Materials for Organic Solar Cells
In organic solar cells, light is absorbed by optically active molecules (chromophores). This generates excitons (electron-hole pairs), which are converted into electric current. Critical to the efficiency of any solar cell is absorbing as much as possible of the solar spectrum. One limitation of solar cell efficiency is the loss of photons with energies below the absorption threshold, which corresponds to the optical gap of the absorber. Optical upconversion of two low-energy photons into one high-energy photon can increase solar cell efficiency by harvesting photons with sub-gap energies. In organic chromophores, upconversion can be achieved by triplet-triplet annihilation (TTA), where two low-energy triplet (same-spin) excitons are converted into one high-energy singlet (opposite-spin) exciton, which can be absorbed by the solar cell.
Another limitation of solar cell efficiency is that typically one exciton is converted into one charge carrier (the Shockley-Queisser limit). This limitation may be overcome by converting one high-energy photon, whose excess energy would otherwise be lost to heat, into two charge carriers. This may be achieved in organic chromophores by singlet fission (SF), the opposite process of TTA, where one singlet exciton is down-converted into two triplet excitons.
Together, TTA and SF can reduce losses at the low-energy end and the high-energy end of the solar spectrum and thus significantly increase solar cell efficiency. However, the realization of solar cells with TTA and SF is hindered by dearth of suitable materials, as very few chromophores are currently known to undergo TTA or SF. Computational exploration of the chemical space may significantly accelerate the discovery of new TTA and SF materials.
To study the excited-state properties of chromophores and their crystals, we use many-body perturbation theory (MBPT). Within this formalism, the GW approximation is used to calculate properties associated with charged excitations, such as ionization potentials and electron affinities. The Bethe-Salpeter equation (BSE) is solved to calculate properties associated with neutral excitations, such as optical absorption spectra, singlet and triplet excitation energies, and exciton wave-functions.
Finding Predictive Models for Singlet Fission by Machine Learning
One of the current challenges for computational materials discovery is that in many cases the properties of interest emerge from complex phenomena and are therefore time-consuming and/or costly to calculate or measure. In such cases, high-throughput screening approaches cannot be straightforwardly applied. Furthermore, the usage of machine learning (ML) techniques is complicated by the limited availability of data and the high cost of data acquisition. One such case is singlet fission (SF). The primary criterion of SF to be possible is the thermodynamic driving force, which is the difference between the singlet exciton energy and the energy of two triplet excitons. Evaluating the driving force for intermolecular SF in the solid state requires calculating the excited-state energetics of molecular crystals, which may have several hundred atoms in the unit cell. This can be done using many-body perturbation theory (MBPT), however these calculations have a very high computational cost. Therefore, alternative models are needed for materials screening, which are fast to evaluate and predictive of the SF driving force.
We tackled this challenge by using the sure-independence-screening-and-sparsifying-operator (SISSO) ML algorithm to identify low-cost predictive models for SF. The input of SISSO is a set of primary features, which are physical descriptors that could be correlated with the target property. SISSO generates a huge feature space by iteratively combining the primary features using linear and nonlinear algebraic operations. Subsequently, linear regression is performed to identify the most predictive models. SISSO essentially performs a computer experiment in which hypotheses are systematically generated and tested against reference data. Physical and chemical knowledge is leveraged in the choice of primary features and in the rules for combining them. An important advantage of SISSO is that it can work well with a relatively small amount of data.
To train SISSO, we have generated a data set of MBPT excitation energies of 101 polycyclic aromatic hydrocarbon (PAH) molecular crystals with up to ~500 atoms in the unit cell. SISSO successfully generated predictive models, the most accurate of which yielded a root-mean-square error (RMSE) below 0.2 eV, on par with MBPT (an example is shown in the top panel). Moreover, the best-performing models had a near-perfect classification accuracy for determining whether or not a given material is a promising SF candidate. Based on considerations of the model accuracy vs. its computational cost, a hierarchical screening approach was proposed to narrow down the candidate pool. Three new promising SF candidates were identified out of the PAH101 set (shown in red in the bottom panel). Our workflow can now be used to screen larger repositories of molecular crystal structures. Moreover, our approach can be replicated for materials discovery in other domains.
