Computer simulation of heterogeneous polymer photovoltaic devices

Figure 2 shows a typical configuration of an OSC. The active layer comprised of a blend of donor and acceptor materials forms the domain under consideration. Figure 1 shows the structure of the active layer as seen under a transmission electron microscope [32]³. The top and bottom boundaries are connected to electrodes. Typically, the anode is transparent to the visible region of radiation spectrum, while the metallic cathode is reflective⁴.

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The active layer of OSC devices (figure 2) consists of a blend of electron donor and electron acceptor materials. The current generation process in OSCs can be broadly divided into three stages. Stage 1—exciton generation and diffusion: the absorption of light resulting in the generation and subsequent diffusion of tightly coupled electron–hole pairs known as excitons toward the DA interface. Stage 2—charge generation: the excitons then separate at the interface to form electrons in the acceptor region and holes in the donor region. Stage 3—charge transport: the movement of electrons and holes toward the cathode and anode, respectively. Each stage is described in detail as follows.

Stage 1. The electron donor in the active layer of the OSCs absorbs the photons reaching it through the transparent electrode. Unlike inorganic photovoltaic devices, where the absorbed photons directly create free electron–hole pairs, organic photovoltaic devices give rise to tightly bound electron–hole pairs known as excitons [17] (figure 3(a)). These excitons, which have a short lifetime of ∼0.5 ns nanoseconds [31], must diffuse to the donor–acceptor interface in order to be separated into free charges.

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Stage 2. At the interface, [33] the difference in the electron affinity of the materials causes the electron in the exciton to jump to the acceptor (figure 3(b)). The hole remains in the donor region. This transfer of electron causes the electron–hole pair to become loosely bound. The loosely bound electron and hole can be separated by the local electric field at the interface.

Stage 3. Once the exciton has separated into a free electron and a hole in the two regions across the interface, the charges travel to the electrodes through the respective materials. That is, the electron is transported through the acceptor-rich region toward the anode, while the hole is transported through the donor-rich regions toward the cathode (figure 3(c)). The transport characteristics are affected by the local electric field [17]. The tortuous structure of the pathways (see figure 1) makes transport of the generated charges toward the electrodes a complex process.

The equations describing the charge transport processes in a heterogeneous OSC are given below. The four fields of interest are the electron density n, hole density p, electrostatic potential φ and exciton density X.

$$\begin{eqnarray} &&\nabla\cdot(\epsilon\nabla\phi) = q(n - p) \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&\nabla\cdot{\bit J_n} = -qf\mathcal{D}_{[\nabla\phi,X]} + qf\mathcal{R}_{[n,p]} \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&\nabla\cdot{\bit J_p} = +qf\mathcal{D}_{[\nabla\phi,X]} - qf\mathcal{R}_{[n,p]} \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&{\bit J_n} = -qn\mu _n\nabla \phi + qV_t\mu_n\nabla n \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &&{\bit J_p} = -qp\mu _p\nabla \phi - qV_t\mu_p\nabla p \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\nabla\cdot(V_{\rm t}\mu_X\nabla X) - f\mathcal{D}_{[\nabla\phi,X]} - \mathcal{R}_{[X]} = -\mathcal{G} - f\mathcal{R}_{[n,p]} \end{eqnarray} \tag{ 6 }$$

where the Poisson equation (equation (1)) incorporates the effect of electron and hole densities on potential φ in the active layer. is the dielectric constant, and is a spatially varying field. Equations (2), (3) are commonly known as the current-continuity equations. The notations for electron and hole current densities are J_n and J_p, respectively. The exciton generation, diffusion and dissociation are represented by equation (6). $\mathcal{G}$ and $\mathcal{D}_{[\nabla\phi,X]}$ denote rate of exciton generation and dissociation, respectively. Exciton relaxation rate is denoted by $\mathcal{R}_{[X]}$.

