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 research is being reported in the journal Physical Review Letters, published June 29.
Jiahao Chen, a postdoc in MIT’s Department of Chemistry and lead author of the report, says that finding this novel approach to simulating the electronic properties of “disordered materials” — those that lack an orderly crystal structure — involved a team of physicists, chemists, mathematicians at MIT and a computer scientist at the Universidad Autónoma de Madrid. The work was funded by a grant from the National Science Foundation aimed specifically at fostering interdisciplinary research.
The project used a mathematical concept known as free probability applied to random matrices — previously considered an abstraction with no known real-world applications — that the team found could be used as a step toward solving difficult problems in physics and chemistry. “Random-matrix theory allows us to understand how disorder in a material affects its electrical properties,” Chen says.
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. But, Chen explains, “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.
“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.” These disordered materials are very difficult to model mathematically, but this new method could be a useful step in that direction, he says.
Essentially, what the method developed by Chen and his colleagues 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, he explains.
Amazingly, the researchers found that their method, although it yields an approximation instead of the real solution, turns out to be highly accurate. When the approximation is plotted on a graph along with the exact solution, “you couldn’t tell the difference with the naked eye,” Chen says.
While mathematicians have used such methods in the abstract, “to our knowledge, this is the first application of this theory to chemistry,” Chen says. “It’s been very much in the domain of pure math, but we’re starting to find real applications. It’s exciting for the mathematicians as well.”
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. 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. “It’s a happy accident that it worked out as well as it did,” he adds.
“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.”
David Leitner, a professor of theoretical and biophysical chemistry and chemical physics at the University of Nevada at Reno who was not involved in this work, says the potential practical impact of this research “is great, given the challenge faced in calculating the electronic structure of disordered materials and their practical importance.” He adds that the key test will be to see if this approach can be extended beyond the one-dimensional systems described in this paper to systems more applicable to actual devices. “Extension to higher dimensions is critical in assessing the work’s significance,” he says.
Such calculations “remain a big challenge,” Leitner says, and further work on this approach to the problem “could be very fruitful.”
In addition to Chen, the team included 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.
Mbinu hiyo mpya hutumia mbinu ya hisabati ambayo haijawahi kutumiwa hapo awali katika fizikia au kemia. Ingawa mbinu hiyo hutumia makadirio badala ya ufumbuzi halisi, utabiri unaotokana na hiyo unalingana na mali halisi za elektroniki za vifaa visivyo vya kioo kwa usahihi mkubwa. Utafiti huo uliripotiwa katika jarida la Physical Review Letters, lililochapishwa Juni 29. Jiahao Chen, mwanafunzi wa postdoc katika Idara ya Kemia ya MIT na mwandishi mkuu wa ripoti hiyo, anasema kwamba kupata mbinu hii mpya ya kuiga mali za elektroniki za vifaa visivyo na utaratibu - zile ambazo hazina muundo wa kioo wenye utaratibu - ilihusisha timu ya wanasaikolojia, wanakemia, wanahisabati katika MIT na mwanasayansi wa kompyuta katika Chuo Kikuu cha Autonoma cha Madrid. Kazi hiyo ilifadhiliwa na ruzuku kutoka kwa Shirika la Kitaifa la Sayansi lililokusudiwa hasa kukuza utafiti wa taaluma mbalimbali. Mradi huo ulitumia dhana ya