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Copper selenide (Cuâ‚‚Se) attracts scientific interest for its thermoelectric ability to convert heat into electricity, but a lack of atomic-level understanding has limited its practical applications for decades.

A new computational framework developed by University of Michigan engineers and European collaborators accurately calculates electronic properties and atomic vibrations within the crystal structure, according to a study in Â鶹ÒùÔºical Review Letters.

"There are so many effects entangled together like a knot, but our new method separates each individually with low computational costs," said Yuxuan Wang, a doctoral student of materials science and engineering at U-M and lead author of the study.

These findings will help develop thermoelectric devices able to harvest electricity from waste heat in nuclear power plants or car exhaust. Because thermoelectrics work in both directions—applying an electrical current creates one hot and one cold side—copper selenide could also power heaters or refrigerators that operate more efficiently and quietly than conventional systems with no noisy turbines, fans, pumps or pistons.

With heat and electricity as the only input and output, these thermoelectric devices would not release hazardous emissions, and copper selenide itself is non-toxic and relatively abundant compared to other thermoelectric compounds.

A computational headache

Copper selenide is particularly difficult to model because the copper ions are superionic, meaning they hop around with liquid-like mobility within a solid. The constant copper ion movement creates a dynamic, asymmetrical structure.

Traditional computational methods exploit crystalline symmetry by modeling a small group of atoms to predict the entire crystal's structure and properties. But copper selenide's mobile ions create countless possible atomic configurations, making calculations prohibitively expensive. These traditional methods also struggle to separate temperature effects (thermal vibrations) from the fundamental quantum behavior.

For these reasons, theoretical predictions of copper selenide's properties don't match experimental measurements. While computational models predict copper selenide should be a metal, experiments show it behaves like a semiconductor.

Researchers have also failed to accurately calculate the band gap, which determines electrical conductivity, and atomic vibration patterns known as the phonon density of states, which impact thermal conductivity. Phonons are quantum units of material vibration.

Modeling mobile copper atoms

The research team built the new framework to predict how copper atoms move, or are displaced, within the material. It is based on the anharmonic special displacement method (ASDM), which uses mathematical equations to predict the distance and direction of copper atom displacements and how they change with temperature.

By calculating the thermodynamic-averaged positions the copper atoms would adopt at a given temperature, the model captures copper selenide's behavior using only one computational snapshot instead of hundreds, improving computational efficiency. The method turns complex dynamic movements to a quasi-static picture.

The new quasi-static framework makes simulating superionic materials more accurate and affordable by calculating the positions of copper ions at each temperature.

Modeling with the "quasi-static polymorphous framework" correctly predicted that copper selenide is a semiconductor and accurately reproduced how the band gap narrows with increased temperature, matching experimental observations.

The new model helped solve a decades-long debate about heat conduction in copper selenide. Thermoelectric materials must conduct electricity well and conduct heat poorly to maintain the temperature differentials, but researchers have disagreed on how copper ion movement creates low thermal conductivity.

"While some expected that long-range copper ion diffusion or anharmonic vibrations caused the low thermal conductivity, those factors contribute only minor effects. Instead, copper ions' local vibrations within the pyramid-shaped spaces selenium forms around them are the primary driver of the material's low thermal conductivity," said Emmanouil Kioupakis, a professor of materials science and engineering at U-M and corresponding author of the study.

The copper ion's vibrations within the pyramid are "overdamped," meaning that instead of forming an organized wave-like motion to transport heat, their chaotic motion scatters heat-carrying phonons instead.

"This new computational framework will help simulate and design even more efficient superionic materials for a variety of energy applications, from silent fridges to solid-state batteries and devices to turn waste heat to electricity," said Pierre Ferdinand Poudeu, a professor of materials science and engineering at U-M and senior author of the study.

The Université de Rennes also contributed to this research.