Remark 1: Donor–acceptor interface. The exciton dissociation $\mathcal{D}_{[\nabla\phi,X]}$ and charge recombination $\mathcal{R}_{[n,p]}$ are confined to a thin region at the DA interface using the factor f. This is similar to the boundary condition for excitons at the interface, used by Martin et al [28]. The factor f takes a value of one in the thin region (∼2 nm thick) across DA interface which is used to approximate the interfacial region. f is zero elsewhere. This finite interfacial region allows transport of both electrons and holes but the electrons are restricted to acceptor regions (and holes to donor regions) using a mobility value 20 orders of magnitude lower than the mobility in acceptor regions (donor regions in case of holes).

The mobilities for electron, holes and excitons are denoted by μ_n, μ_p and μ_X, respectively. The thermal voltage V_t = k_BT/q, where k_B is the Boltzmann constant, T the temperature and q denotes elementary charge. The Onsager dissociation rate [26] $\mathcal{D}$ is given by

$$\begin{equation} \mathcal{D}_{[\nabla\phi,X]} = \frac{3\gamma}{4\pi a^3}{\rm e}^{-E_{\rm b}/K_{\rm B} T}\frac{J_1(2\sqrt{-2b})}{\sqrt{-2b}}X. \end{equation} \tag{ 7 }$$

In this expression E_b = q²/(4πx) is the exciton binding energy, $b = q^3 |\nabla\phi|/(8\pi\epsilon k_{\rm B}^2 T^2)$ is the field parameter and J₁ is the first order Bessel function given by

$$\begin{equation} \frac{J_1(2\sqrt{-2b})}{\sqrt{-2b}} = \sum_{m=0}^{\infty}\frac{(2b)^m}{m!(m+1)!}. \end{equation} \tag{ 8 }$$

The charge recombination $\mathcal{R}_{[n,p]} = \gamma np$, where γ is the Langevin recombination parameter. In this study, the following form for Langevin parameter $\gamma = \frac{q}{\epsilon}(\mu_n + \mu_p)$ [2] is used. The exciton relaxation $\mathcal{R}_{[X]} = X/\tau_X$ [2], where τ_X is the average lifetime of an exciton. Generation rate of excitons is given by $\mathcal{G}_{[x]}=\alpha_0\Gamma_0 {\rm exp}(-\alpha_0x)$, where α₀ is the absorption coefficient, Γ₀ the photon flux, and x is the distance from the top of the active layer which is in contact with the transparent electrode.

Remark 2. Boundary conditions. Assuming that the cathode and anode line up with the conduction and valence band, respectively, the boundary conditions [34, 35] for the charge densities are given by

$$\begin{eqnarray} &&n_{[\rm C]} = N_{\rm C} \nonumber\\ &&p_{[C]} = N_{\rm V} {\rm exp}\left(\frac{-E_{\rm gap}}{V_{\rm t}}\right) \end{eqnarray} \tag{ 9 }$$

for the cathode, and

$$\begin{eqnarray} &&n_{[\rm V]} = N_{\rm C} {\rm exp}\left(\frac{-E_{\rm gap}}{V_{\rm t}}\right) \nonumber\\ &&p_{[\rm V]} = N_{\rm V} \end{eqnarray} \tag{ 10 }$$

for the anode. Here, E_gap is the band gap and N_C, N_V denote the effective density of states of conduction and valence band, respectively. The built-in potential V_bi is given by qV_bi = φ_an − φ_cat, where φ_an and φ_cat are the work functions for the anode and the cathode, respectively. Alignment of electrodes with the conduction and valence bands gives V_bi = E_gap. Hence, the boundary condition for potential at the electrodes is given by

$$\begin{equation} \phi _{\rm C} - \phi _{\rm V} = E_{\rm gap} - V_{\rm a} \end{equation} \tag{ 11 }$$

where V_a is the applied voltage.

Remark 3: Current–voltage plot. The solution of the excitonic drift–diffusion equations (1)–(6) is used to calculate the electron and hole current density vectors using equations (4) and (5)⁵. The current density available from the device is obtained by integrating the normal component of J_n, J_p at the electrodes. The variation of current density is plotted for a range of applied voltages—zero volts to open-circuit voltage—and is popularly known as the current–voltage characteristic plot.

