Jim Chelikowsky and PhD graduate Kai-Hsin Liou
Jim Chelikowsky and recent Oden Institute PhD graduate, Kai-Hsin Liou, sitting in the Professor's Oden Institute office.

A new record has been set by the Oden Institute’s Center for Computational Materials for calculating the energy distribution function, or “density of states,” for over 100,000 silicon atoms, a first in computational materials science. Calculations of this kind enable greater understanding of both the optical and electronic properties of materials.

Jim Chelikowsky leads the Center for Computational Materials, which set a new standard for the number of atoms that can be modeled. They didn’t just raise the bar though. They smashed it – multiplying the previously held record number by a factor of 10. 

Chelikowsky along with Oden Institute PhD graduate, Kai-Hsin Liou and postdoctoral fellow, Mehmet Dogan, led the team behind this significant technical advancement in atomic modeling. Working with silicon atoms, they increased the number that could be modeled simultaneously from around 10,000 to over 100,000.

One mathematical way to approach such complex systems is by describing solutions in sines and cosines. This is useful for crystalline matter because it is periodic and we know that the properties of a little piece of a crystal will inform the whole crystal.

The sine and cosine approach does not play well with high performance computing machines though. “Too much time is spent in communicating data between processors as opposed to computational operations. So, this approach fails for large systems,” Chelikowsky explained.  

Secondly, if we want to examine systems that are not crystalline, such as clusters, surfaces or amorphous solids, the sine-cosine approach can only be used by artificially replicating the system of interest. This replication operation is cumbersome and has additional technical limitations. 

The researchers used what’s known as real space methods – laying down a grid of points, and then solving equations based on the data from those points. 

Modeling materials in this way isn’t easy. Most solids (or condensed matter), have around 1023 atoms per cubic centimeter. Things get even more complex at the quantum scale. In a quantum description, each electron is characterized by a state consisting of 31,000,000 grid points for the case at hand. The number of states is around 250,000 for a 100,000-atom system of silicon. We need to compute roughly 1013 pieces of information to a high degree of accuracy. If this were not a quantum system, we would need only 3 numbers per atom (not 31 million grid points).  In ohter words, the quantum calculation is roughly ten million times more complicated by that measure.

The equations of interest combine three physical concepts methods to help simplify a solution for the quantum mechanical properties of materials.

“First we look at how the electrons are distributed by assuming whatever happens for an atom will be the same for a solid,” said Chelikowsky.  “This is a well-established way to achieve a good starting point, albeit somewhat approximate. Second, we use density functional theory, another extremely popular method. There are literally hundreds, if not thousands, of papers published each year using this technique, which was recognized by a Nobel Prize in Chemisty in 1998.”  

Finally, the scientists implement a “pseudopotential model” of the material in question.  The pseudopotential model of an atom refers to a method for providing simplified descriptions of complex systems. In this case, the pseudopotential of the silicon atoms’ is found by looking at its chemically important electrons only. 

With so much complex mathematics involved in research of this kind, the team relied heavily on resources at Texas Advanced Computing Center (TACC). “We are very fortunate to have such an incredible resource on campus,” said Chelikowsky.

Unlike the Edisonian approach - testing dozens of materials in a lab, Chelikowsky is using an Einsteinian approach testing orders of magnitude more materials on the computer. Einstein would have been proud.

— Professor Marvin L Cohen, UC Berkeley.

The Rise of Computational Physics

Over the last half century, one of the most significant advances in the study of the physics of materials has been the use of computational tools to explain and predict properties of materials. The successful predictions of new materials and material properties have led to new insights in fundamental science, the production of useful materials and the creative manipulations of known materials. 

Since Einstein first established our understanding of quantum mechanics in the early 20th century, the theory has formed the basis for explaining all the properties of subatomic particles such as protons, neutrons and electrons.

A century later, materials scientists have developed atomic models for countless materials – with varying degrees of success – to better understand why and how, for example, some exhibit electrical and/or mechanical properties that others lack. 

Having access to reliable atomic data has many applications – from astrophysics to x-ray lithography and, in this case, materials science.

Silicon, which is present in sand and glass, is best known as the semiconductor material central to electronic components – making it perhaps one of the most useful materials on the planet. 

“Scaling our capacity to model silicon atoms allows us to study how they interact in larger numbers, what kinds of bonds they form,” said Jim Chelikowsky.   

But, Chelikowsky also notes that this is a technical advancement, meaning the real-world applications can’t really be known until other scientists apply the modeling technique to specific research questions. “It is one of those things where we can’t know what could be done with 100,000 atomic models of silicon until we had 100,000 atomic models of silicon.”   

Technical or not, it is significant, according to Marvin L Cohen, Professor of Physics at UC Berkeley. Cohen is a leading expert in the field of condensed matter physics, especially the electronic structure of solids, and was also Jim Chelikowsky’s former PhD supervisor.

“James Chelikowsky and his team are leaders in the field of applying real space methods to solve for the electronic properties of solids,” he said. “His recent success in calculating the energy distribution function or “density of states” for over 100,000 silicon atoms marks our entry into a new era whereby using real space approaches will enable the exploration of systems given to us by Nature and beyond.”

Materials never before found in the laboratory, says Cohen, can now be tested to see if they can be stable and have useful properties. Those that are useful may then be fabricated for use by humankind.

Computational modeling techniques are enabling new discoveries too problematic to be made physically in a conventional lab setting.

“Unlike the Edisonian approach which would suggest testing dozens of materials in a lab, Professor Chelikowsky is using an Einsteinian approach testing orders of magnitude more materials on the computer,” said Cohen. “Einstein would have been proud.”

 

By John Holden