Charge-generating mid-gap trap states define the thermodynamic limit of organic photovoltaic devices

Detailed balance is a cornerstone of our understanding of artificial light-harvesting systems. For next generation organic solar cells, this involves intermolecular charge-transfer (CT) states whose energies set the maximum open circuit voltage VOC. We have directly observed sub-gap states significantly lower in energy than the CT states in the external quantum efficiency spectra of a significant number of organic semiconductor blends. Taking these states into account and using the principle of reciprocity between emission and absorption results in non-physical radiative limits for the VOC. We propose and provide compelling evidence for these states being non-equilibrium mid-gap traps which contribute to photocurrent by a non-linear process of optical release, upconverting them to the CT state. This motivates the implementation of a two-diode model which is often used in emissive inorganic semiconductors. The model accurately describes the dark current, VOC and the long-debated ideality factor in organic solar cells. Additionally, the charge-generating mid-gap traps have important consequences for our current understanding of both solar cells and photodiodes – in the latter case defining a detectivity limit several orders of magnitude lower than previously thought.

Substrate preparation: Commercial patterned ITO coated glass substrates from Ossila were used for all devices in this work. All the substrates were cleaned in an Alconox (detergent) aqueous solution bath at 60 °C, followed by sequential sonication in deionize (DI) water, acetone and 2-propanol for 10 minutes each. The cleaned substrates were dried with nitrogen and then treated in UV-Ozone cleaner (Ossila, L2002A2-UK).
Electron/Hole Transport Layer (ETL/HTL) deposition: Solar cells were fabricated with either a conventional or inverted architecture. For the conventional devices, PEDOT:PSS was used as the HTL. A PEDOT:PSS solution was first filtered through a 0.45 μm PVDF filter, then it was spin-coated (6000 rpm for 30s resulting in a thickness of 30 nm) onto ITO substrates and annealed at 155 °C for 15 minutes. For the inverted devices, ZnO was used as the ETL. A ZnO solution was prepared by dissolving 200 mg of zinc acetate dihydrate in 2-methoxyethanol (2ml) and ethanolamine (56µl). The solution was stirred overnight under ambient conditions and was spin-coated onto ITO substrates (4000 rpm resulting in a thickness of approximately 30 nm). The substrates were annealed at 200 °C for 60 minutes.