The use of normalized units helps one to avoid numerical overflow/underflow and improves the efficiency of the algorithms. Using the scaling strategy of Markowich et al [37], the basic equations transform into the following:

$$\begin{eqnarray} &&\hat{\nabla}\cdot(\lambda^2\hat{\nabla}\hat{\phi}) = (\hat{n} - \hat{p}) \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &&\hat{\nabla}\cdot(-\hat{\mu_n} \hat{n}\hat{\nabla} \hat{\phi} + \hat{\mu_n}\hat{\nabla} \hat{n}) = -f\hat{\mathcal{D}} + f\hat{\mathcal{R}}_{[n,p]} \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} &&\hat{\nabla}\cdot( \hat{\mu_p} \hat{p}\hat{\nabla} \hat{\phi} + \hat{\mu_p}\hat{\nabla} \hat{p}) = -f\hat{\mathcal{D}} + f\hat{\mathcal{R}}_{[n,p]} \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} &&\hat{\nabla}\cdot(\hat{\mu_X}\hat{\nabla}\hat{X}) = f\hat{\mathcal{D}} + \hat{\mathcal{R}}_{[X]} - \hat{\mathcal{G}} - f\hat{\mathcal{R}}_{[n,p]} \end{eqnarray} \tag{ 15 }$$

where $\hat{ }$ denotes non-dimensionalization. $\hat{ }$ will not be used in the following for notational convenience.

Finite-element formulation. We use a finite-element formulation to solve this set of equations (12)–(15). The numerical instabilities associated with the presence of drift (convective) term in the current-continuity equations is taken care using the streamline upwind Petrov–Galerkin (SUPG) formulation [38]. The weak form of the problem defined by equations (12)–(15) can be stated as follows.

Given a domain Ω, with boundary ∂Ω, find φ, n, p, X  H¹(Ω), φ = φ_D, n = n_D, p = p_D, X = X_D on ∂Ω_D such that

$$\begin{eqnarray} &&\int_{\Omega} \left(\nabla\cdot(\lambda ^2\nabla\phi) - (n-p)\right)w\,{\rm d}\Omega = 0 \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\ms \int_{\Omega} \left(\nabla\cdot {\bit J_n} - (-f\mathcal{D}+f\mathcal{R}_{[n,p]})\right)(w + q^*)\,{\rm d}\Omega = 0 \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\ms \int_{\Omega} \left(\nabla\cdot {\bit J_p} - (+f\mathcal{D}-f\mathcal{R}_{[n,p]})\right)(w + q^*)\,{\rm d}\Omega = 0 \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\ms \int_{\Omega} \left(\nabla\cdot(\mu_X\nabla X) - f\mathcal{D} - f\mathcal{R}_{[X]} + \mathcal{G} + \mathcal{R}_{[n,p]}\right)w\,{\rm d}\Omega = 0 \end{eqnarray} \tag{ 19 }$$

for all w  H¹(Ω), w = 0 on ∂Ω_D, where H¹(Ω) represents the Sobolev space. φ_D, n_D, p_D and X_D represent the Dirichlet boundary values for the potential φ, charge densities n and p, and exciton density X, respectively. q^* is the SUPG function which modifies the Galerkin test function w to a Petrov–Galerkin test function (w + q^*) [38]. On the remainder of the domain boundary ∂Ω/∂Ω_D, a zero boundary condition for the normal component of ∇φ, J_n and J_p is implicitly applied.

Discretization. In this stage we replace the infinite-dimensional solution space H¹(Ω) with a finite-dimensional approximate H^h(Ω) representing the discrete solution set on a grid. The finite-dimensional counterparts of the solution variables φ, n, p, X are denoted by φ_h, n_h, p_h, X_h. The Galerkin form of equations can be stated as follows.

Find φ_h, n_h, p_h, X_h  H^h(Ω), satisfying the essential boundary conditions such that