An energetics perspective on why there are so few triplet-triplet annihilation (TTA) chromophores
In order for TTA to be thermodynamically favorable, the singlet energy, S1, must be smaller than twice the triplet energy, T1 (2T1 > S1). In order to increase the TTA quantum yield (QY), the main competing process, which is conversion to a higher triplet state, T2, should be less energetically favorable than TTA (2T1-T2 < 2T1-S1). This places stringent requirements on the excited-state energetics of chromophores. We used GW+BSE to evaluate 59 chromophores of different chemical families as candidates for TTA. Most of the experimentally known TTA chromophores are concentrated in the region colored in yellow, where the competing triplet pathway is still somewhat more favorable than TTA. Very few chromophores are found in the region colored in green, where TTA is more favorable. This explains from the energetics perspective why there are so few known TTA chromophores and why the QY of TTA is typically low. That said, our results also indicate that the performance of chromophores from known chemical families may be improved by chemical modifications, such as functionalization with side groups, and that new chemical families could be explored to discover more TTA chromophores.
Computational discovery of new candidates for inter-molecular singlet fission in the solid state
In order for a material to undergo singlet fission (SF) there must be a sufficient thermodynamic driving force, which means that the singlet exciton energy, ES, should be greater than twice the triplet exciton energy, ET. A singlet exciton with a charge transfer (CT) character, where the electron probability density is distributed on neighboring molecules to the molecule with the hole, is considered favorable for inter-molecular SF. Based on this, we formulated a two-dimensional descriptor for expected SF performance with the primary descriptor, ES-2ET plotted on the x-axis and the degree of CT character plotted on the y-axis. We evaluated several molecular crystals with respect to this descriptor using GW+BSE. The most promising candidates are found in the top right corner of the plot, close to pentacene, which is currently the best known SF material. Materials experimentally known to undergo inter-molecular SF in the solid state are colored in red. So far, we have identified several promising new candidates, including monoclinic rubrene, quaterrylene, phenylated acenes, pyrene-fused acenes, and zethrenes.
Crystal Engineering Communications 18, 7353 (2016); The Journal of chemical physics 148, 184101 (2018); The Journal of Physical Chemistry C 123, 5890 (2019); Journal of Physics: Condensed Matter 32, 184001 (2020); J. Phys. Chem. C 124, 26134 (2020)
Double-Bader analysis of exciton character
To evaluate the degree of charge transfer character (%CT) of an exciton wave-function produced from a GW+BSE calculation, we have developed the double-Bader analysis (DBA) method. Bader analysis is a popular charge partitioning scheme for electron density distributions with one spatial variable. We have extended it to exciton wave-functions with two spatial variables (electron and hole) by performing nested sums over electron probability distributions with respect to different hole positions. The dbaAutomator code streamlines the performance of DBA and evaluates the convergence of exciton wave-functions produced by the BerkeleyGW code with respect to the fine k-point grid.
Effect of crystal packing on the excitonic properties of molecular crystals
When molecules form a solid the single molecule properties are altered by the environment of surrounding molecules. The fundamental gap of a molecule is the difference between the ionization potential (IP) and the electron affinity (EA). In a molecular crystal, the dielectric screening stabilizes charges, which decreases the IP and increases the EA. this leads to polarization induced gap narrowing. In a molecular crystal the discrete molecular orbital energies evolve into dispersed bands. The band dispersion further reduces the fundamental gap. The way molecules are packed into a crystalline solid affects the electronic coupling between neighboring molecules. As a result, different crystal structures of the same molecule (polymorphs) may have significantly different electronic and optical properties. For example, in the orthorhombic form of rubrene backbone overlap between neighboring molecules leads to strong electronic coupling and dispersed bands. In the monoclinic form of rubrene there are no co-facial interactions between neighboring molecules, which leads to weak electronic coupling and flat bands. Based on GW+BSE calculations, the monoclinic form of rubrene has a higher thermodynamic driving force for SF and a singlet exciton with a high degree of charge transfer character. The excitonic properties of molecular crystals can be modulated by adding side groups that modify the crystal packing either by creating steric hindrance or by changing the intermolecular interactions.