[Paper Review] Electron-K-Phonon Interaction In Twisted Bilayer Graphene

Chao‐Xing Liu, Yulin Chen+2 more|arXiv (Cornell University)|Mar 27, 2023

Graphene research and applicationsMaterials Science10 references3 citations

TL;DR

This paper develops an analytic theory of electron-K-phonon interaction in twisted bilayer graphene (TBG), showing that a single optical K-phonon (~160 meV) mediates superconductivity via symmetry-protected pairing channels. It identifies a singlet s-wave inter-Chern-band pairing with the highest critical temperature, alongside a gapless nematic d-wave intra-Chern-band state, stabilized in the chiral flat band limit, with phonon-mediated attraction analytically derived using topological heavy fermion mapping.

ABSTRACT

We develop an analytic theory to describe the interaction between electrons and K-phonons and study its influence on superconductivity in the bare bands of twisted bilayer graphene (TBG). We find that, due to symmetry and the two-center approximation, only one optical K-phonon (~ 160meV) of graphene is responsible for inter-valley electron-phonon interaction. This phonon has recently been found in angular-resolved photo-emission spectroscopy to be responsible for replicas of the TBG flat bands. By projecting the interaction to the TBG flat bands, we perform the full symmetry analysis of phonon-mediated attractive interaction and pairing channels in the Chern basis, and show that several channels are guaranteed to have gapless order parameters. From the linearized gap equations, we find that the highest Tc pairing induced by this phonon is a singlet gapped s-wave inter-Chern-band order parameter, followed closely by a gapless nematic d-wave intra-Chern-band order parameter. We justify these results analytically, using the topological heavy fermion mapping of TBG which has allowed us to obtain an analytic form of phonon-mediated attractive interaction and to analytically solve the linearized and T=0 gap equations. For the intra-Chern-band channel, the nematic state with nodes is shown to be stabilized in the chiral flat band limit. While the flat band Coulomb interaction can be screened sufficiently enough - around Van-Hove singularities - to allow for electron-phonon based superconductivity, it is unlikely that this effect can be maintained in the lower density of states excitation bands around the correlated insulator states.

Motivation & Objective

• To understand the role of electron-K-phonon interaction in mediating superconductivity in the flat bands of twisted bilayer graphene (TBG).
• To identify the symmetry-protected pairing channels induced by the K-phonon in the TBG flat bands.
• To determine whether electron-phonon coupling can overcome Coulomb repulsion in the correlated insulator regime.
• To provide an analytic solution for phonon-mediated attraction using the topological heavy fermion mapping of TBG.
• To assess the impact of Coulomb screening on effective interaction strength near Van Hove singularities.

Proposed method

• Develops a deformation potential theory for electron-phonon coupling in TBG, focusing on intra-layer interactions around the Dirac points.
• Uses the Bistritzer-MacDonald model to describe the electronic structure of twisted bilayer graphene in the Moiré Brillouin zone.
• Applies the two-center approximation and symmetry analysis to show that only one optical K-phonon mode (~160 meV) contributes to inter-valley scattering.
• Projects the electron-phonon interaction onto the flat bands and performs a full symmetry analysis in the Chern basis.
• Employs the linearized gap equation and T=0 BCS theory to solve for pairing channels analytically.
• Uses RPA screening to compute the effective Coulomb interaction, with dielectric constant and screening strength evaluated numerically as a function of Fermi energy.

Figure 1: (a) Phonon dispersion of graphene. The irreps for phonon modes at $\Gamma$ and $\mathbf{K}_{D}$ are labelled. Inset: BZ of graphene. (b) MBZ of TBG. (c) and (d) shows the momentum dependence of the normalized gap function $|\Delta_{\mathbf{k}}|$ for the inter-Chern-band $A_{1}$ singlet (or

Experimental results

Research questions

• RQ1 Which phonon modes in TBG mediate electron-electron attraction via electron-phonon coupling?
• RQ2 What are the symmetry-protected pairing channels induced by the K-phonon in the flat bands of TBG?
• RQ3 Can electron-phonon coupling induce superconductivity in the presence of strong Coulomb repulsion, particularly near Van Hove singularities?
• RQ4 How does Coulomb screening affect the effective interaction strength in the flat band and correlated insulator regimes?
• RQ5 What is the relative stability and critical temperature of different pairing channels, including gapped and gapless states?

Key findings

• Only one optical K-phonon mode (~160 meV) contributes to inter-valley electron-phonon coupling due to symmetry and the two-center approximation.
• The highest critical temperature pairing is a singlet s-wave inter-Chern-band order parameter, with a gapped superconducting state.
• A gapless nematic d-wave intra-Chern-band pairing channel is also strongly supported and stabilized in the chiral flat band limit.
• The phonon-mediated attractive interaction is analytically computed using the topological heavy fermion mapping, matching well with numerical results.
• Coulomb screening significantly reduces the effective interaction strength near Van Hove singularities, with RPA screening yielding an effective interaction of ~0.4 meV.
• The screening effect is strongly energy-dependent, with the dielectric constant reaching ~65.4 near the DOS peak, but likely insufficient in the lower DOS regime near correlated insulator states.

Figure 2: (A) The superconductor order parameter amplitudes $|\Delta_{\pm}|$ (red circles and blue crosses) and the ground state energy (black dots) as a function of $\mu$ . The superconducting phase has nodes in the shadowed regime. (B) and (C) show the BdG spectrum with and without nodes at $\mu=0