Organic solar cells

Riccardo Volpi and Mathieu Linares

DOI: 10.1039/9781782626862-00001

1 Introduction

As climate change and energy sufficiency are progressively becoming more pressing issues, environmentally friendly and cheap energy sources need to be developed. Solar energy production, with inorganic solar cells, has progressed greatly in recent years and is already able to compete with traditional energy sources. Photovoltaic solar cells convert photons into electric current. Although traditional photovoltaic technology based on inorganic materials has become commercially successful, it faces some limitations that can be overcome by organic solar cells. Organic solar cells are low-cost and easy to process, furthermore they possess innovative properties being potentially lightweight, flexible, and transparent. However, organic solar cells still have lower efficiencies and shorter lifetimes than traditional inorganic solar cells. While the best organic solar cells have reached around 11% efficiency, the best single junction crystalline silicon solar cells and thin film CdTe cells have efficiencies of around 25% and 22%, respectively. Furthermore, the lifetime of organic solar cells is still short in comparison to the lifetimes of inorganic solar cells, so stability challenges must be addressed for organic solar cells to compete with conventional photovoltaics on the market. The efficiency of an organic solar cell is determined by the efficiency of the different steps from photon absorption to charge collection, as shown in Fig. 1. Photons are absorbed in either the acceptor or the donor phase with efficiency ?abs, generating excitons (1). These excitons then either diffuse toward the interface and form a Charge Transfer (CT) state (2) or decay (6). The CT state is defined as an electron-hole pair Coulombically bound, composed usually by an electron on an acceptor molecule and a hole on a donor molecule. In an organic solar cell CT states (3) are formed at the interface from exciton dissociation. The efficiency of the excitons forming CT states is ?ex, and it is high when the distance to the interface is small. The CT states then split (4) into free electrons and holes with efficiency ?diss. After the charge carriers are freed, they may still move back to the interface and recombine in a CT state (8), or even to the ground state (7) and (9). ?trans is the efficiency of the free charge carriers being collected at the electrodes (5). The product of these different efficiencies gives the external quantum efficiency (EQE), the ratio of charge carriers collected to the number of incoming photons.

EQE(E) = ?abs?ex?diss?trans (1)

The efficiency of organic solar cells is greatly impacted by the processes that occur at the interface between the donor and acceptor. The electron and hole have a strong Coulombic attraction, which must be overcome for the charges to separate. Organic solar cells have much lower dielectric constants and more localized electronic states than inorganic solar cells, so excited states are localized and the Coulombic barrier to overcome for the charges to dissociate is large.

The amorphous nature of these materials make analytical studies difficult and theoretical investigations of mobility and charge transport are to a large extent based on simulations: drift diffusion and Kinetic Monte-Carlo simulations (KMC). Drift-diffusion models employ a classical picture, they are very suitable to simulate the whole solar cell and give some macroscopic information. Kinetic Monte-Carlo simulations have potentially the capability to model the quantum phenomena happening at the nanoscale, but the more details are included in the KMC scheme the more computational effort is required. Detailed KMC schemes are thus mainly limited in studying some interesting portions of a solar cell. Several studies employing both approaches showed the significant contribution of morphology to the efficiency of solar cells. Different structures for organic solar cells have been researched, the simplest consisting of a single layer of an organic semiconductor between two electrodes. However, solar cells with this structure have low efficiency, and the performance of the device is improved by a bilayer structure of two organic materials: a donor and an acceptor materials. With this design, a trade-off must be made between light absorption and exciton dissociation. The thicker the layers are, the more light will be absorbed, increasing the efficiency of the solar cell. However, the excitons formed in the single material, will have in average a greater distance to the donor-acceptor interface, where they can dissociate in a CT state. If the domain size is too large, the exciton will decay before reaching the interface, thereby lowering the efficiency of the solar cell. A solution to this problem is to combine the donor and acceptor phases so that the distance to the interface is short even with thick films. This can be best achieved with interdigitated structures, providing good exciton dissociation and at the same time a clear path to the electrodes for the free charge carriers. The interdigitated interface is very promising but also very difficult to obtain in practice. Another type of interface commonly used in the lab employs a blend of donor and acceptor materials forming a three-dimensional interpenetrated structure called bulk heterojunction (BHJ) solar cell.