Active layer and top electrode deposition
The deposition methods of the active layers are described below for each sample. All top electrodes were deposited by thermal evaporation under a vacuum of 10 Tor with an appropriate mask (from Ossila) to define a 0.04 cm cell area for each Pixel.
BQR:PC70BM devices were fabricated with a conventional architecture (ITO/PEDOT:PSS/BQR:PC70BM/Ca/Al). For as cast devices, BQR and PC70BM were dissolved in toluene (24 mg/ml with the donor:acceptor ratio of 1:1) and stirred at 60 °C for 3 hours. Then BQR:PC70BM solution was spin coated (1000 rpm) on the PEDOT:PSS layer to achieve a film thickness of 100 nm. For solvent annealed (SVA) devices, the BQR:PC70BM films were further exposed to a Tetrahydrofuran (THF) environment in a closed petri dish for 20s and then thermally annealed (90 °C) for 10 mins. For both SVA and as cast devices, 20 nm of calcium (Ca) and 100 nm of Aluminium (Al) were evaporated as the top electrodes. The solution was spin-coated using a spin rate of 800 rpm to obtain an active layer thickness of 90 nm. 7 nm of MoO3 and 100 nm of Ag were then evaporated as the top electrode.
PCDTBT:PC70BM devices were fabricated with a conventional architecture (ITO/PEDOT:PSS/PCDTBT:PC70BM/PDINO/Ag). PCDTBT and PC70BM were dissolved in Dichlorobenzene (DCB) with the donor:acceptor ratio of 1:4, and the thicknesses of the active layers were adjusted by changing the concentration of the solution and the speed of spincoating (30 mg ml -1 DCB solution with 1500 rpm for 54 nm active layer, 40 mg ml -1 DCB solution with 1500 rpm for 85 nm active layer, 40 mg ml -1 DCB solution with 1000 rpm for 105 nm active layer, 40 mg ml -1 DCB solution with 600 rpm for 155 nm active layer, 50 mg ml -1 DCB solution with 600 rpm for 185 nm active layer, 60 mg ml -1 DCB solution with 600 rpm for 315 nm active layer, 60 mg ml -1 DCB solution with 400 rpm for 585 nm active layer). 10 nm of PDINO was cast on the active layer from a methanol solution (1 mg ml -1 ), then 100 nm of Ag was deposited on the PDINO to form a cathode.
PM6:Y6 devices were fabricated with an inverted architecture (ITO/ZnO/PM6:Y6/MoO3/Ag). PM6:Y6 was dissolved in a CF solution (14 mg ml -1 with 0.5 vol.% CN) with a donor:acceptor ratio of 1:1.2, and spin-coated (3000 rpm) on ZnO to form 100 nm film. The cast active layers were further treated with thermal annealing at 110 o C for 10 min. 7 nm of MoO3 and 100 nm of Ag were evaporated as the top electrode.
PBDB-T:ITIC devices were fabricated with an inverted architecture (ITO/ZnO/PBDB-T:ITIC/MoO3/Ag). PBDB-T: ITIC was dissolved in a CB solution (14 mg ml -1 with 0.5 vol.% DIO) with a donor:acceptor ratio of 1:1, and spin-coated (800 rpm) on ZnO to form 100 nm film. The active layers were further treated with thermal annealing at 100 o C for 10 min. 7 nm of MoO3 and 100 nm of Ag were evaporated as the top electrode.
PBDB-T:PC70BM devices were fabricated with an inverted architecture (ITO/ZnO/PBDB-T:PC70BM/MoO3/Ag). PBDB-T:PC70BM was dissolved in a CB solution (14 mg ml -1 with 3 vol.% DIO) with a donor:acceptor ratio of 1:1.4, and spin-coated (1000 rpm) on ZnO to form 100 nm film. Then the as-cast films were rinsed with 80 μL of methanol at 4000 rpm for 20 s to remove the residual DIO. 7 nm of MoO3 and 100 nm of Ag were evaporated as the top electrode.
PTB7-Th:PC70BM devices were fabricated with an inverted architecture (ITO/ZnO/PTB7-Th:PC70BM/ MoO3/Ag). PTB7-Th:PC70BM was dissolved in a CB solution (14 mg ml -1 with 3 vol.% DIO) with a donor:acceptor ratio of 1:1.5, and spin-coated (600 rpm) on ZnO to form 100 nm film. Then the as-cast films were rinsed with 80 μL of methanol at 4000 rpm for 20 s to remove the residual DIO. 7 nm of MoO3 and 100 nm of Ag were evaporated as the top electrode. For the ultra-sensitive EQE fittings, the following expression, in accordance with Equations 2 and 3 in the main text, were used to fit the sub-gap features: where on the right-hand side the first term corresponds to EQE , ( ) while the second term corresponds to EQE , ( ). The fitting parameters for each material system are presented below.

Supplementary Note 2: Modified SRH Theory
The generation and recombination rates involving optical generation and radiative transitions of free electrons and holes taking place via trap states can be understood in terms of modified Shockley-Read-Hall (SRH) statistics. 1 After accounting for radiative transitions, the modified SRH net generation-recombination rate via traps reads 2 =̃ ̃ [ − * * * * ] [ + * * ] +̃ [ + * * ] (2) where and is the free electron and hole density, respectively, is the trap density, while ̃ ( ) = ( ) + ( ) is the coefficient for the transition of an electron (hole) between trap state and the conduction (valence) level being composed of radiative and non-radiative components ] , while and are the maximum optical generation rates for electrons and holes via traps, respectively, both depending linearly on the light intensity; is the energy of the trap state, is the energy of the conduction level, and is the energy of the valence level. Note that the conduction and valance level corresepond to acceptor LUMO and donor HOMO levels, respectivly. In accordance with detailed balance, we furthermore have 2 where ( ) is the corresponding absorption cross section for electrons (holes) and Φ is the black-body spectrum of the environment. Finally, the associated net recombination-generation current density via traps is given by where is the elementray charge and is the active layer thickness.

Derivation of the dark current density
For transitions predominately taking place via mid-gap states, we expect = = and ≈ ≈ exp( 2 ⁄ ), where is the intrinsic carrier density. Then, assuming ̃ =̃ = and that non-radiative transitions dominate over raditive ones (i.e. ( ) ≫ ( ) ), the associated current density in the dark ( = = 0) simplifies as with =̃ 2 ⁄ being the corresponding dark saturation current density. Furthermore, we assume optical transitions via mid-gap states to be governed by Marcus-type charge transfer with = = . Accordingly, an absorption cross section of the form is expected. Here, is a prefactor that depends on the oscillator strength. On the other hand, the absorption coefficient for optical trap generation can be expressed as = , where is the occupancy of the trap states which for mid-gap states is ≈ 1 2 ⁄ . Then, after noting that for weakly absorbing states the EQE may be approximated as EQE , = , we finally obtain where EQE , = ( ) ( ) + ( ) ⁄ denotes the radiative efficiency of the states, while It should be noted that is generally dependent on the ligh intensity (via and ). However, owing to the extremely weak absorption of traps in our case, the rate is dominated by injected carriers in forward bias ( ≫ * * ); hence, the expressions for and derived for dark conditions remain valid under open-circuit conditions (at 1 sun).

