Theory of phonon-mediated superconductivity in twisted bilayer graphene

Abstract: We present a theory of phonon-mediated superconductivity in near magic angle twisted bilayer graphene. Using a microscopic model for phonon coupling to moir\'e band electrons, we find that phonons generate attractive interactions in both $s$ and $d$ wave pairing channels and that the attraction is strong enough to explain the experimental superconducting transition temperatures. Before including Coulomb repulsion, the $s$-wave channel is more favorable; however, on-site Coulomb repulsion can suppress $s$-wave pairing relative to $d$-wave. The pair amplitude varies spatially with the moir\'e period, and is identical in the two layers in the $s$-wave channel but phase shifted by $\pi$ in the $d$-wave channel. We discuss experiments that can distinguish the two pairing states.

• The paper introduces a microscopic framework for phonon-mediated electron pairing in TBG at a magic angle near 1.05°.
• The paper shows that while s-wave pairing is initially dominant, on-site Coulomb repulsion can favor d-wave symmetry under certain conditions.
• The paper predicts measurable phase shifts and gap uniformity in the moiré superlattice that can validate the proposed pairing mechanisms.

Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene

In this paper, the authors develop a theoretical framework for understanding phonon-mediated superconductivity in twisted bilayer graphene, specifically when the twist angle approaches the "magic angle" where superconducting properties are experimentally observed. By employing a microscopic model of phonon interactions with moiré band electrons, the authors address the pairing mechanisms in $s$ and $d$ wave channels, considering the role of both phonon-generated attractive interactions and Coulomb repulsion.

Superconductivity in Twisted Bilayer Graphene

The discovery of superconductivity in twisted bilayer graphene has generated significant interest due to the intriguing interplay between electronic correlation and moiré band structure effects. Previous works have observed superconductivity and insulating states near the magic angle, attributed to the flatness of energy bands, which leads to high density-of-states and interaction-driven phenomena. However, the specific mechanism driving the superconducting state remains elusive.

The experiments show superconductivity occurs when the twist angle is close to the largest magic angle of approximately $1.05^\circ$. The twist introduces moiré superlattices, which significantly modify the electronic properties due to the resultant flat bands. This flat band structure is critical as it enhances the density-of-states, facilitating interaction-driven superconducting states.

Phonon-Mediated Pairing

The authors propose a scenario where phonons in the individual graphene sheets mediate electron pairing, generating attractive interactions in both $s$ and $d$ wave channels. They explore several phonon modes and demonstrate that in-plane optical phonons contribute substantially to the electron pairing. The coupling constants and resultant interactions are calculated assuming continuum models, with a focus on in-plane $E_2$, $A_1$, and $B_1$ phonon modes known for their strong electron-phonon coupling.

The derived phonon-mediated attractive interaction is robust enough, given the high density-of-states near the magic angle, to account for experimentally observed superconducting transition temperatures. Furthermore, they find that while $s$-wave pairing is energetically favorable initially, on-site Coulomb repulsion suppresses it more effectively than $d$-wave pairing, which may, under certain conditions, become the dominant pairing mechanism.

Pairing Symmetries

The paper examines the spatial modulation of the pairing amplitudes over the moiré pattern. For $s$-wave pairing, the pair amplitude is found to be constant across layers, while $d$-wave pairing shows a $\pi$ phase shift between layers—indicative of an inter-sublattice pairing symmetry. The $d$-wave pairing is characterized by chiral components ($d_\pm$), with the chiral symmetry being preserved due to the presence of a uniform gap in the superconductor's excitation spectrum, potentially leading to topological superconductivity.

Discussion and Implications

This theory offers several predictions for experimental verification, such as phase-sensitive measurements to detect the distinct pairing symmetries and gauge their dependence on the twist angle, chemical potential, and external perturbations like pressure. It suggests the superconducting state should be sensitive to strain, uniaxial stress, and environmental screening, each altering the competition between $s$ and $d$ wave instabilities.

The authors acknowledge the notable particle-hole asymmetry observed experimentally, which is underexplored in the current model, suggesting further investigation into screening effects and other possible corrections. Specifically, they address how Coulomb repulsion enhances the theoretical understanding of superconductivity in these systems and propose a minimal model for studying the competing superconducting and insulating phases.

Conclusion

The theoretical insights presented deepen the understanding of twisted bilayer graphene, proposing phonon-mediated mechanisms as a plausible basis for superconductivity near the magic angle. The results open avenues for further studies to resolve the discrepancies between theoretical predictions and experimental findings and to explore new phenomena emerging from such van der Waals heterostructures. This work serves as a foundation for deciphering the interplay of lattice structure, phonon dynamics, and electronic interactions in fostering high-temperature superconducting states.