$$\begin{eqnarray} &&\fl \int_{\Omega} \Bigl( -\lambda^2\nabla\phi_h\cdot\nabla w_h - (n_h-p_h)w_h \Bigr)\,{\rm d}\Omega = 0 \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\ms \fl \int_{\Omega} \left( n_h\nabla \phi_h \cdot \nabla w_h - \nabla n_h \cdot \nabla w_h - \nabla n_h \cdot \nabla \phi_h q^*_h\right)\,{\rm d}\Omega = \int_{\Omega} (-\mathcal{D}+\mathcal{R}_{[n,p]})(w_h + q^*_h)/\mu_n \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\ms \fl \int_{\Omega} \left( p_h\nabla \phi_h \cdot \nabla w_h + \nabla p_h \cdot \nabla w_h - \nabla p_h \cdot \nabla \phi_h q^*_h\right)\,{\rm d}\Omega = \int_{\Omega} (+\mathcal{D}-\mathcal{R}_{[n,p]})(w_h + q^*_h)/\mu_p \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\ms \fl -\int_{\Omega}\mu_X\left(\nabla X\cdot\nabla w\right)\,{\rm d}\Omega = \int_{\Omega}\left(\mathcal{D} + \mathcal{R}_{[X]} - \mathcal{G} - 0.25\mathcal{R}_{[n,p]}\right)w\,{\rm d}\Omega \end{eqnarray} \tag{ 23 }$$

for all w_h  H^h, where φ_h, n_h, p_h and X_h are the Galerkin approximations to the electrostatic potential, electron density, hole density and exciton density, respectively. The discretized equations (20)–(23) generate a coupled system of non-linear algebraic equations.

Overview. Given the microstructural description of the device, the morphological details are mapped onto a computational mesh. This mesh is used by the excitonic drift–diffusion equation solver to obtain the charge density and potential distribution in the device. The current–voltage characteristic curve is post-processed from the results.

Framework setup and domain decomposition. The drift–diffusion equation solver module is implemented in C++ using an in-house parallelized finite-element library which is based on PETSC [39].

Determining the electrical properties of heterogeneous polymer devices with small feature sizes requires fine mesh resolution. Additionally, the relatively large device dimensions (1000 nm × 100 nm) considered in this study results in large degrees of freedom. Furthermore, we need to solve the non-linear excitonic drift diffusion equations at different values of applied voltages in order to obtain the current–voltage characteristic curve. This necessitates parallelization of the framework for quick simulations. We utilize domain decomposition strategy to partition the computational mesh among the processors of a cluster using ParMETIS [40].

Solvers. The non-linear problem is solved utilizing the 'scalable nonlinear equation solver' (SNES) module of the PETSC library. The linear system at each iteration of the non-linear problem is solved using the MUMPS [41] library.

Scaling analysis. We consider a heterogeneous two-dimensional morphology for the timing study. The non-linear excitonic drift–diffusion equations were solved using Newton's method with relative tolerance for convergence set to 10⁻¹². The code was compiled with optimization options for the compiler set on. The timing tests were performed on the 'Cystorm' cluster at Iowa State University which consists of 3200 computer processor cores and high-speed infiniband interconnection.

Figures 4(a) and (b) show the results of the scaling study for different number of degrees of freedom. The plots allow us to determine the optimum number of processors to use for a problem with specific number of degrees of freedom. A problem size of 1.28 million degrees of freedom solved on around 60 processors would result in maximum speedup. These plots (figure 4) show a reduction in solution time with increase in processors used, and this reduction is improved as the problem size increases.

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The three-dimensional simulations scale similar to their two-dimensional counterparts with some deterioration due to increased bandwidth of the stiffness matrix.

We utilize the computational framework to capture the influence of realistic active layer morphology on the electrical properties of a photovoltaic device. Figure 5(a) shows the vertical cross-section of the active layer of a heterogeneous organic photovoltaic device composed of donor and acceptor materials. The thickness of the active layer is 100 nm. The top and bottom are connected to the anode and cathode, respectively.

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The horizontal length of the microstructure is about 1000 nm. This allows us to analyse the effect of a wide range of morphological feature types on the photovoltaic behavior of the device. The active layer is composed of P3HT as the electron-donating polymer and PCBM as the electron acceptor (see figure 1). The P3HT : PCBM system has been extensively investigated due to its optimal HOMO–LUMO gap, good charge mobilities, and ease of manufacturability [42–45]. In this class of devices light is absorbed by the electron donor to generate excitons⁶. To accurately represent this feature, the exciton generation is limited to the donor region. Light intensity decays with depth due to absorption. The resultant exciton generation rate is shown in figure 5(b). As the light is shining through the transparent anode at the top of the active layer, exciton generation rate $\mathcal{G}$ is maximum near the anode.