Conditions when optical generation via mid-gap states dominates: EQE PV vs PL
The optical generation via traps becomes dominant under special conditions when the influence of injected carriers from the contacts is negligible and the generation of free charge carriers by direct optical transitions are absent. For mid-gap states, assuming thermal generation to be negligible ( ≫ ), the following simplified rate equation for free charge carriers can be obtained assuming = , = = and = (̃ ) , where ̃=̃ =̃ ≫ = ; moreover, is the charge collection time and is the band to band recombination coefficient. Here, the term on the left-hand-side represents the charge extraction rate, while the first and second term on the right-hand-side corresponds to trap-assisted net generationrecombination rate (based on modified SRH theory) and the band to band recombination rate, respectively.
Under short-circuit conditions, the carrier density is expected to be small and the recombination terms negligible. Subsequently, the short-circuit current density, ∝ ⁄ , takes the form ∝ (13) being linear with the light intensity. Hence, we expect the photocurrent induced by mid-gap states to be linear with light intensity at short-circuit (at low generation levels). This is also seen experimentally in Supplementry Figure 5.
In PL measurements, on the other hand, charge-extracting electrodes are absent, corresponding to = ∞. Under these conditions, everything that is generated ultimately recombines; after neglecting third-order terms for the carrier density, we find PL spectra count at the peak (1.45 eV) is plotted versus laser excitation power. The increase of the laser power leads to a quadratic growth of the PL intensity at lower power, while a linear dependence is observed at higher power. This behaviour is consistent with the behaviour expected from modified SRH theory (see above), strongly supporting the presence of optical release.

Supplementary Note 3: The Two-Diode model
The total bulk recombination current, taking place via CT and the low-energy sub-gap channels, is described by two parallel currents and being the CT-induced recombination current and the trap-induced recombination current, respectively. The total dark current is given by where is the additional leakage current induced by an external shunt resistance (caused by non-idealities in the device fabrication). This system can be described in terms of the equivalent circuit shown in Supplementary Figure 7.
Accordingly, the diode current only involves the direct recombination between free electrons and holes, being governed by their respective quasi-Fermi levels ( for electrons, and for holes). The associated current-voltage (J-V) characteristics is governed by the quasi-Fermi level difference = − between electrons and holes (at the electrodes); hence, where is the corresponding dark saturation current which is given by in accordance with the reciprocity principle; Φ is the black-body spectrum at = 300K ( is the Boltzmann constant and is the absolute temperature).
On the other hand, the recombination (and dark generation) of free electrons and holes via trap states is composed of a two-step process: (i) the transition involving a free hole and a trap, and (ii) the transition involving a free electron and a trap. Furthermore, in accordance with Shockley-Read-Hall statistics, 2 trapped carriers occupying the mid-gap states can be described by their own quasi-fermi level . Subsequently, the diode current induced by trap-assisted recombination between free electrons and holes can be described by two diode components which are in series with each other: the first diode current being governed by the quasi-Fermi level difference = − (i), and the second by = − (ii); hence, where and are the corresponding dark saturation currents associated with process (i) and (ii), respectively. For the case when mid-gap traps are dominant, we expect = = 2 ⁄ . Then, noting that = = (because of the series connection), it follows that = exp 2 − 1 (20) with being the corresponding dark saturation current for this recombination channel, which in general is composed both radiative and non-radiative transitions. An explicit expression for can be derived based on the modified SRH theory which takes radiative transitions via traps into account (see Supplementary Note 2). We find = EQE , EQE PV,t Φ (21) where EQE , and EQE PV, t are the electroluminescent and photovoltaic external quantum efficiencies associated with recombination and absorption of mid-gap states, respectively.
Supplementary Figure 7. Two-diode model for describing the dark J-V characteristics.
(a) The equivalent circuit of the assumed two-diode model. Two recombination current and are shown in the circuit, where is described by two diodes which are in series with each other. (b) The schematic energy levels at the donor-acceptor interface are shown together with the quasi-Fermi levels of electrons ( ), holes ( ) and the traps ( ). The band to band recombination current and the trap assisted recombination currents and are shown with downwards arrows.