All posts by Anubhav

Here be dragons: should you trust a computational prediction?

In a sense, computational materials science performs virtual “measurements” of a material given a description of its structure. While many such computational measurement techniques have been developed, density functional theory (DFT) is now the most popular approach to rapidly screen compounds for technology. DFT solves quantum mechanics equations that determine the behavior of electrons in a material (approximately); these electron interactions ultimately determine the material’s chemical properties. Compared to other theories, DFT strikes a good balance in terms of being accurate, transferable (not needing too many tweaks for different materials), and low in computational cost[1].

Yet, it is all too easy to list shortcomings of density functional theory. One can only model a repeating unit cell containing about 200 atoms; beyond this, the computational cost of using DFT very rapidly becomes unattainable. Thus, with DFT the richness of materials behavior at larger length scales is lost. Even for materials that can be described with so few atoms, DFT calculations are often subject to large inaccuracies. Unfortunately, there is no theory to tell you how accurate or inaccurate a particular DFT calculation will be. Finally, with computational screening you are usually unsure if the materials you designed in a computer can ever be synthesized in the lab.

With all these limitations, how should a materials researcher interpret DFT results?

Materials cartography

One might view the results of density functional theory predictions in the same way he or she might set sail with an old map of the world. That is, as a useful guide that should be taken with a healthy grain of salt. For example, we can draw parallels between Ptolemy’s 2nd century world map and the current state of the art in high-throughput calculations.

In many ways, high-throughput computations give us a map that is not unlike the early maps of the world. [Ptolemy World Map,]
Certain regions of the world are mapped better than others. Ptolemy’s map gets the northern Mediterranean roughly correct, but then starts failing catastrophically in the southern region. The Black Sea is quite accurate; the Caspian Sea is entirely wrong. Similarly, DFT works quite well for metals and many semiconductors and insulators, but starts to become very unreliable for strongly correlated materials (such as superconductors). How much you trust DFT really depends on what region of the “materials world” you are in.

Sometimes, the overall trend is correct, but the fine details are lacking. The shape of the Arabian Peninsula in Ptolemy’s map is approximately correct but doesn’t get the details right. This is often the case for certain properties computed by density functional theory. For example, when optimizing the coordinates of the atoms in a material’s structure, DFT often slightly under-predicts or over-predicts the bond lengths, but usually gets the overall solution remarkably close to that seen in experiments.

Other times, the trend can’t be trusted, but some fine details can still be recovered. The width of the Red Sea on Ptolemy’s map is not represented faithfully, particularly how it widens severely at the bottom. Yet, certain details like the Gulf of Aden can be seen from the map, and the coastline (particularly the eastern one) is not too bad. Similarly, with some materials properties (such as the band structure), DFT often misses the big picture (width of the  energy gap between conduction and valence bands) but recovers some useful details (e.g. general shape of the bands on either side, and their overall curvature).

Entire swaths of the world are completely missing. Just as the sailors of Ptolemy’s day had not ventured out to all parts of the world yet, DFT computations have not yet been performed across all potential materials. With high-throughput DFT, we are essentially sending hundreds of ships in every direction to look for entire new materials continents filled with hidden riches. However, we can’t say that the map is complete.[2]

If you are an experimentalist and are going to set sail with a DFT-based map, you’d better have a sense of adventure and realize you might have to correct course based on what you see. You might also want to get some previous guidance about the sketchier areas of the materials world. But, you’d probably rather have the map than rely only on your personal knowledge of the seas.

Slaying the dragons and sea monsters

In earlier days of mapmaking, it was not uncommon to place drawings of monsters over uncharted regions, warning the seafarer to proceed at their own risk (hence the popular phrase, “Here be Dragons”[3]). A similar warning might apply today to experimentalists interpreting DFT maps for excited state properties and strongly correlated materials.

Yet, the DFT version of the materials world is growing closer to the truth with time. While certain “monsters” remain unslayed for decades, brave DFT theorists have wounded many of them (and been knighted with a PhD). Over time, as DFT techniques improve, the monsters will start to disappear from DFT maps[4]. And as high-throughput computations set sail for new materials in new directions, more classes of interesting technological materials will become known to us. Our maps are not perfect, but they will continue to improve; in the meantime, experimentalists will have to retain their sense of adventure!