We consider the effect of the random and highly interpenetrating structure of the DA interface on the distribution of excitons, charges and electric field under short-circuit conditions.

The ohmic boundary conditions results in maximum electron and hole densities at the cathode and anode, respectively (figures 5(c) and (d)). The electron density decreases from the cathode to the anode throughout the domain, but the decrease is much more pronounced in the donor than the acceptor region. Similarly, hole density values drop from the anode to the cathode with a greater rate of decrease in the acceptor region.

X is maximum in the inner regions of donor material and its value goes to zero while approaching the interface where the excitons dissociate into free charges. An exponential reduction in the incident light available for absorption along the thickness of the active layer results in higher exciton density near the anode at the top.

Figures 5(f) and (g) show the contour of the electron and hole current densities components normal to the electrodes. The electron current density (figure 5(f)) is negligible in the donor region and increases in the acceptor region while going from the anode to the cathode. A similar effect is observed with hole current density (figure 5(g)), where maximum values are observed in donor material close to the anode. The electrons are observed to be more dispersed toward the center of the active layer than the holes. This is due to greater likelihood of exciton dissociation near the anode at the top of the domain. The holes generated at the top of the active layer drift upward toward the anode, while the region in the bottom half remains largely hole free. But electrons created at the top of the domain drift downward toward the cathode and hence are more widely distributed along the thickness.

The spatial variation of electron and hole concentrations results in variation of electric field in the x-direction. Figure 6(a) shows the x-component of the electric field with regions of positive values in the acceptor material and negative values in the donor material which are the electron and hole transporting media, respectively. The y-component of the electric field (figure 6(b)) is an order of magnitude larger than the x-component. The relative variation of the y-component across different materials is less pronounced than the x-component due to the potential created by the difference in work functions of the electrodes. Moreover, a larger value of y-component of the electric field can be seen in the electron rich acceptor regions compared with donor regions. Hence the spatial variation of charge densities across different materials results in variation in electric field, but the effect is more visible for the x-component than the y-component as the internal potential dominates.

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Figure 7 shows the current–voltage characteristic plot obtained by simulating the morphology shown in figure 5(a) over a range of applied voltages. Table A1 in the appendix lists the values used for the various parameters involved in the simulation. The current–voltage curve shown here is similar to typical experimentally obtained curves for P3HT–PCBM based systems [47, 48].

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Table A1. Overview of the parameters used.

Parameter	Symbol	Numerical value	Units
Electron effective density of states	NC	2.5 × 1025	m−3
Hole effective density of states	NV	2.5 × 1025	m−3
Electron zero-field mobility	μn	2.5 × 10−7	m2 V−1 s−1
Hole zero-field mobility	μp	3.0 × 10−8	m2 V−1 s−1
Band gap	Eg	1.34	eV
Photon flux	Γ0	4.31 × 1021	m−2 s−1
Donor relative dielectric constant	D	6.5	—
Acceptor relative dielectric constant	A	3.9	—
Absorption coefficient	α0	2 × 107	m−1
Average exciton lifetime	τX	1 × 10−6	s
Boltzmann constant	kB	1.3806503 × 10−23	m2 kg s−2 K−1
Room Temperature	T	300	K
Elementary charge	q	1.60217646 × 10−19	As

This section analyses the electrical properties of a sawtooth morphology with device dimensions, feature size and interface area identical to the realistic morphology analyzed in the previous section. This provides a means to assess the performance of a realistic heterogeneous polymer solar cell against an idealized morphology.

Figure 8(a) shows a sawtooth morphology with 35 teeth each of donor and acceptor. The device dimensions are maintained same as that of figure 5(a). The number and height of teeth is calculated to get an average feature size of 15 nm and the same interface area. The material parameters and simulation settings are same as that for realistic morphology simulation.

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The charge densities (figures 8(c) and (d)) are observed to be maximum at the respective electrodes (anode for holes and cathode for electrons) and decrease toward the opposite electrode. Away from the electrodes, the maximum values of charge density are found near the DA interface. As shown in figure 8(e), maintaining an exciton generation rate distribution similar to realistic morphology case gives an exciton density distribution with maximum values in the donor region lying between the anode and DA interface.

