. B 82, 155306 (2010)]. These results provide a general physical framework for interpreting the $J\text{−}V$ characteristics and understanding the efficiency of both small molecule and polymer organic, planar and bulk HJ solar cells.

Figure 1

(Color online) (a) Energy-level diagram showing the anode and cathode work functions, $W{F}_a$ and $W{F}_c$, and their associated injection barriers ${\varphi }_a$ and ${\varphi }_c$, respectively. The interfacial gap, $\Delta E_{HL}$, is the energy difference between the highest occupied molecular orbital energy of the donor and the lowest unoccupied molecular orbital energy of the acceptor. Current is unipolar in the donor $\left(J_p\right)$ and acceptor $\left(J_n\right)$ layers and is determined from generation/recombination in the HJ region, roughly defined by the spatial extent, $a_0$, of the polaron pair distribution at the interface. (b) Processes occurring within the HJ region. Excitons diffuse, with current density, $J_X$, to the HJ and undergo charge transfer to form polaron pairs. These may recombine, at rate $k_{\text{PP}r}$, or dissociate with rate, $k_{\text{PP}d}$, as determined by the Onsager-Braun model (Ref. 10). The current density, $J$, contributes to the interfacial free electron $\left(n_I\right)$ and hole $\left(p_I\right)$ densities, which bimolecularly recombine to form polaron pairs at rate $k_{rec}$.

Figure 2

(Color online) (a) Calculated dark current density $\left(J\right)$-voltage characteristics over the temperature range from 120 to 300 K in 20 K increments using Eq. (13) and the parameters in Table . The slope in reverse bias is due to the field-dependent dissociation of thermally generated polaron pairs. In forward bias, three regions are apparent. At $V_a<0.3\text{V}$, trap-limited recombination involving free acceptor electrons and trapped donor holes dominates, and the slope is proportional to the donor ideality factor, $n_{\text{D}}$. At higher bias, the inverse process dominates (i.e., free donor holes recombine with trapped acceptor electrons) and the slope is proportional to the acceptor ideality factor, $n_{\text{A}}$. Series resistance $\left(R_s\right)$ limits the current at $V_a>0.8\text{V}$. (b) Diode parameters $n$ and $n\text{ln}\left(J_s\right)$ corresponding to the dark currents in (a). Both ideality factors increase with decreasing temperature, though $n_{\text{A}}$ does so only slightly.

Figure 3

(Color online) (a) Intensity dependence of the open-circuit voltage $\left(V_{oc}\right)$, short-circuit current $\left(J_{sc}\right)$, and $FF$ derived from the same model and parameters as in Fig. 2. (b) Temperature dependence of $V_{oc}$, $FF$, and polaron pair dissociation efficiency at short circuit, ${\eta }_{\text{PP}d}\left(J_{sc}\right)$.

Figure 4

(Color online) Dark current density vs forward voltage for (a) $\text{CuPc}/{\text{C}}_{60}$ and (b) $\text{SubPc}/{\text{C}}_{60}$ devices recorded for $T=296$, 275, 247, 218, 193, 171, 155, 145, 128, and 114 K. Bold (red) lines indicate fits to Eq. (18) in the text. Thin (black) lines connect the data points and serve as a guide to the eyes. Both data sets are refit using the generalized Shockley equation in (c) and (d), where the difference between data and theory is most pronounced at low voltage and temperature. Several intermediate temperatures have been omitted in (c) and (d) for clarity.

Figure 5

(Color online) (a) Temperature dependence of the ideality factors derived from the $J\text{−}V$ data of Fig. 4. The ideality factors increase with decreasing temperature in all cases. Values of $n_{\text{D}}$ and $J_{s\text{D}}$ cannot be extracted for $\text{CuPc}/{\text{C}}_{60}$ at $T>220\text{K}$ because the second exponential term in Eq. (18) is insignificant. (b) Plot of $n\text{ln}\left(J_s\right)$ vs $1/k_{b}T$, where $k_{b}T$ is the thermal energy, for the $\text{CuPc}/{\text{C}}_{60}$ and $\text{SubPc}/{\text{C}}_{60}$ devices. Note the similarity in $n_{\text{A}}$ and $n_{\text{A}}\text{ln}\left(J_{s\text{A}}\right)$ between the two cells, which share a common ${\text{C}}_{60}$ acceptor whereas $n_{\text{D}}$ and $n_{\text{D}}\text{ln}\left(J_{s\text{D}}\right)$ clearly differ due to the different donors.

Figure 6

(Color online) (a) Dependence of the open-circuit voltage, $V_{oc}$, on the intensity of simulated AM1.5G solar illumination at room temperature. Solid lines are calculated from Eq. (18) using values of $n$ and $J_s$ from Fig. 5 along with the short-circuit current densities measured at each intensity. (b) Temperature dependence of $V_{oc}$ under laser illumination at $\lambda =496\text{nm}$, and at an intensity of $30\text{mW}/{\text{cm}}^2$. Solid lines are calculated using the temperature-dependent $n$ and $J_s$ of Fig. 5 and the short-circuit current density measured at room temperature. Deviation between the calculation and the data at low temperature indicates a reduced polaron pair dissociation efficiency, ${\eta }_{\text{PP}d}\left(T\right)$, shown on the right-hand axis, and normalized to its value at room temperature. Dashed lines connecting the data points are a guide to the eyes.

Figure 7

(Color online) (a) Dependence of open-circuit voltage, fill factor, and short-circuit current ($V_{oc}$, $FF$, $J_{sc}$, respectively), and normalized power-conversion efficiency, ${\eta }_P$, on the polaron pair recombination rate, $k_{\text{PP}r}$. An exciton current of $q{J}_X=10\text{mA}/{\text{cm}}^2$ is assumed. (b) Fourth quadrant $J\text{−}V$ characteristics at several values of $k_{\text{PP}r}$ resulting in the trends shown in (a).

Figure 8

(Color online) (a) Current-voltage characteristics calculated for $V_{bi}$ ranging from 0.1 to 0.7 V in 0.2 V increments using the same model as in Fig. 7b with $k_{\text{PP}r}={\left(20\text{ns}\right)}^{-1}$. (b) Schematic illustrating the physical basis for the S-kink observed in (a). The top panel shows the dependence of the internal field, $F_I$, and polaron pair dissociation efficiency, ${\eta }_{\text{PP}d}$, on the applied bias, $V_a$. Dissociation is aided by $F_I<0$. The lower panel shows the corresponding photocurrents, which sum with the dark current to give the total current (dashed lines). When $V_{bi}$ is small, the internal field reaches zero at small $V_a$, resulting in a rapid change in photocurrent before the dark current becomes appreciable, leading to the S-kink as shown.