A multidisciplinary team of researchers at MIT and in Spain has found a new mathematical approach to simulating the electronic behavior of noncrystalline materials, which may eventually play an important part in new devices including solar cells, organic LED lights, and printable, flexible electronic circuits.
The new method uses a mathematical technique that has not previously been applied in physics or chemistry. Even though the method uses approximations rather than exact solutions, the resulting predictions turn out to match the actual electronic properties of noncrystalline materials with great precision, the researchers say.
The project used a mathematical concept known as free probability applied to random matrices, previously considered an abstraction with no known real-world applications. The team found it could be used as a step toward solving difficult problems in physics and chemistry because randommatrix theory allowed them to understand how disorder in a material affects its electrical properties.
Typically, figuring out the electronic properties of materials from first principles requires calculating certain properties of matrices — arrays of numbers arranged in columns and rows. 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 to begin with, this is a very difficult mathematical problem to solve. However, random-matrix theory gives a way to short-circuit all that, using a probability distribution instead of deriving all the precise values.
The new method makes it possible to translate basic information about the amount of disorder in the molecular structure of a material — that is, just how messy its molecules are — into a prediction of its electrical properties.
Essentially, what this new method does is take 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, thus sidestepping the complex calculations that would be required to solve the original problem. The researchers found that their method, although it yields an approximation instead of the real solution, turns out to be highly accurate. The incredible accuracy of the method, which uses a technique called free convolution, led the team to investigate why it was so accurate, which has led in turn to new mathematical discoveries in free probability theory.
This work was done by Jiahao Chen, a postdoc in MIT’s Department of Chemistry; MIT associate professor of chemistry Troy Van Voorhis; chemistry graduate students Eric Hontz and Matthew Welborn; postdoc Jeremy Moix; MIT mathematics professor Alan Edelman; graduate student Ramis Movassagh; and computer scientist Alberto Suárez of the Universidad Autónoma de Madrid. For more information, Click Here