In the present chapter we will focus on the KMC approach used to model charge transport in organic materials. The properties of organic materials are different from those of crystals; their intrinsic disorder tends to localize the charge carriers on one or few molecules. The conduction in these materials is therefore temperature activated, in contrast to the band conduction of crystalline materials. Two methods are predominantly used in literature to study charge transport using KMC simulations: the Miller-Abrahams and the Marcus hopping rates. The Miller-Abrahams formula is one of the simplest ways to couple temperature to the hopping rate. This is achieved by means of a Boltzmann factor helping to overcome the energy barrier for the transport. The Marcus hopping rate can be derived from the Fermi golden rule and takes into account also the reorganization energy after each hop.

In this book chapter, we will show how to model the steps (3), (4) and (5) of Fig. 1, namely, how the CT state split and the charges are subsequently transported to the electrodes. We will thus not consider the transport of excitons or the coupling between the excited state, the CT state and the ground state and we refer the reader to the appropriate literature. After a brief discussion on the method to obtain realistic interfaces with atomistic details, we will focus on the transport through Marcus equation and on the way to calculate the different parameters involved in it. We will also analyze the effect of the electric field on CT state splitting and conduction of free charge carriers. In particular, in order to optimize the device efficiency, we will illustrate how to choose the electric field to favor simultaneously both processes, if this is possible.

2 Morphology

The morphology of the donor-acceptor interface in a BHJ organic solar cell has a large impact on the efficiency of the solar cell. Excitons can only diffuse 10-20 nm before decaying, so the donor and acceptor should be sufficiently mixed so that the excitons can reach the interface before decaying. However, once separated, the charge carriers need pathways to their respective electrodes. If, for example, an electron is in an acceptor domain that is completely surrounded by the donor, then the electron will have no path to the electrode and it will eventually decay through recombination. In a bilayer structure, there is always a pathway for free charges to travel to the electrode, but many excitons will decay before they can reach the interface and thus have a chance to split. In the bulk heterojunction structure, however, it is much easier for excitons to reach the interface before decaying, but some of the charges will get trapped in domains that have no path to the electrode or recombine with other charges when traveling by an interface. Two types of recombination can occur: geminate or nongeminate. Geminate recombination is when two charge carriers resulting from the absorption of the same photon recombine (7 in Fig. 1). Nongeminate recombination (9 in Fig. 1) occurs when two free charge carriers originating from different photons recombine with each other at the interface. When there is high interpenetration, free charge carriers are likely to travel close to an interface and recombine with other free charge carriers, increasing the non-geminate recombination rate. It is important to optimize the interpenetration of the donor and acceptor materials in order to extract as many charge carriers as possible from the photogenerated excitons.

Charges hop between molecules or polymer segments, and this hopping occurs with a certain probability, making probabilistic methods such as KMC appropriate for modeling charge transport. In early works the molecules of an organic material were modeled only as a lattice of molecular sites, and the structure of the molecules was neglected. More recent works have improved upon these models by including an atomistic description of the molecules. To study the complex interplay of geminate and non-geminate recombination and thus obtain a meaningful simulation of charge transport in the solar cell, a realistic morphology is needed. Several methods can be used to model the interface between donor and acceptor materials. As a first approximation, the crystal structure of the two materials can be brought into contact and the interdistance can be optimized without relaxing the molecule inside each phase. In order to add more disorder, it is then possible to relax the interface by using molecular dynamics simulations like it has been done in several studies. However, if this technique will allow to introduce disorder and create mixing at the interface, this latter will remain quite planar and it will not reach a real three-dimensional interpenetrated structure. In order to obtain a realistic BHJ interface we are currently working on another approach. The idea is to start from volumes obtained with the Ising model developed by Heiber et al. with different degrees of intermixing, and subsequently fill these volumes with donor and acceptor molecules as illustrated in Fig. 2. From those initial structures, molecular dynamics in the NPT ensemble can be performed to relax the different boxes that can be then used for Kinetic Monte Carlo simulations.