hisabati inayojulikana kama uwezekano wa bure uliotumika kwa matrices za nasibu - hapo awali ilichukuliwa kuwa abstraction bila matumizi ya ulimwengu halisi - ambayo timu iligundua inaweza kutumika kama hatua kuelekea kutatua matatizo magumu katika fizikia na kemia. "Nadharia ya matrix ya nasibu inatuwezesha kuelewa jinsi machafuko katika nyenzo yanavyoathiri mali zake za umeme", Chen anasema. Kwa kawaida, kugundua mali za elektroniki za vifaa kutoka kwa kanuni za kwanza inahitaji kuhesabu mali fulani za matrices <unk> safu za nambari zilizopangwa katika safu na safu. Nambari katika matrix inawakilisha nishati ya elektroni na mwingiliano kati ya elektroni, ambayo hutokea kutoka kwa njia molekuli zimepangwa katika nyenzo. Ili kuamua jinsi mabadiliko ya kimwili, kama vile mabadiliko ya joto au kuongeza uchafu, itaathiri vifaa kama hivyo, kwa kawaida inahitaji kubadilisha kila nambari katika matrix, na kisha kuhesabu jinsi hii inabadilisha mali ya matrix. Kwa vifaa vya kutofautiana, ambapo maadili ya namba katika matrix hayajulikani kwa usahihi kuanza na, hii ni tatizo ngumu sana la hisabati kutatua. Lakini, Chen anaelezea, "Random-matrix nadharia inatoa njia ya mfupi-mzunguko wote kwamba", kutumia usambazaji uwezekano badala ya kupata maadili yote sahihi. Mbinu hiyo mpya inaruhusu kutafsiri habari za msingi kuhusu kiasi cha machafuko katika muundo wa molekuli ya nyenzo - yaani, jinsi molekuli zake zinavyochanganyikiwa - katika utabiri wa mali zake za umeme. "Kuna maslahi mengi katika jinsi semiconductors kikaboni inaweza kutumika kufanya seli za jua" kama uwezekano wa gharama ya chini mbadala wa seli za jua silicon, Chen anasema. Katika aina fulani za vifaa hivi, molekuli zote, badala ya kuwa zimepangwa kikamilifu, zimevurugwa. Vifaa hivi visivyo na utaratibu ni vigumu sana kuiga kwa hisabati, lakini njia hii mpya inaweza kuwa hatua muhimu katika mwelekeo huo, anasema. "Kimsingi, njia iliyoundwa na Chen na wenzake inachukua tatizo la matrix ambalo ni ngumu sana kutatua kwa njia za kawaida za hisabati na ""kuipunguza kwa mchanganyiko wa matrix mbili ambazo mali zake zinaweza kuhesabiwa kwa urahisi,"" hivyo kuepuka mahesabu magumu ambayo yangehitajika kutatua tatizo la awali." Kwa kushangaza, watafiti walipata kwamba mbinu yao, ingawa inatoa takriban badala ya suluhisho halisi, inageuka kuwa sahihi sana. Wakati approximation ni plotted juu ya grafu pamoja na ufumbuzi halisi, "huwezi" kusema tofauti na jicho uchi, "Chen anasema. Wakati wanahisabati wametumia mbinu kama hizo kwa njia isiyo ya kawaida, "kwa ujuzi wetu, hii ni matumizi ya kwanza ya nadharia hii kwa kemia", Chen anasema. <unk>Ilikuwa sana katika uwanja wa hisabati safi, lakini tunaanza kupata matumizi halisi. "Ukweli wa ajabu wa mbinu hiyo, ambayo hutumia mbinu inayoitwa ""convolution ya bure,"" ulisababisha timu hiyo kuchunguza kwa nini ilikuwa sahihi sana, ambayo imesababisha ugunduzi mpya wa hisabati katika nadharia ya uwezekano wa bure." Chen anasema, mbinu iliyopatikana kwa kukadiria kiasi cha kupotoka kati ya hesabu sahihi na makadirio ni mpya, "kuendeshwa na maswali yetu" kwa wanahisabati katika timu. "Ni bahati mbaya kwamba ilifanikiwa kama ilivyofanikiwa", anaongeza. "Matokeo yetu ni hatua ya kwanza ya kuahidi kuelekea ufumbuzi sahihi sana wa mifano ya kisasa zaidi", Chen anasema. "Kutumia mbinu hii, David Leitner, profesa wa kemia ya nadharia na biofizikia na fizikia ya kemikali katika Chuo Kikuu cha Nevada huko Reno, anasema athari ya utafiti huu ""ni kubwa, ikizingatiwa changamoto inayokabiliwa katika kuhesabu muundo wa elektroniki wa vifaa vya machafuko na umuhimu wao wa vitendo.""" "Upanuaji wa vipimo vya juu ni muhimu katika kutathmini umuhimu wa kazi", anasema. "Lakini, ""hatua hii ya kwanza ni ya kipekee, na inapaswa kuonekana kuwa ya kipekee kwa wote,"" alisema Leitner, ambaye pia ni mwanafunzi wa chuo kikuu cha MIT, na pia ni mwanafunzi wa chuo kikuu cha MIT, na pia ni mwanafunzi wa chuo kikuu cha MIT, na pia ni mwanafunzi wa chuo kikuu cha MIT, na pia ni mwanafunzi wa chuo kikuu cha MIT, na mwanafunzi wa chuo kikuu cha MIT, Ramis Moves."