The Theoretical Influence of the Difference Between the LUMO Energy Levels of Donor and Acceptor in Organic Photovoltaic Triplejunction Solar Cells

In organic photovoltaic solar cells, light absorption does not immediately lead to free charge carriers. Instead, an exciton is created. The highest efficiency is reached when the lowest unoccupied molecular orbital (LUMO) of the donor is as close as possible to the LUMO of the acceptor. However, a necessary condition for efficient dissociation of the created excitons is that the difference between the LUMOs of donor and acceptor is higher than the exciton binding energy. The value of the exciton binding energy in different materials is a subject of discussion. The excess of this necessary minimum of the LUMO-difference corresponds with an energy loss. Moreover, it is often not possible to optimize suitable material combinations for organic photovoltaic cells to an ideal (low) LUMO difference. Another energy loss in organic solar cells is caused by their narrow absorption windows, compared to the absorption band of inorganic solar cells. A way to capture a wider band of the solar radiation is using more solar cells with different bandgaps in a row. In this article, we study three organic cells in a row, i.e. a triple-junction. More specifically, we study the theoretical influence of the difference between the LUMO energy levels of donor and acceptor for an organic triple-junction solar cell. We study as well the monolithic as the stacked configuration.


Introduction
A characteristic of organic solar cells is th eir narrow absorption w indow, compared to the absorption band of inorganic sem iconductors [1].A possible way to capture a wider band of the solar spectrum -and thus in creasing the power con version efficiency -is using m ore solar cells with different bandgaps in a row, referred to as a m ulti-junction solar cell.In this article, we will focus on triple-junction solar cells, i.e. three cells in a row.The absorber of the first single solar cell in such a triple-junction cell has a large bandgap E g1 .High-energy photons with an energy hν > E g1 are absorbed by the first cell.The second cell, with a lower bandgap E g2 , absorbs the m iddle-energy photons with energy between E g1 and E g2 .The third cell absorbs the low-energy photons between E g2 and E g3 (Figure 1).In this configurati on, the photon energy is used m ore efficiently: the voltage at which electrical charge is collected in each subcell is closer to the energy of the photons absorbed in that subcell.In the ideal configuration, the subc ells are electrically separated.This is called the stacked or 6terminal configuration (Figure 1a).However, experimental and commercial multi-junction solar cells are usually of the monolithic type (Figure 1b).This means that they are not only optically in series, but also electrically in series.This configuration will never reach an effici ency that is higher than that of a stacked (6-terminal) triple-junction cell, b ecause all single cells canno t be operating at their optim al working point at the same time (unless they have an equal maximum-power current).

Assumptions
The active material in a single organic bulk heteroj unction solar cell consists of an interpenetrating network of an electron accepto r (e.g.fullerene de rivatives) and an electron dono r (e.g.conjugated polymers), sandwiched between two electrodes with different work functions.The optical bandgap E g is defined as the difference between the lowest u noccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) of the absorber material.
We consider a 6 -terminal triple-junction solar cell, consisting of three single organic cells.We assume that in each sing le cell, only one m aterial absorbs light.Usually, most of the light is abso rbed by the dono r; this is the case we will con sider here onwards.Because we assum e full absorp tion in each subcell, we neglect interference and optical coupling of the subcells, thu s overestimating the efficiency potential.The organic cell with the widest absorber bandgap is at top (at the side of the sun), thus E g1 > E g2 > E g3 .The distance between th e HOMO of th e donor and the LUMO of acceptor is considered as the therm odynamic limitation of the useful energy [2].We call this value the interf ace bandgap E i .For an organic solar cell with ohm ic contacts, the ope n circuit voltage V oc is linearly dependent on the interface bandgap E i .For a cell with non-ohm ic contacts, the V oc is dependent on the work function difference of the electrodes.In these calculations, we assume a cell with ohmic contacts.
For our simulation, the following funda mental assumptions are m ade about the stacked triplejunction cell (Figure 1a): (i) every photon with energy hν higher than the bandgap E g1 is absorbed by the first cell and leads to a useful energy E i1 .This assumption im plies that each absorbed photon eventually leads to a free electron and a free hole, w ith an energy difference of E i1 between them.(ii) every photon with energy hν between E g1 and E g2 is absorbed by the second cell and leads to a useful energy E i2 .(iii) every photon with energy hν between E g2 and E g3 is a bsorbed by the th ird cell and leads to a u seful energy E i3 .(iii) photons with energy hν lower than E g3 are fully transmitted.The maximum efficiency η max is therefore given by: with N(E) the incident photon flux.For all our sim ulations, we use the AM 1.5 experimentally measured solar spectrum [3].In this ideal s cenario, the open circuit voltage V oc of the subcells will b e given by E ij /q (j=1,2,3) with q the electric charge.The fill factor FF of the subcells is assumed to equal unity, as well as the ex ternal quantum efficiency EQE of the first cell for wavelengths below the cutoff wavelength λ g1 (corresponding with E g1 ).Similar conditions apply to the second and third cell.
In a monolithic triple-junction solar cell (Figure 1b), the individual cells are electrically connected in series.This means that the total voltage over the cell is the sum of the voltages over each indiv idual cell, and thus equals the sum of the interface ba ndgaps of the single cells.Furtherm ore, the sam e current flows through all single cells.Hence, the m aximum efficiency η max for a monolithic organic triple-junction cell is given by: with min(x,y,z) the minimum of x, y and z.The open circuit voltage V oc of the whole m onolithic cell will be giv en by ( E i1 +E i2 +E i3 )/q, the fill factor FF equals unity, as does the external quantum efficiency EQE for wavelengths below the cut-off wavelength.
In organic bulk heterojunction solar cells, light absorption does not immediately lead to free charge carriers.Instead, an exciton is created.In an ideal scenario, the highest efficiency is reached when the LUMO of the donor is as close as possible to the LUMO of th e acceptor.However, a necessary condition for efficient dissociation of the created excitons is that the difference between the LUMOs of donor and acceptor (ΔLUMO) is higher than th e exciton binding energy [4].The value of the exciton binding energy (and the minimal ΔLUMO) in different materials is a subject of discussion, but a value of 0.3 eV was put forward as an empirical threshold necessary for exciton dissociation [5].The excess of this necessary minimum of the LUMO-difference corresponds with an energy loss.
In the next section, we calcu late the theoretical influence of the diff erence between the LUMO energy levels of donor and accep tor for an org anic stacked and m onolithic triple-junction solar cell.The absolute value of the maximum efficiency is only relevant for academic purposes.It is the relative difference between the efficiencies that is impor tant in quantif ying the inf luence of the LUMO differences.