3 Transport through Marcus equation: kinetic Monte Carlo

Kinetic Monte Carlo simulations are one of the prominent tools for the simulation of charge carriers (single or multiple) in organic materials. In the notation used in this chapter, a particular molecular orbital level in the system is identified by an upper case letter M = (i,m), i.e. the orbital level m of the molecule i. The molecular orbital M will always implicitly belong to the molecule i, unless explicitly stated. In the same way, the orbital N = (j, n) belongs to the molecule j (where i and j are different molecules). The flowchart of the KMC algorithm is presented in Fig. 3. Since organic materials exhibit usually a disordered amorphous structure, it is common practice in the literature to obtain a realistic structure as outcome of a MD simulation (see previous Section 2). Once the structure has been defined, every charge carrier a involved in the simulation is placed in its respective initial positions (step 1). At every time step the hopping rates for all the charge carriers are calculated with Marcus formula (step 2). If a charge carrier is situated on the molecular orbital M, its probability to hop to the orbital N is

[MATHEMATICAL OMITTED] (2)

Marcus formula involves temperature T (through the Boltzmann constant kB) and three main physical quantities depending on the particular hop: the transfer integral HNM, the site energy difference ?ENM and the reorganization energy ?NM. The evolution of the system is determined thanks to a weighted random selection algorithm (WRS), in which the probability of selecting a particular hop depends on its associated weight. First, the hopping charge carrier is selected using as weight for every charge carrier a the total escape rates [MATHEMATICAL OMITTED] [(step 3). This selects the charge that is jumping, thus identifying the origin of the hop (step 4). Let us call c the selected hopping charge and let us assume such charge is placed on the orbital M. Now, among all the possible hops that charge c can make, we will select the one determining the evolution of the system through another WRS using as weights the Marcus rates wMN (step 5). This hop will be considered in the simulation as happening after a time ?t sampled from an exponential distribution ~RM exp(– RM?t), with RM being the total escape rate from the orbital M, i.e. RM = [sN WMN (step 6).

At step 7 the system is updated with the selected hop and the time of the simulation is increased by the estimated time for such hop calculated at step 6. The simulation can be stopped due to several stopping criteria. A stopping criterion can be the simulation time or the number of hops. Also positional criteria can be used, like for example checking if a charge carrier is passing through a particular plane in space. If no stopping criterion is respected (step 8), we go back at step 2 and continue the simulation. Periodic boundary conditions can be used in x, y and/or z. In the following of this section, we will explain how the three main parameters in Marcus equation (the transfer integral HNM, the site energy difference ?ENM and the reorganization energy ?NM) can be calculated with a combination of quantum chemistry and classical mechanics tools.

3.1 Site energy difference

The site energy difference is the difference in energy of the charge carrier before and after the jump. The energy of a charge carrier can be calculated with several methods. For instance, it can be sampled from a Gaussian distribution with a certain width reflecting the energetical disorder of the system. This technique was initially proposed in the pioneering work by Bassler and it allows to phenomenologically recover the broadening of the density of states (DOS) of a charge carrier in the bulk. This approach allowed to gain great insight into the charge transport mechanisms of organic materials and also to achieve qualitatively the correct mobility temperature dependence. However, to achieve the correct mobility field dependence instead, the correlation of the energy landscape is mandatory as demonstrated by Novikov and Vannikov and subsequently showed in a KMC scheme.

To model a realistic correlation of the energy landscape in the bulk, it is possible to consider in a more sophisticated way the energy of a charge placed on the molecular orbital M as composed by four terms

[MATHEMATICAL OMITTED] (3)

[member of] M is the energy of the orbital M = (i, m]) and the [+ or -] sign depends on the type of charge carrier, if the energy of the orbital has to be added (electron) or subtracted (hole). This can be evaluated either at the semi-empirical or quantum mechanical level for each and every molecule in the box in order to consider the energetic shift due to specific conformation. The other terms represent the effect of the environment and they are depending on the positions and the quantum states of the molecules around. The array s = (s1, s2, ..., si, sj, ...) represents the state of the system and it is composed by the quantum states of all the molecules. After every charge hopping, the state of the system will change. If for example an electron jumps from the LUMO (Lowest Unoccupied Molecular Orbital) of molecule i to the LUMO of molecule j, the state passes from s1 = (0, 0, ..., LUMO, 0, ...) before the jump, to s2 = (0, 0, ..., 0, LUMO, ...) after the jump. The term Eperm-eli(s) considers the Coulombic interactions with permanent atomic charges within a cut-off distance, so that a charge carrier located on molecule i interacts only with the molecules inside its interaction set I(i), i.e. the set composed by the molecules nearer than the cut-off (Fig. 4). The interaction of the charge carrier with the permanent atomic charges of the surrounding molecules can be expressed as

[MATHEMATICAL OMITTED] (4)

where [member of]0 is the vacuum dielectric constant, h is a molecule belonging to the interaction set I (i) and a, b are atoms. Every atom a has a position [??]a and a permanent atomic charge qa(si) (which depends on the quantum state of molecule i).