Figure 9(a) shows the x-component of the electric field calculated under short-circuit conditions. This picture is similar to the results presented by Buxton and Clarke [2]. Similar to the realistic morphology case, the y-component (figure 9(b)) of the electric field is an order of magnitude larger than the x-component.

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The comparison of current density distribution plots for the realistic morphology (figures 5(f) and (g)) and idealized morphology (figures 8(f) and (g)) clearly reveals the superiority of the sawtooth structure which, unlike the heterogeneous structure, allows all the charges generated at the DA interface to be collected at the electrodes. Additionally, full contact of donor material with anode and acceptor with cathode augments the charge collection process even more. Figure 10 highlights this difference in performance between sawtooth and realistic device structures.

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In order to further tune the realistic device morphologies to obtain high short-circuit current densities, we investigate the effect of gradual increase in the feature size while maintaining the same blend volume fraction, on the sub-processes involved in current generation process.

In this section, we study the effect of systematically changing the interface area and in turn the feature size of the devices on photovoltaic properties. A similar study delineating the effect of increasing interfacial length on the current density was undertaken by Buxton and Clarke [2]⁷.

We consider realistic device morphologies as shown in figure 11⁸. for the present analysis. The different morphologies shown in figure 11 show variation in the (a) interfacial area, (b) feature size and (c) electrode contact area (acceptor with cathode and donor with anode) simultaneously. An increase in the interfacial area corresponds to a decrease in feature size. The electrode contact area remains relatively unchanged, as the volume fraction of each material in the blend is same. A large interface area would result in greater opportunity for the excitons to dissociate. However, it also deteriorates the charge transport due to high intertwining of pathways and greater number of isolated islands and dead-ends.

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The material and simulation parameters are identical to the previous section. Figure 12 shows the current–voltage characteristic curves obtained for the morphologies shown in figure 11. This shows that the morphology with highest interfacial area has the best performance suggesting a dominant role played by exciton dissociation rate in deciding the final performance. To further understand the role played by the DA interface, the relative effect of interfacial area on the sub-processes involved in light absorption to charge collection stage (generation, dissociation and relaxation of excitons, charge recombination) is analyzed. Figure 13 shows increasing interfacial area plotted against the integral of $\mathcal{G}$, $\mathcal{R}_X$, $\mathcal{D}$ and $\mathcal{R}$. The overall exciton generation rate $\mathcal{G}$ (figure 13(a)) remains relatively unchanged for different interface areas. This is due to using the same device dimensions and material volume fractions. Figure 13(b) shows a sublinear increase in exciton dissociation rate with increase in interfacial area suggesting a reduced increase in $\mathcal{D}$ for any further increase in the interfacial area. A reverse trend is observed for exciton relaxation rate $\mathcal{R}_X$. The charge recombination $\mathcal{R}$, which limits the device performance, shows an almost linear increase with interface area.

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The increase is charge recombination rate $\mathcal{R}$ is overshadowed by increase in exciton dissociation rate and decrease in exciton relaxation rate $\mathcal{R}_X$, thereby resulting in a net increase in device performance. The charge recombination increases super-linearly with interface area, whereas the increase in exciton dissociation is sublinear, which would lead to diminishing performance improvements with further increase in interface area.

We showcase the capability of the framework to simulate 3D realistic morphologies with figures 14 and 15. Figure 14(b) shows the morphology of the device used in this study. The calculated electron and hole current densities are shown in figures 14(e) and (h), respectively. The cathode and anode facing views are also included to aid visualization. Figure 15 shows the corresponding morphology and current density plots for ideal morphology. As can be observed from figures 14 and 15, the current density for ideal morphology is much larger than that for real morphology due to better charge transport pathways and electrode contact area (cathode contact area for acceptor and anode contact area for donor).

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For real morphology, some features which appear as isolated 'islands' in two dimensions may have direct pathways to electrode in 3D, and in general, 3D structures may provide more number of pathways from charge generation site to charge collection site than an equivalent 2D device structure. Hence 3D simulation helps visualize and analyze the charge transport properties which are not easily seen to 2D simulations.