These Mathematical Equations Are Slashing America’s Electric Bills

Using the landscape function, engineers closed the so-called ‘green gap’ in which power-efficient LEDs could produce red or blue light but not green. All three colors combine to form white light, meaning that the improvements to green LEDs enabled by the landscape function reduced the energy demands of many lighting applications.

Filoche emphasizes that the incredible breakthroughs were only possible because of fundamental research. He brings up a quote from business professor and author Oren Harari: “The electric light did not come from the continuous improvement of candles.” By extension, “you didn’t invent the LED by trying to improve the bulb,” says Filoche, a principal investigator of the collaboration and Centre National de la Recherche Scientific research director at the Langevin Institute of the École Supérieure de Physique et de Chimie Industrielles (ESPCI) Paris.

The landscape function’s impact is now spreading to other fields as well, such as studies of the vibrations of proteins and of how air flows through our lungs. The concept’s wide-reaching importance, Mayboroda says, is an example of what physicist Eugene Wigner called “the unreasonable effectiveness of mathematics.”

The revolutionary advancements — and smaller energy bills — wouldn’t have been possible without investments in math and basic science by the U.S. government — particularly the Department of Energy — and from private sources such as the Simons Foundation, Mayboroda says. “The funding of fundamental research made it happen,” she says. “In order to make a fundamental disruption, in order to really invent something, you have to step away from an explicit effort to achieve existing engineering goals.”

The incandescent lightbulb, when introduced, was its own revolution. It was a massive improvement over the candles, oil lamps and gas lights that had long illuminated our homes. But the lightbulb’s design meant that it wasn’t very efficient.

In an incandescent bulb, electrons move across a thin bit of electrical wire called a filament. Like balls rolling downhill, the electrons start with high electric potential energy and end up with lower electric potential energy due to the filament resisting the flow of electrons. That difference in energy must go somewhere — in this case, the electric energy is converted into heat. In a standard incandescent bulb, the filament quickly heats up to around 2,500 degrees Celsius. This scorching temperature causes the filament to glow in the visible part of the light spectrum, though almost all the energy is wasted as heat.

LEDs — short for light-emitting diodes — take a more direct approach to producing light. They aim to convert electrical energy directly into bits of light called photons. They do this by pairing the electrons with something called a hole. A hole is a vacancy in an atom or material where an electron could be but isn’t. Although holes aren’t particles in the conventional sense, physicists can treat them like they are.

When an electron loses energy in the right conditions in the presence of a hole, it releases the energy as a photon. When an electron loses energy and there is no hole nearby, the energy typically turns into a bit of heat. The goal, therefore, is getting the holes and electrons together in the right way. This is done through layers of an extraordinary type of material scientists call a quantum well. These materials concentrate electrons and holes in clusters, meaning that more electrons join up with holes to produce light rather than heat.

Dialing in the properties of these quantum wells is tricky. You want to gather particles together, but you don’t want to do such a good job that they become permanently stuck in place (turning the material into an electrical insulator). For much of the history of LEDs, engineers developed quantum wells through laborious trial and error, testing various compositions and fabrication methods. The dream became to one day be able to use computers to predict how well a given material would work as a light-emitting quantum well. Unfortunately, the calculations required are so numerous and difficult — even using modern supercomputers — that the computational approach was infeasible for most systems. Unless, of course, a mathematical breakthrough came that changed the game.

Predicting how electrons and holes move about a given material requires an understanding of waves. In the strange and difficult-to-understand world of quantum physics, particles such as electrons can also behave as waves (a concept called wave-particle duality). Just like ocean waves, electron waves have crests and troughs. When two sets of waves meet, they add together. If two crests combine, they create an extra-tall peak. If a crest and a trough overlap, they cancel out.

In the 1950s, American physicist Philip Anderson of Bell Laboratories discovered that waves that should move freely through a material would sometimes get stuck in one place. The phenomenon happened with electrons, light and other waves. He realized that this seemed to happen when the material had a lot of disorder — that is, the atoms in the material didn’t have a uniform layout.

Anderson went on to win the 1977 Nobel Prize in physics for this discovery, which became known as Anderson localization. Even though physicists knew that such localization was possible thanks to Anderson’s work, they couldn’t figure out how to predict when and where it would happen. Even decades later, they still couldn’t crack the problem. The harnessing of Anderson localization, while elusive, seemed like it would revolutionize applications such as the quantum wells in LEDs where you want to trap waves in place.

Fast forward to a 2009 research conference in France at which Mayboroda presented her mathematical research on biharmonic functions, which have applications to many problems, including the deformation of thin plates. Filoche, watching in the audience, realized that Mayboroda’s work had many similarities to his studies of the localization of vibrations. Filoche’s research had already contributed to the invention of ‘fractal walls’ — barriers with irregular shapes that could be built along highways to reduce traffic noise.