Results
To study the influence of ΔLUMO, we calculate the m aximum efficiency in this ideal scenario by changing this parameter, and determ ine for each ΔLUMO the optim al bandgaps for the different subcells.First, we only change ΔLUMO 1 (the ΔLUMO of the firs t subcell) and k eep ΔLUMO 2 and ΔLUMO 3 constant at 0.3 eV (the empi rical threshold necessary for exciton dissociation).If there is no energy difference between the LUMOs of the first s ubcell, the maximum efficiency reaches 62% and 61% for a stacked and monolithic configuration respectively (Figure 2a).The efficiency at ΔLUMO 1 = 0.3 eV, the m inimum threshold for exciton dissociati on, is 56% and 55% respect ively, a decrease of 10% relative compared to no LUMO difference.The e fficiency decreases 1 to 3% relative per 0.1 eV.This relative decrease is higher for lower values of ΔLUMO 1 .The optimal bandgap E g1 increases with increasing ΔLUMO 1 for both the stacked and the m onolithic configuration.The higher the LUMO difference, the sm aller the part of the incom ing spectrum that is bein g absorbed.This reduces the relative decrease per 0.1 eV.The op timum of all three b andgaps increase with higher ΔLUMO 1 .This was to be expected.Ind eed, a h igh ΔLUMO 1 of the f irst subcell will lo wer significantly the u seful energy of the absorbed photons in this first subcell.This is compensated by increasing E g1 .As a result, a broader part of the solar spectrum is transmitted to the other two subcells, leading to a rearrangement of the optim al bandgaps of those subcells to hi gher values.The maxim um efficiency will never decrease below 49.5%, because this is the efficien cy of a ta ndem cell (i.e. a m ulti-junction with two subcells) where both ΔLUMOs are 0.3 eV.The bandgap of the first solar cell will then be that big that it will no longer absorb any photons and the triple-junction will act as a tandem cell.
We now consider the influence of ΔLUMO 2 (with ΔLUMO 1 = ΔLUMO 3 = 0.3 eV).The efficiency drops from 64% / 61% at 0 eV to 56% / 55% for 0. 3 eV and 50% / 43% for 1.0 e V for the stacked / monolithic configuration respectively (Figure 2b).We notice a sharp decline in the beginning which decreases for higher ΔLUMO 2 values.The explanation is analogous as for ΔLUMO 1 .For higher ΔLUMO 2 values, this decrease dim inishes fast.Analogous conclusions as for ΔLUMO 1 can be drawn for the optim al bandgaps: the ideal bandgap of the second subcell in creases with higher ΔLUMO 2 values to co mpensate for the en ergy loss caus ed by the LUMO difference.As a result, the o ptimal bandgap of the first subcell decreases whereas E g3 increases.This reduces the influence of the second (less efficient) subcell.At high ΔLUMO 2 values, the optimal values of E g1 and E g2 coincide, reducing the triple junction to a tandem cell.Analogous conclusions can be drawn for ΔLUMO 3 (Figure 2c).

Figure 1 .
Figure 1.(a) A stacked or 6-term inal triple-junction solar cell: the first single cell absorbs photons with energy higher than E g1 .The second and third cell absorb photons with energy between E g1 and E g2 , respectively, E g2 and E g3 .Photons with energy below E g3 are no t absorbed.The three su bcells are electrically s eparated.(b) A m onolithic or 2-terminal triple-junction solar cell: the single cells are electrically connected in series.