Microscopic determination of the charge-transfer-state Coulomb binding energy and dielectric screening at organic donor-acceptor heterojunctions and their influence on the open-circuit voltage of organic solar cells

Organic photovoltaic cells generate their photovoltage at donor-acceptor (D-A) heterojunctions, where the interface energy offset between the donor ionization potential and the acceptor electron affinity is converted into a charge-transfer (CT) state whose Coulomb binding energy, reorganization energy, and coupling to the dark current jointly set the open-circuit voltage. Rand, Burk, and Forrest (Phys. Rev. B 75, 115327 (2007)) modeled this offset-to-voltage conversion by combining Marcus theory of interfacial electron transfer with generalized Shockley theory, but treated the Coulomb binding term of their voltage expression phenomenologically, using plausible values of the dielectric constant and the electron-hole separation, and acknowledged further open issues: extension to disordered and bulk-mixed morphologies, the ultimate efficiency limits under their predicted offset tradeoffs, the illumination-dependent parallel conductance that masks the open-circuit voltage at high intensity, and a spectroscopically validated, temperature-dependent determination of the Marcus parameters at real interfaces. Later literature has reframed the voltage question through detailed-balance CT-state spectroscopy, absorbed the two-state Marcus description into vibronic treatments, and experimentally surpassed the 2007 efficiency ceilings with low-offset non-fullerene acceptor devices. What survives as an open problem is a quantitative, system-specific, first-principles determination of the interfacial CT-state binding energy, dielectric screening, and relaxation partition, integrated with the interfacial electron-transfer kinetics, so that the offset-to-voltage conversion is predictive for real, including disordered and bulk-mixed, heterojunctions; and, secondarily, the microscopic origin and mitigation of the illumination-dependent parallel conductance.

Background

In an organic photovoltaic (OPV) cell, light is absorbed by small-molecule or polymer semiconductors and produces excitons — neutral, tightly bound electron-hole pairs — because organic semiconductors have low dielectric constants and weak screening. These excitons dissociate at a donor-acceptor (D-A) heterojunction: an electron transfers from the donor (a material that readily gives up electrons, characterized by its ionization potential IP, the energy to remove an electron) to the acceptor (which readily takes electrons, characterized by its electron affinity EA). The immediate product is an interfacial charge-transfer (CT) state — an electron on the acceptor Coulomb-attracting a hole on the donor across the interface. The key electrical output is the open-circuit voltage $V_{OC}$, the voltage at which the photogenerated current is exactly balanced by the recombination (dark) current. A central empirical fact of the field is that $qV_{OC}$ is always substantially smaller than the offset $\mathrm{IP}_D - \mathrm{EA}_A$ ($q$ is the elementary charge), the deficit being attributed to the Coulomb binding of the geminate (born-together) electron-hole pair, to thermalization of the CT state, and to recombination losses. The 2007 paper by Rand, Burk, and Forrest modeled the dark current of a D-A junction by combining generalized Shockley diode theory (the standard $J(V)$ description of a solar cell, with the diode current written as $J_S\exp(qV/kT)$, where $J_S$ is the dark-current prefactor, $k$ the Boltzmann constant, and $T$ the temperature) with Marcus theory of electron transfer — the classical theory in which a transfer rate depends on the driving force, on the reorganization energy $\lambda$ (the energy associated with rearranging nuclear geometry and polarization upon charge transfer), and on the electronic coupling $V_{if}$, and in which rates decrease again at very large driving force (the 'Marcus inverted region'). Their fit to 14 small-molecule D-A pairs reproduced the observed increase of $V_{OC}$ with light intensity and its decrease with temperature, and their voltage expression (their Eq. (10)) reads schematically $qV_{OC}^{max} = \mathrm{IP}_D - \mathrm{EA}_A - q^2/(4\pi\varepsilon_0\varepsilon_r r_{DA})$ minus further loss terms, where the Coulomb term uses a phenomenological electron-hole separation $r_{DA}$ and dielectric constant $\varepsilon_r$. Their simulations suggested single-junction efficiency limits near 12% for planar double heterostructures and near 16% with mixed layers and tandem stacks, and they attributed deviations of $V_{OC}$ from the ideal intensity dependence to a parasitic, illumination-dependent parallel conductance (a shunt resistance $R_p$ that depends on incident power $P_0$), attributed to bulk photoconductivity via trap-assisted exciton dissociation. Two structural terms used below: a planar (bilayer) heterojunction is one in which donor and acceptor form two adjacent films, whereas a bulk heterojunction is a blend in which donor and acceptor interpenetrate so that the D-A interface is distributed throughout the film volume; and energetic disorder denotes the statistical spread of local electronic site energies induced by packing and conformational variations, whose width and shape differ between material classes. Subsequent work reshaped the picture: Vandewal and co-workers showed, using sensitive measurements of CT-state absorption and emission together with the principle of detailed balance (which links absorption and emission of the same transition) — absorption being measured as the external quantum efficiency (EQE, the wavelength-resolved fraction of incident photons converted to collected charge) and emission as electroluminescence (EL, light emitted when the diode is driven under forward bias) — that $V_{OC}$ is determined by the energy of the charge-transfer state itself; Chen and co-workers developed a unified description of the non-radiative part of the voltage loss, combining temperature-dependent electroluminescence with dynamic vibronic simulations and going beyond the two-state Marcus treatment; and the non-fullerene-acceptor (NFA) era — acceptor molecules replacing fullerenes — produced low-offset devices with non-radiative voltage losses below 0.2 V and certified efficiencies near 20%, far above the 2007 ceilings, although very small offsets do again suppress charge generation, as Müller and co-workers demonstrated by showing that CT-state dissociation efficiency itself can limit low-offset cells. Reviews of CT-state characterization and of recombination-related voltage losses remain active, and device-design recipes still list a high dielectric constant (to screen the CT binding) as a requirement, indicating that the microscopic determination of the CT binding and screening is not considered settled.