After the talk, Filoche and Mayboroda chatted about those similarities and began collaborating. Over the next three years, they found that the modeling of thin plates was the breakthrough needed to predict Anderson localization. Mayboroda’s mathematics showed how a thin plate with a non-uniform shape would buckle and warp when you exert pressure on one side. Together, Filoche and Mayboroda discovered that this deformation also shows how vibration waves would travel through the material.

The two ultimately invented a mathematical object called the landscape function that reveals how waves move about the terrain of disordered materials, including where Anderson localization will happen.

“It turned out to be an immensely useful tool, something that we definitely didn’t see at that point,” Mayboroda says.

Over the coming years, Filoche and Mayboroda brought additional researchers to the project and improved the landscape function. This included Filoche’s then-colleague at France’s École Polytechnique, Claude Weisbuch, who was the first to realize that the landscape function could be applied to disordered semiconductors. In collaboration with David Jerison of the Massachusetts Institute of Technology, Douglas Arnold of the University of Minnesota and Guy David of the University of Paris-South, Filoche and Mayboroda crafted a variation of the landscape function that exactly predicts where electrons will localize in a system and how much electric potential energy they’ll have when they do.

“The same mathematical equation or concept can describe very diverse phenomena,” Mayboroda says. “In particular, we realized that the concept we discovered, the concept of the landscape, pertains to the quantum world as well.”

That was the breakthrough the physics community had been waiting for. Running the localization function was a faster way to predict localization in materials than previous methods that required solving complex quantum physics equations over and over for every possible wave frequency. What once took a year on a workstation computer was now possible in a matter of days, says Speck.

“Everything fell into place amazingly fast,” he says. “We started adopting these landscape computations in what we do on a daily basis” to predict localization in materials.

With the new computations, physicists made surprising discoveries about what made a material effective at localizing electrons for LEDs. In an LED, layers of different materials are stacked like layers in a cake. “The quantum wells are like the frosting,” Speck says. Previously, physicists assumed that the layers of atoms in the quantum wells should be flat. With the help of the landscape function, they discovered a superior design: Pyramids with six-sided bases could pierce into the quantum wells, helping the electrons and holes get into position. Although such features had long been known to occur in the materials, they had been treated as defects to be avoided.

“It was heresy to have these kinds of morphological features in the structure; they were considered bad,” Speck says. “Now we work extremely hard, using the landscape function and in experiments, to engineer these defects to give the best electrical efficiency. That’s what filled in the green gap.” Green LEDs, he says, are now around 33 percent more efficient than before.

The impact of the LED revolution extends beyond shrinking our electric bills. Power-hungry incandescent bulbs aren’t often an option in parts of the world that lack power grids. Instead, people rely on the old way of generating light: fire. However, getting the fuel required is time-consuming or expensive, and flames pose a serious safety hazard. LEDs, on the other hand, require so little energy that they can run off a battery attached to a small solar panel.

“It’s shocking how much of the income in the truly underdeveloped world went to light and how many people died in fires due to burning fuels for light before LED lighting became available,” Speck says.

The availability of power-efficient LED lights has been “truly life-changing in communities without reliable electricity,” says Aaron Cohen, executive director of Light Up the World, a nonprofit that brings solar technologies to remote communities in developing countries. LEDs “reduce or, in many cases, eliminate the reliance on harmful fuel-based lighting, improve safety and enable critical activities like studying, working and caring for families after dark. The ripple effects are felt in education, healthcare and economic opportunity in a profound way.”

More LED innovations await, Speck says. He and other material scientists are using the landscape function to see just how far they could push the efficiency of LEDs. Though it may initially sound physically impossible, he predicts that LEDs with 100 percent efficiency are possible. That is, LEDs that produce as much light as they use in electrical energy. But that’s not the limit. On the horizon could be LEDs so efficient that they steal heat from their surroundings to produce even more light. “That’s the dream,” he says, “a light-emitting refrigerator.”

The Simons Collaboration on Localization of Waves ends this year. The collaboration has been by all measures a success. Collaboration member Hugo Duminil-Copin won the 2022 Fields Medal, and fellow collaboration member Alain Aspect received the 2022 Nobel Prize in physics. The localization work won’t end with the collaboration, though. Filoche, Mayboroda and their colleagues will continue pushing the boundaries of mathematics and basic science.

“For me, the high is mathematical discovery and collaborating with the best people in the world,” Mayboroda says. “The best moment is still when you discover something that you haven’t seen before, regardless of whether it’s applicable to any real-world problem. We managed to create something disruptively different, whether you look at it from the point of view of mathematics, or physics, or material engineering or hopefully many, many other applications that are going to come.”