[1] The amount of time needed to compute the properties of a single material depends on the material’s complexity and on the number and type of properties desired. A reasonable range is between 100 CPU-hours (1 day on your laptop) and 10,000 CPU-hours (100 days on your laptop).
[2] For more details on why we can’t compute everything, see the previous post on “The Scale of Materials Design”.
[3] Despite being a great phrase, “Here be dragons” apparently wasn’t used that much on old maps.
[4] For an example of how DFT is progressing into difficult territory, see this recent prediction of a new superconductor.

The Scale of Materials Design

How many potential materials might there be? For those of us trying to compute our way to the next technological breakthrough, it’s important to grasp the scale of the problem. The first step is to define the term material.

The scale of materials

The objects that we interact with every day are composed of about 1023 atoms. Rather than describe all those atoms, however, materials scientists classify a material’s structure into different length scales.

Materials Scales
Materials exhibit structure on many length scales, each of which influences the overall properties.

At the fundamental scale, solid objects are composed of atoms that repeat in a pattern defined by a unit cell, much like a three-dimensional M.C. Escher tessellation.[1] The positions of atoms in the unit cell constitutes the crystal structure, whereas the elemental identity of the atoms define the chemistry. We will refer to some combination of crystal structure and chemistry as a chemical compound. In other words, a compound is a specific arrangement of atoms with chemical identities (e.g., Si, Na, Fe, etc.) in a repeating box.

A material encompasses much more than a compound, and includes additional structure at the micro- and macro- scales. At the microstructure level, the compound’s unit cells are arranged into a regular pattern within a grain. The properties of the grains and their boundaries as well as defects and impurities within the grain can have a large effect on a material’s properties. Materials can also form composites of multiple compounds at this scale. A large part of materials science concerns engineering microstructure to achieve certain properties without changing the fundamental compound(s). Similarly, the structure of the material at the macro scale – such as their surfaces – can further influence properties.

For the purposes of counting, we will restrict ourselves to counting compounds, not materials. This restriction anticipates that when we screen materials computationally, we will model all 1023 atoms of the material as one infinitely repeating unit cell. The number of possible materials would be much bigger than our estimation of the number of compounds.

Estimation method I: A googol of compounds in a box

To count compounds, we might arrange atoms in a box representing the unit cell. Let’s position 30 atoms into a 10x10x10 grid of points, selecting one of 50 elements from the periodic table[2] for each of the 30 atoms.

Packing materials in a box
Tossing 30 atoms into a 10x10x10 box using 50 possible elements gives a googol of possibilities!

It’s now straightforward to calculate the number of compound combinations. The number of potential choices of grid points for our 30 atoms is 1000C30, or about 1057, representing the possible crystal structures. Next, we assign elements to each of those atoms to get chemistry. Since we are independently assigning one of 50 elements to 30 atoms, there are 5030 ways to pick, or about 1051 possibilities. Multiplying those two numbers together, we obtain a result of about 10108 different compounds, or over a googol (10100)!

A googol is a truly unimaginable number. It is not the number of grains of sand on all the beaches on earth (~1021), or the number of hair widths to the sun (~1016). It is even more than the number of known atoms in the universe (~1080). There is no way we can compute all the combinations of 30 atoms on a 10x10x10 grid.

Estimation method II: Modify the known crystal prototypes

You wouldn’t estimate the space of potential 100,000-word novels by counting all the possibilities for words randomly strewn across pages.[3] Like a novel, a legitimate chemical compound obeys rules and patterns[4][5]. A more reasonable guess at the number of compounds might apply a few rules:

  • Rather than be arranged arbitrarily on grid points, the atoms in compounds tend to recur in the same crystal structures. New materials are very likely to reuse an old arrangement of atoms, but with different elements substituted in.
  • Known compounds, or at least those that are not nightmarish to synthesize, tend to be composed of only a few distinct elements, generally 5 or less per compound.

Applying these rules completely changes our estimation. As of this writing, the Inorganic Crystal Structure Database has classified about 6500 crystal structure prototypes, or known arrangements of atoms in a box. In terms of chemistry, if we choose 5 element compounds from a set of 50 elements, that’s 505 or 312 million combinations per prototype[6]. Multiplying those numbers together, we get about 2 trillion compounds.

At this point, we could start splitting hairs. Certainly, we don’t know all the crystal structure prototypes yet. However, even if we’ve found only 1% of crystal structures so far, we are only changing our final number by a factor of 100 (to 200 trillion). The story is similar if we want to try more than 50 elements or more than 5 element compounds. We could instead go the other way, and add further heuristic rules that reduce the estimation down from a trillion. The order of trillions of possible permutations is perhaps a reasonable middle ground, keeping in mind that our goal is just a basic estimation.

