From the point of EMT, we have considered the active layer of the ternary blend solar cell, as a threecomponent nanocomposite material in which the PCBM polymer is the host and the PM6 fullerene and Y6 nonfullerene acceptors are the first and second hosts, respectively. In this method, considering the weight ratio and size of each guest, an effective dielectric function is calculated for the whole nanocomposite layer, which includes the optical and morphological properties of all three components.
One of the most applicable extensions of the wellknown MaxwellGarnett (MG) theory, is the MaxwellGarnettMie (MGM) model, computes the effective dielectric function, by calculating the electric dipole polarizability of a composite from the Mie scattering theory. MGM theory, considers both extrinsic dynamic and intrinsic confinement effects as shown in Eq. 1[24]:
\(\frac{{{\varepsilon _{eff}}  {\varepsilon _h}}}{{{\varepsilon _{eff}}+2{\varepsilon _h}}}=\frac{{3i{V_1}{\lambda ^3}}}{{16{\pi ^3}\bar {R}_{1}^{3}\varepsilon _{1}^{{3/2}}}}a_{{M1}}^{L}({\bar {R}_1})+\frac{{3i{V_2}{\lambda ^3}}}{{16{\pi ^3}\bar {R}_{2}^{3}\varepsilon _{2}^{{3/2}}}}a_{{M2}}^{L}({\bar {R}_2}){\text{ (1)}}\)
Where\({V_1}\), \({\bar {R}_1}\) and\({V_2}\),\({\bar {R}_2}\) are the volume fraction and mean radius of each type of guest components respectively and \(\lambda\) is the wavelength. \(a_{{M1}}^{1}({\bar {R}_1})\), \(a_{{M1}}^{2}({\bar {R}_1})\) are the first electric Mie coefficient for first and second guest component calculated from Eq. 2.a, b:
\(\begin{gathered} a_{{M1}}^{1}({{\bar {R}}_1})=\frac{{\frac{{n_{{NP}}^{1}}}{{{n_h}}}{\psi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_1})\psi _{1}^{'}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_1})  {\psi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_1})\psi _{1}^{'}(\frac{{2\pi n_{{NP}}^{1}}}{\lambda }{{\bar {R}}_1})}}{{\frac{{n_{{NP}}^{1}}}{{{n_h}}}{\psi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_1})\xi _{1}^{'}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_1})  {\xi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_1})\psi _{1}^{'}(\frac{{2\pi n_{{NP}}^{1}}}{\lambda }{{\bar {R}}_1})}}{\text{ (2}}{\text{.a)}} \hfill \\ a_{{M1}}^{2}({{\bar {R}}_1})=\frac{{\frac{{n_{{NP}}^{1}}}{{{n_h}}}{\psi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_2})\psi _{1}^{'}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_2})  {\psi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_2})\psi _{1}^{'}(\frac{{2\pi n_{{NP}}^{1}}}{\lambda }{{\bar {R}}_2})}}{{\frac{{n_{{NP}}^{1}}}{{{n_h}}}{\psi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_2})\xi _{1}^{'}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_2})  {\xi _1}(\frac{{2\pi {n_h}}}{\lambda }{{\bar {R}}_2})\psi _{1}^{'}(\frac{{2\pi n_{{NP}}^{1}}}{\lambda }{{\bar {R}}_2})}}{\text{ (2}}{\text{.b)}} \hfill \\ \end{gathered}\)
Where \({\psi _1},{\xi _1},\psi _{1}^{'},\xi _{1}^{'}\) are the firstorder RiccatiBessel functions and their derivations respectively.
Using effective medium approach and assuming the blend of Donor/Acceptor1/Acceptor2 nanocomposite as an effective united active layer of the organic solar cell, we use the Koster’s augmented Driftdiffusion, formalism for the active layer of OSC. Using the uniform electric field approximation which is applicable for sufficiently thin active layer thickness, the electrostatic field is simply could be written as in Eq. 3[25].
\(E=\frac{{{V_{app}}+\Delta {\Phi _2}  \Delta {\Phi _1}}}{L}{\text{ (3)}}\)
Where is the active layer thickness, \(\Delta {\Phi _1}\)and\(\Delta {\Phi _2}\)are respectively the workfunction difference between the front and bottom electrode and the active layer. \({V_{app}}\) is the applied voltage between anode and cathode. Figure 2 demonstrates a schematic view of the studied ternary blend organic solar cell.
Following Koster and coworker’s, augmented drift diffusion formalism, the steadystate continuity equation for holes can be written as Eq. 4[26].
\( {V_{th}}{\mu _p}\frac{{{d^2}p}}{{d{x^2}}}+E{\mu _p}\frac{{dp}}{{dx}}=P(E){G_{opt}}  (1  P(E))R{\text{ (4)}}\)
Where \({V_{th}}={\raise0.7ex\hbox{${KT}$} \!\mathord{\left/ {\vphantom {{KT} q}}\right.\kern0pt}\!\lower0.7ex\hbox{$q$}}\) is the thermal voltage, \({\mu _p}\)is the mobility of holes, \({G_{opt}}\) is the optical carrier generation rate and \(P(E)\)is the exciton’s dissociation probability, calculated according to the OnsagerBraun theory, presented in Eq. 5ac.
\(\begin{gathered} P(T,E,{k_f},a)=\frac{{{k_d}}}{{{k_d}+{k_f}}}{\text{ (5}}{\text{.a)}} \hfill \\ {k_d}(T,E,a)=\frac{{3\mu e}}{{4\pi \varepsilon {a^3}}}{e^{  [\Delta \varepsilon (a)/KT]}}\frac{{8\pi \varepsilon {\varepsilon _0}{K^2}{T^2} \times {J_1}[2\sqrt {  2b(T,E)} ]}}{{  2{e^3}E}}{\text{ (5}}{\text{.b)}} \hfill \\ \Delta \varepsilon (a)=\frac{{{e^2}}}{{4\pi \varepsilon {\varepsilon _0}a}}{\text{ (5}}{\text{.c)}} \hfill \\ \end{gathered}\)
In Eq. 5.b. the mobility total mobility of electrons and holes \(\mu ={\mu _p}+{\mu _n}\), \({J_1}\) is the Bessel function of the first kind of order of 1, is the initial pair separation distance of the charge transfer exciton, and \(\Delta \varepsilon (a)\) is the pair binding energy. The bimolecular (or Langevin’s) recombination rate \({\text{R}}\)is given by Eq. 6[27].
\({\text{R=}}\frac{{{\text{q}}\mu (np  n_{i}^{2}){\text{ }}}}{{2\varepsilon }}{\text{ (6)}}\)
Where\({{\text{n}}_i}\)is the intrinsic carrier density of electrons and holes. Finally after solving the continuity equation with applying the boundary conditions, the complete answer to the Eq. 4 could be written as presented in Eq. 7[28].
\(\begin{gathered} {\text{p(x)=[}}{N_V}\exp (\frac{{  {\varphi _2}}}{{q{V_t}}})  \frac{{{N_V}\exp (\frac{{{E_g}+{\varphi _1}}}{{{\varphi _2}}})  \frac{{PGL}}{{E{\mu _p}}}}}{{\exp (\frac{{qEL}}{{KT}})  1}}]+ \hfill \\ {\text{ + }}[\frac{{{N_V}\exp (\frac{{{E_g}+{\varphi _1}}}{{{\varphi _2}}})  \frac{{PGL}}{{E{\mu _p}}}}}{{\exp (\frac{{qEL}}{{KT}})  1}}{\text{ }}\exp (\frac{{qEx}}{{KT}})]+\frac{{PGx}}{{E{\mu _p}}}{\text{ (7)}} \hfill \\ \end{gathered}\)
Where \({N_V}\)is the effective density of states in the valence band. The steadystate continuity equation of electrons could be written in a similar method. Applying an analogous calculation for the electrons, the total current density, which is the sum of the drift and diffusion current density of electron and holes, obtains.
In order to take into account the effect of each component’s optical parameters, we calculated the absorption coefficient of each of them and also the total effective absorption coefficient of active layer via dielectric function of each component and the calculated effective dielectric function respectively. The absorption coefficient in this method could be defines as Eq. 8.
\(\alpha (\lambda )=\frac{{4\pi }}{\lambda }{\left\{ {  \operatorname{Re} ({\varepsilon _{eff}})+\frac{1}{2}\left[ {\operatorname{Re} (\varepsilon _{{_{{eff}}}}^{2})+\operatorname{Im} (\varepsilon _{{_{{eff}}}}^{2})} \right]} \right\}^{\frac{1}{2}}}{\text{ (8)}}\)
In table.1 we have presented the experimental values that we have used for mobility of electron and hole of PM6:Y6: PCBM with different values of Y6 percent.
Table 1
electron and hole mobility for different values of Y6 percent[29].
Y6 (%)

