High-temperature superconductivity: Probing electron-phonon quadratic coupling

A new study published in Physical Review Letters (PRL) explores the potential of quadratic electron-phonon coupling to enhance superconductivity through the formation of quantum bipolarons.

Electron-phonon coupling is the interaction between electrons and lattice vibrations called phonons. This interaction is essential for superconductivity (resistanceless electrical conduction) in certain materials because it facilitates the formation of Cooper pairs.

Cooper pairs are pairs of electrons bound together through attractive interactions. When these Cooper pairs condense into a coherent state, we get superconducting properties.

Electron-phonon coupling can be categorized based on its dependence on phonon displacement, meaning how much the lattice vibrates. The most commonly considered case is when the electron density is linearly related to the lattice shifts, causing a distortion of the lattice surrounding each electron.

The researchers wanted to study whether superconductivity could be enhanced in materials exhibiting quadratic coupling, which is when the interaction energy is proportional to the square of the phonon displacement.

Phys.org spoke with the study’s co-authors, Zhaoyu Han, Ph.D. candidate at Stanford University and Ph.D. Pavel Volkov, Assistant Professor in the Department of Physics, University of Connecticut.

Speaking about his motivation for doing this research, Han said, “It has been one of my dreams to identify and design new mechanisms that can help achieve high-temperature superconductivity.”

Dr. Volkov said: “The superconductivity of doped strontium titanate was discovered more than 50 years ago, however its mechanism remains an open question, with conventional mechanisms unlikely. Therefore, I began to look for alternative electron-phonon coupling mechanisms.”

Linear coupling and its challenges for superconductivity

As mentioned earlier, the connection can be categorized as linear or quadratic connection.

Linear coupling refers to the scenario where the coupling is proportional to the phonon displacement. On the other hand, the quadratic coupling depends on the square of the phonon displacement.

They can be identified by studying the symmetry of the material, experimental observations and theoretical frameworks. However, their implications for superconductivity appear to be quite different.

Linear coupling, seen in most superconducting materials, is widely studied due to its prevalence in many materials and has a theoretical framework.

However, conventional superconductors with linear electron-phonon coupling face limitations. These materials have a low critical temperature, which is the temperature below which the material can exhibit superconductivity.

Han explained, “The critical temperatures for these superconductors are typically below 30 Kelvin, or -243.15 degrees Celsius. This is partly because the Cooper pair binding energy and kinetic energy are exponentially suppressed in the weak and strong binding regimes.”

In weak binding, the electron-phonon interactions are weak due to the low binding energy. With strong coupling, the interactions are stronger, leading to a higher effective mass of Cooper pairs, which suppresses superconductivity.

However, the suppression hinders any efforts to improve the critical temperatures in such materials by simply increasing the coupling strength, encouraging researchers to explore materials with quadratic electron-phonon coupling, which are not as well understood.

The Holstein model and quantum bipolarons

The Holstein model is a theoretical framework used to describe the interaction between electrons and phonons. It was previously used to study the general physics of linear electron-phonon coupling.

The researchers extended the Holstein model to include quadratic electron-phonon coupling in their study.

Holstein’s model helps calculate quantities such as the binding energy of Cooper pairs and the critical temperature of superconductors.

In conventional materials, phonon-mediated electron binding leads to the formation of Cooper pairs.

The interaction is linear, meaning that the bond strength increases with the amplitude of the lattice oscillations. This interaction can be understood using the principles of classical physics and is well supported by experimental observations such as isotope effects.

In the case of quadratic coupling, it is completely different. By extending the Holstein model to include the dependence of the second-order coupling on the phonon displacement, the researchers account for the quantum fluctuations (random motion) of the phonons and the zero-point energy (phonon energy at 0 Kelvin).

Electrons interact with quantum phonon fluctuations to create “quantum bipolarons”. Unlike linear coupling, the origin of attractive interactions is purely quantum mechanical.

Superconductivity in the weak and strong binding limits

The researchers found that when the electron-phonon interaction is weak, the mechanism by which electrons pair to form Cooper pairs is not efficient, similar to the linear case. This leads to a low critical temperature, which can be affected by the mass of the ions (isotope effect), but in a different way than in the linear case.

In other words, the (low) critical temperature of a material can vary significantly with different atomic weights.

In contrast, when electron-phonon interactions are strong, quantum bipolarons are formed, which can become superconducting at a temperature given by their effective mass and density.

Below the critical temperature, the condensate of quantum bipolarons can move freely without disturbing the crystal. Greater mobility leads to a superconducting state that is more stable and has a higher critical temperature. Unlike the linear mechanism, quantum bipolar matter is only slightly increased with coupling, allowing for higher critical temperatures.

“Our work shows that this mechanism allows for higher transition temperatures, at least for strong binding. What’s also good is that this mechanism does not require any special prerequisites to be functional, and there are quite realistic conditions where it will be dominant,” he explained. Dr. Volkov.

Han predicted, “Based on the fundamental physical constants relevant to solid materials, an optimistic estimate of the critical temperature achievable by this mechanism may be on the order of 100 Kelvin.”

Future work

“The potential consequence would be primarily an increase in the temperature of the superconducting transition. Superconductivity also depends sensitively on the properties of the electrons, so to achieve strong coupling we propose to use specially designed electron superlattices,” explained Dr. Volkov.

The researchers say that in theory, the next step would be to find the optimal binding force regime for superconductivity. The researchers also hope that experimenters will explore superlattice materials with large quadratic electron-phonon couplings.

“Experimentally, creating superlattices through patterning or using interfaces between twisted materials could be a promising route to realizing the type of superconductivity we predicted,” said Dr. Volkov.

Han also emphasized, “It is crucial to identify materials with large quadratic electron-phonon couplings from ab initio calculations, as this has not been systematically explored.”

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