June 22, 2012 — A multidisciplinary research team at Massachusetts Institute of Technology (MIT) and the Universidad Autónoma de Madrid in Spain developed a new mathematical approach to simulating the electronic behavior of noncrystalline materials, with applications in organic light-emitting diodes (OLEDs), flexible printable organic (FPO) electronic circuits, and solar cells.

This mathematical technique, free convolution (a form of free probability applied to random matrices), has not previously been applied in physics or chemistry. It uses approximations rather than exact solutions, yet the resulting predictions match the actual electronic properties of noncrystalline materials with great precision.

The method takes a matrix problem that is too complex to solve easily by traditional mathematical methods and “approximates it with a combination of two matrices whose properties can be calculated easily,” without the complex calculations that would be required to solve the original problem, explained Jiahao Chen, a postdoc in MIT’s Department of Chemistry.

Simulating materials that lack an orderly crystal structure with random-matrix theory allows researchers to couple disorder in a material with its effect on electrical properties, Chen said. Typically, figuring out the electronic properties of materials from first principles requires calculating certain properties of matrices. The numbers in the matrix represent the energies of electrons and the interactions between electrons, which arise from the way molecules are arranged in the material.

To determine how physical changes, such as shifting temperatures or adding impurities, will affect such materials would normally require varying each number in the matrix, and then calculating how this changes the properties of the matrix. With disordered materials, where the values of the numbers in the matrix are not precisely known, this is a very difficult mathematical problem to solve.

Random-matrix theory’s probability distribution makes it possible to translate basic information about the amount of disorder in the molecular structure of a material into a prediction of its electrical properties.

While mathematicians have used such methods in the abstract, “to our knowledge, this is the first application of this theory to chemistry,” Chen says. The team also investigated why free convolution was so accurate, which led to new mathematical discoveries in free probability theory. The method derived for estimating the amount of deviation between the precise calculation and the approximation is new, Chen says, “driven by our questions” for the mathematicians on the team.

“Our results are a promising first step toward highly accurate solutions of much more sophisticated models,” Chen says. Ultimately, an extension of such methods could lead to “reducing the overall cost of computational modeling of next-generation solar materials and devices. There is a lot of interest in how organic semiconductors can be used to make solar cells” as a possible lower-cost alternative to silicon solar cells, Chen says. In some types of these devices, “all the molecules, instead of being perfectly ordered, are all jumbled up.”

The research is reported in the journal *Physical Review Letters*, to be published June 29.

The team included Chen, MIT associate professor of chemistry Troy Van Voorhis, chemistry graduate students Eric Hontz and Matthew Welborn and postdoc Jeremy Moix, MIT mathematics professor Alan Edelman and graduate student Ramis Movassagh, and computer scientist Alberto Suárez of the Universidad Autónoma de Madrid.

The work was funded by a grant from the National Science Foundation aimed specifically at fostering interdisciplinary research.

Courtesy of David Chandler, MIT News Office. Learn more at www.mit.edu.