Exciton dissociation in organic solar cells: An embedded charge transfer state model

- We've added paragraph 5 to the introduction, as follows: A first analysis of charge injection was presented in our previous work \cite{chika2022model}. Here we enlarge our approach by considering several types of recombination processes, various type of environments, attractive or repulsive potential and by analysing more deeply the process of injection and its interdependence with phonon emission on the CTS

-We've added paragraph 2 to the section 2.3, as follows: Several mechanisms can lead to the genimate recombination between the electron in the acceptor side and the hole left in the donor side. This can be for example photon emission or multiple emission of low energy phonons which lead to a transfer of the electron into localized states near the interface and then to recombination with the hole [14].

-We've added paragraph 1 to the section 2.4, as follows: The Hamiltonian $H$ of our model takes into account a tight-binding hopping model for the electron and includes the essential coupling with local vibration modes, as is common for molecular solids \cite{bredas2004charge}

-We've added the final paragraph to section 2.4, as follows: As stated above these processes can be represented by a coupling to a continuum of states and in the formalism used here we just need to model the self-energy $\Delta_R (z)$ that represents this coupling to a continuum. We consider the cases where the continuum is a wide band ($ \Delta_R(z) = -i \Gamma_R $) and narrow band as discussed in Section 5.2. All energies are expressed in units of $J$, which is typically of the order of $0.2$ eV. All parameters are chosen to fall within a realistic experimental range \cite{zheng2019charge,d2016electrostatic,faber2011electron,castet2014charge,antropov1993phonons,d2014electronic,richler2019influence}.

-We've added the final paragraph to section 5, as follows: Note that the parameters of the Hamiltonian are given in units of $J$ which is typically of the order of $0.2$ eV. All parameters are chosen such that they are in a realistic experimental range to model the prototypical PCMB and C60 acceptor systems \cite{zheng2019charge,d2016electrostatic,faber2011electron,castet2014charge,antropov1993phonons,d2014electronic,richler2019influence}.These molecular solids have electronic bands which are narrow (typically 4J which is less than one eV) and have molecular vibration modes of high frequency which can be of the order of 0.1/0.2 eV.

-We've added the final paragraph to section 6, as follows: In an even higher energy range, we find a tunneling regime. For the electron to be injected into the environment beyond the CTS, its initial energy \(\varepsilon_I\) must first be reduced by exciting phonons on the CTS to get in resonance with the acceptor band. This situation is analogous to the crossing of a potential barrier, and the concept of Wigner time for barrier crossing can be applied here. It indicates that in this regime the electron spends more time on the CTS and can therefore be more sensitive to recombination. Additionally, as the density of states on the CTS decreases at high energies \(\varepsilon_I\), the injection time from the donor increases, according to Fermi's golden rule. Thus, in this tunneling regime, the residence times of the electron on the donor side and on the CTS are longer. Therefore if recombination processes are fast enough, they will reduce the quantum yield of the injection and possibly lead to a cold CTS. In the wide band recombination model the time for recombination is constant and will always become smaller than the Wigner time for residence on the CTS when \(\varepsilon_I\) increases. Yet in the narrow-band recombination model, the recombination time also increases with \(\varepsilon_I\). Therefore, once injected into the CTS, the electron may still have a significant injection yield on the acceptor side (see Figure \ref{figenvirV0dos} b))

-We've added the final paragraph to section 7, as follows: This study shows that the coupling between electrons and vibrational modes affects quantum efficiency, introducing the concepts of hot and cold charge transfer states. Our findings indicate that interface parameters are crucial for optimizing cell performance and that the interface's impact is more significant than environmental variations beyond the CTS. In addition, the electrostatic potential can play a moderate role in the injection process, even when it is attractive and creates localized bound electron-hole states at the interface. The charge injection process is described as a transfer into a continuum at an energy \(\varepsilon_I\). When \(\varepsilon_I\), is in the bottom part of the continuum an efficient resonant injection regime occurs with a cold CTS. At higher values of \(\varepsilon_I\) the electron must first loose energy by emitting phonons before going in the acceptor side. In this tunneling regime the electron spends more time on the donor orbital and then on the CTS site and the recombination can be favored.
Although a detailed comparison with a given system is beyond the scope of this work, we believe that the present model offers a simplified but comprehensive approach to advanced analyses of experimental results \cite{bassler2015hot}. It should be helpful for a better understanding of the exciton dissociation process and for finding ways to improve the organic solar cells. The present embedded charge transfer state model could still be improved by considering several orbitals and several modes per site and even finite temperature. We emphasize that the modeling of the dynamics of the recombination process is also an essential ingredient.