High-temperature superconductivity: Exploring quadratic electron-phonon coupling

A new study published in Physical review papers (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 vibrations in a lattice called phonons. This interaction is essential for superconductivity (electrical conduction without resistance) in certain materials as it facilitates the formation of Cooper pairs.

Copper 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 the phonon shift, meaning how much the lattice vibrates. The most commonly considered case is when the electron density is linearly related to the lattice displacements, causing a distortion of the lattice to surround each electron.

The researchers wanted to study whether superconductivity can be enhanced for materials that exhibit 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 Dr. Pavel Volkov, Assistant Professor in the Department of Physics, University of Connecticut.

Speaking about his motivation behind pursuing this research, Han said, “It has been one of my dreams to identify and propose 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. That’s why I started looking at alternative mechanisms of electron-phonon coupling.”

Linear coupling and its challenges for superconductivity

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

Linear coupling refers to the scenario when 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. Their implications for superconductivity, however, appear quite different.

Linear coupling, seen in most superconducting materials, has been 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, respectively. “

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

However, the suppression hinders any attempt to improve the critical temperatures in such materials by only 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 has previously been used to study the generic physics of linear electron-phonon coupling.

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

The Holstein 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, which means that the coupling strength increases with the amplitude of the lattice vibrations. This interaction can be understood using classical principles of physics and is well supported by experimental observations such as isotope effects.

In the case of quadratic coupling, this is completely different. Extending the Holstein model to include the second-order dependence of the coupling on the phonon displacement, the researchers calculate the quantum fluctuations (random motions) of the phonons and the zero-point energy (phonon energy at 0 Kelvin).

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

Superconductivity at the limit of weak and strong coupling

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

In other words, the critical (lower) temperature of the material can vary significantly with different atomic masses.

In contrast, when the electron-phonon interactions are strong, we get the formation of quantum bipolarons, which can become superconducting at a temperature determined by their effective mass and density.

Below the critical temperature, the condensate of quantum bipolarons can move freely without disturbing the crystal. More mobility leads to a superconducting state, which is more stable and has a higher critical temperature. Unlike the linear mechanism, the bipolaron quantum mass is only slightly enhanced by coupling, allowing for higher critical temperatures.

“Our work shows that this mechanism allows for higher transition temperatures, at least for strong coupling. What’s also nice is that this mechanism doesn’t require any special preconditions to be functional, and there are quite realistic conditions where be dominant,” he explained. Dr. Volkov.

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

Future work

“The possible impact, first of all, would be an increase in the temperature of the superconducting transition. Superconductivity also depends significantly on the properties of the electrons; thus, to achieve strong coupling we propose the use of specially designed superlattices for electrons,” he explained Dr. Volkov.

The researchers mention that theoretically, the next step would be to find the optimal coupling force regime for superconductivity. The researchers also hope that experimentalists will explore superlattice materials with large quadratic electron-phonon couplings.

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

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

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