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A new study published in Natural physics introduces a theory of electron-phonon coupling that is affected by the quantum geometry of electronic wave functions.

The movement of electrons in the lattice and their interactions with lattice vibrations (or phonons) play a major role in phenomena such as superconductivity (conduction without resistance).

Electron-phonon coupling (EPC) is the interaction between free electrons and phonons, which are quasi-particles representing the vibrations of a crystal lattice. EPC leads to the formation of Cooper pairs (electron pairs) responsible for superconductivity in certain materials.

The new study explores the realm of quantum geometry in materials and how they can contribute to the power of EPCs.

Phys.org spoke with the study’s first author, Dr. Jiabin Yu, a Moore postdoctoral fellow at Princeton University.

Speaking about the motivation behind the research, Dr Yu said: “My motivation is to go beyond common wisdom and understand how the geometric and topological properties of wave functions affect the interactions in quantum materials. In this work, we focus on EPC, one of the most important interactions in quantum materials.”

Electronic wave functions and EPC

A quantum state is described by a wave function, a mathematical equation that contains all the information about the state. The electron wave function is basically a way of measuring the probability of where an electron is in the lattice (the arrangement of atoms in a material).

“In condensed matter physics, people have long used energy to study the behavior of materials. Over the past few decades, a paradigm shift has led us to realize that the geometric and topological properties of wave functions are crucial to understanding and classifying realistic quantum materials,” explained Dr. Yu.

In the context of EPC, the interaction between the two depends on the location of the electron in the crystal lattice. This means that the electron wave function to some extent governs which electrons can couple with phonons and affect the conduction properties of that material.

The researchers in this study wanted to investigate the effect of quantum geometry on EPC in materials.

Quantum geometry

The wave function, as mentioned above, describes the state of a quantum particle or system.

These wave functions are not always static and their shape, structure and distribution can evolve in space and time, just as a wave in the ocean changes. But unlike waves in the ocean, quantum mechanical wave functions follow the laws of quantum mechanics.

Quantum geometry studies this variation of spatial and temporal characteristics of wave functions.

“The geometric properties of single-particle wave functions are called band geometry or quantum geometry,” explained Dr. Yu.

In condensed matter physics, the band structure of materials describes the energy levels available to electrons in a crystal lattice. Think of them as rungs on a ladder, with energy increasing the higher you go.

Quantum geometry affects the band structure by affecting the spatial extent and shape of the electron wave functions in the lattice. In simple words, electron distribution affects the energy structure or arrangement of electrons in the crystal lattice.

The energy levels in the lattice are critical because they determine important properties such as conductivity. Also, the tape structure will vary from material to material.

Gaussian approximation and jumping

The researchers built their model using a Gaussian approximation. This method simplifies complex interactions (such as those between electrons and phonons) by approximating the distribution of variables such as energies as Gaussian (or normal) distributions.

This facilitates mathematical processing and drawing conclusions about the influence of quantum geometry on EPC.

“The Gaussian approximation is essentially a way to relate the hopping of electrons in real space to the quantum geometry of momentum space,” said Dr. Yu.

Electron hopping is a phenomenon in crystal lattices in which electrons move from one place to another. For efficient hopping to occur, the wavefunctions of electrons at adjacent sites must overlap, allowing the electrons to tunnel through the potential barriers between the sites.

The researchers found that the overlap is affected by the quantum geometry of the electron wave function, thereby affecting the hopping.

“EPC often comes from the change of hopping with respect to lattice vibrations. So naturally EPC should be enhanced by strong quantum geometry,” explained Dr. Yu.

They quantify this by measuring the EPC constant, which indicates the strength of the bond or interaction, using the Gaussian approximation.

To test their theory, they applied it to two materials, graphene and magnesium diboride (MgB₂).

Superconductors and applications

The researchers chose to test their theory on graphene and MgB₂ as both materials have superconducting properties governed by EPC.

They found that for both materials the EPC was strongly affected by geometrical contributions. Specifically, the geometric contributions were measured to be 50% and 90% for graphene and MgB₂respectively.

They also discovered the existence of a lower bound or bound on the contributions due to quantum geometry. In simple terms, there is a minimal contribution to the EPC constant due to the quantum geometry and the rest of the contribution is from the energy of the electrons.

Their work suggests that raising the superconducting critical temperature, which is the temperature below which superconductivity is observed, can be done by improving the EPC.

Some superconductors such as MgB₂ are phonon-mediated, meaning that the EPC directly affects their superconducting properties. According to the study, the strong quantum geometry implies a strong EPC, opening a new avenue for the search for relatively high-temperature superconductors.

“Even if EPC cannot mediate superconductivity by itself, it can help cancel some of the repulsive interaction and help generate superconductivity,” added Dr. Yu.

Future job

The theory developed by the researchers has only been tested for certain materials, meaning it is not universal. Dr Yu believes the next step is to generalize this theory to make it applicable to all materials.

This is particularly important for the development and understanding of various quantum materials (such as topological insulators) that can be influenced by quantum geometry.

“Quantum geometry is ubiquitous in quantum materials. Researchers know that it must affect many quantum phenomena, but they often lack theories that clearly capture this effect. Our work is a step toward such a general theory, but we are still far from fully understanding it,” concluded Dr. Yu.