Problem Statement

For organic donor-acceptor heterojunctions, establish a quantitative, system-specific microscopic determination of the charge-transfer-state Coulomb binding term $q^2/(4\pi\varepsilon_0\varepsilon_r r_{DA})$ that appears in the offset-to-voltage relation of Rand, Burk, and Forrest (Phys. Rev. B 75, 115327 (2007)) — that is, of the system-specific electron-hole separation $r_{DA}$, the dielectric screening at the interface, and the relaxation/polarization partition between donor and acceptor — so that, combined with the interfacial electron-transfer kinetics (reorganization energies $\lambda_{if}$, electronic couplings $V_{if}$, and their temperature dependence, spectroscopically validated at real interfaces), the open-circuit voltage of a given material pair can be predicted from microscopic properties rather than fitted. The determination must be demonstrably valid not only for planar small-molecule heterojunctions but also for bulk heterojunctions (donor-acceptor blends), highly disordered interfaces, and polymer systems with different energetic disorder statistics. A secondary, still-unresolved part of the source question is the microscopic origin and mitigation of the illumination-dependent parallel conductance $R_p(P_0)$ (attributed in the source to bulk photoconductivity via trap-assisted exciton dissociation), which masks the true open-circuit voltage at high illumination intensity. The 2007 efficiency-ceiling projections (~12% single-junction, ~16% with mixed layers and tandem stacks) are part of the source question but have been experimentally superseded (certified single-junction efficiencies near 20% are now published), and the offset tradeoff they encoded survives only in modified form at very low offsets, where CT-state dissociation limits charge generation. The determination is to be predictive rather than fitted: the intensity- and temperature-dependent open-circuit voltage of a given material pair must follow from microscopic properties, and it must hold for planar small-molecule heterojunctions, bulk heterojunctions, highly disordered interfaces, and polymer systems. The microscopic origin and mitigation of $R_p(P_0)$ is a secondary sub-problem that stands on its own.

References

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