Narrowing it down

The current state-of-the-art in high-throughput computations can test about 10,000 to 100,000 materials candidates, depending on the complexity of the materials involved. So, even materials scientists armed with supercomputers must first narrow down about 1,000,000,000,000 possibilities down to 10,000 using chemical intuition[7]. Computations can narrow down that list to maybe 100. Even fewer of those will see experimental success, and fewer still will be commercialized.

Our estimate for the number of chemical compounds is not too far from the number of humans that have ever lived (~100 billion). Identifying the handful of technologically relevant compounds, then, is truly liking picking out the Da Vincis, Shakespeares, and Beethovens amongst the materials world.

[1] Many materials do not fall neatly into this description, in particular polymers and glasses. (11/20/2013)
[2] Although there are more than 100 elements on the periodic table, many of them are rare, radioactive, synthetic, or extremely expensive. 50 is a nice round number to use.
[3] However, Jorge Luis Borges already performed a very similar thought experiment in a short story titled “The Library of Babel” (4/25/2014)
[4] For example, structures tend to display symmetry; symmetric configurations tend to form an extremum in the crystal structure energy landscape which is often a minimum energy point.
[5] For examples of rules exhibited by novels, check out this wonderful article by Brian Hayes on early work by Markov. (11/26/2013)
[6] Not all prototypes support 5 elements, but we are giving them the benefit of the doubt…
[7] Hopefully, informatics will assist with this process in the future.

Why hack materials?

Although we are surrounded by the products of materials science, few of us consider the history and design of the material world. How does sand (SiO2) transform into the silicon-wafer CPUs that beat you at chess?[1] This journey is not only interesting but also important. Most technologies are in some sense limited by the materials from which they are composed.

solar panel figure
A solar panel on your roof, made up of repeating patterns of silicon atoms that give rise to a band structure. The band structure helps estimate light capture properties.

Computer chips are not the only technology employing the silicon in sand. Solar panels on a roof are likely composed of silicon atoms arranged in a repeating configuration. Those silicon atoms – arranged in just that way – give rise to fundamental materials properties like the band structure. Materials scientists relate such fundamental properties to how efficiently a material can capture light and overall device performance. What other arrangements of atoms could we put in solar panels, how would they perform, and could they lead to an energy revolution?

The needle in the haystack

Unfortunately, discovering the right combination of atoms for a technological application is a “needle in the haystack” problem; the vast majority of atom combinations are not only technologically uninteresting but also impossible to synthesize. Materials scientists and chemists therefore rely on knowledge and insight to guide the search for the few technologically relevant materials. Unfortunately, this process generally results in months or years of dead ends. Like the needle in the haystack, new materials are not easily found, and breakthroughs are rare.

Digging in the haystack

It’s difficult to find the next breakthrough material by simply testing the “haystack” of candidates. [Original image source unknown]
How might we find the “needle in the haystack” more quickly and reliably? One strategy might be to increase manpower. An analogue of this idea in the materials world is high-throughput computing. In this technique, some of the world’s most powerful supercomputers predict the properties of tens of thousands of materials by solving physical equations. The best results from the computers are retained for synthesis. Much of the “sorting through the haystack” is performed by machine, while scientists target the most promising candidates.

Army in the haystack
Hiring an army is one approach to search large haystacks faster. Replace the humans with CPUs and you have high-throughput computing. [Original source unknown, Iraqi National Guard]
Finally, might we do even better than an army digging through haystacks for needles? Indeed, even with the world’s most powerful computers, the fact remains that the number of possible materials combinations is more than we can ever compute. That’s where materials informatics might play a role – to equip the army of CPUs with metal detectors, or rather to focus computational power towards the chemical spaces most likely to yield breakthroughs. As more materials are computed, a materials informatics approach would learn what materials are likely to be successful and adapt the search accordingly. These insights could also be passed on to human researchers, who might otherwise never discover these chemical design rules.

Metal Detector Haystack [U.S. Marine Corps photo by Lance Cpl. James Purschwitz/Released]
Materials informatics is like a metal detector for finding new materials. [U.S. Marine Corps photo by Lance Cpl. James Purschwitz/Released]
It is an exciting time for materials design, one in which computations are starting to play a greater role in guiding new discoveries.  Perhaps soon, there will not only be materials scientists, but also materials hackers that apply skills from the information revolution to the hunt for materials.

[1] If you’re curious about “sand to CPU”, Intel has produced an eccentric video of the process and TechRadar has a nice article about it.