Polymeric organic semiconductors are being developed as possible materials for organic electronics and photocells, to take advantage of modern polymer synthesis and processing. These materials are very different from traditional “hard” semiconductors like silicon.  They are made of flexible long-chain molecules with conductivity primarily along chains, arranged in a nanoscale mixture of ordered and disordered domains.

In the ordered domains, straight chains are regularly packed like pencils in a box, while in disordered domains the chains are randomly packed, like cooked spaghetti.  This disorder can impede the motion of charges along the chains.  So predicting electronic properties of these materials is a big theoretical challenge, because we must treat the effects of disorder, which means averaging over many large disordered conformations…

Tight binding model

“How does disorder along a polymer semiconductor impede charge motion?”

To make predictions for electronic properties of semiconducting polymers with disorder, we use a “coarse-grained” approach.  Semiconducting polymers typically consist of a chain of covalently bonded aromatic rings, conjugated along the chain so electrons can flow.

This structure suggests a coarse-grained model in which each ring corresponds to one site along the chain.  We then write a “tight-binding” model, in which electrons can hop between neighboring rings. 

We fix the hopping parameter by comparing to more atomic-level quantum calculations called density functional theory (DFT).  We can compare the energy of different “standing waves” of electrons along a small repeating piece of chain using DFT, and adjust the hopping parameter to get the same result with the tight-binding model.  

The hopping parameter decreases when we twist the chain between adjacent rings, which tends to break the conjugation between rings that allows electrons to flow.  Using DFT results for twisted chains, we can find how the hopping parameter depends on twist angle.

To test this model against experiment, we focus on how a single semiconducting polymer chain in solution absorbs light.  If the chain were straight, the absorption spectrum would show a sharp feature where incoming light of the right frequency promotes an electron from standing waves of the HOMO (highest occupied molecular orbital on each ring) to standing waves of the LUMO (lowest unoccupied molecular orbital).  

But in solution, the chain is randomly twisted, which randomly disrupts the extended HOMO and LUMO standing waves.  The frequency of the light that matches the energy to promote the electron now depends on the shape of the chain.  This smears out the sharp absorption peak of a straight chain, into a broad peak.  Averaging over the many shapes the chain can make in solution, we can predict the width of this peak.  Our prediction roughly matches experiment, which suggests we have accounted properly for the effects of disorder on the electrons along the chain.

These kinds of calculations are only possible because our tight-binding model is simple enough that we can efficiently compute the energies of electronic states of a long disordered chain, not only just once but averaged many thousands of chain shapes. 

Bombile, J. H., Janik, M. J., and Milner, S. T. “Tight Binding Model of Conformational Disorder Effects on the Optical Absorption Spectrum of Polythiophenes” Physical Chemistry Chemical Physics 18, no. 18 (2016): 12521–12533. doi:10.1039/c6cp00832a

Polarons

“How does a charge in a polymer semiconductor interact with its surroundings?”

When a charge is present in a polymeric semiconductor, the charge interacts with its surroundings in several ways.  A ring along the chain with a localized extra electron tends to distort its shape.  Dipoles on neighboring rings tend to point towards an extra electron.  And the electronic “cloud” of neighboring molecules tends to shift slightly away from an extra electron.  

All three of these interactions can be described as the nearby medium reorganizing and polarizing in response to the added charge, i.e., tending to move like charges away from the added charge.  These polarizing effects lower the total energy.  

For its own part, the added charge tends to spread out, because of the quantum-mechanical tendency to hop between conjugated rings.  A localized charge has a higher kinetic energy.  But the more spread out a charge is, the less the medium polarizes in response, and the lower the reorganization energy.  So if a charge localizes, the change in kinetic energy is positive, but the favorable interaction with the medium (the “reorganization energy”) is negative.  

There is thus a competition between kinetic and reorganization energy, which may result in an optimum length scale for confining the charge.  If this happens, the resulting configuration is called a “polaron”.  In effect, the added charge “digs a hole for itself” by polarizing the nearby medium into a more favorable configuration in which to reside.

We have investigated the three different polarizing interactions for polymer semiconductors using our tight-binding approach, and conclude that the strongest of these is the shift of the electronic cloud of neighboring molecules away from an added charge, called “dielectric” polarization.  We are able to estimate the size and binding energy of dielectric polarons.  This result important for describing how charges move about in the semiconductor, and for understanding how pairs of equal and opposite charges created when the polymer absorbs light can be separated to produce electricity.

Bombile, J. H., Janik, M. J., and Milner, S. T. “Polaron Formation Mechanisms in Conjugated Polymers” Physical Chemistry Chemical Physics 20, no. 1 (2018): 317–331. doi:10.1039/C7CP04355D