\({\mu _n}({\raise0.7ex\hbox{${{m^2}}$} \!\mathord{\left/ {\vphantom {{{m^2}} {V.s}}}\right.\kern0pt}\!\lower0.7ex\hbox{${V.s}$}})\)

\({\mu _p}({\raise0.7ex\hbox{${{m^2}}$} \!\mathord{\left/ {\vphantom {{{m^2}} {V.s}}}\right.\kern0pt}\!\lower0.7ex\hbox{${V.s}$}})\)

10

\(4.2 \times {10^{  14}}\)

\(2.02 \times {10^{  8}}\)

20

\(1.42 \times {10^{  12}}\)

\(1.82 \times {10^{  8}}\)

30

\(8.01 \times {10^{  9}}\)

\(1.5 \times {10^{  8}}\)

40

\(2.1 \times {10^{  9}}\)

\(2.82 \times {10^{  8}}\)

50

\(2.82 \times {10^{  8}}\)

\(2.82 \times {10^{  8}}\)

60

\(2.09 \times {10^{  8}}\)

\(2.82 \times {10^{  8}}\)

70

\(2.09 \times {10^{  8}}\)

\(1.41 \times {10^{  8}}\)

At the next step, we investigated the effects of changes in the percentage of nonfullerene acceptor on the quantum efficiency of the studied structure.
\(\Delta EQ{E_x}=\frac{{EQ{E_x}  EQ{E_0}}}{{EQ{E_0}}}{\text{ (9)}}\)
Where is the percent of Y6 as nonfullerene acceptor and \(EQ{E_0}\)is the external quantum efficiency of the system of binary PM6: PCBM blend.
In order to consider the physical constraint due to the size of the domains by weight percentage, we considered a slab with dimensions of \(100 \times 100 \times 100\)nm3 and calculated the distribution of domains with maximum average dimensions of PCBM acceptor with a fixed weight ratio for Y6 acceptor at 0.5.