Microscopic theory for electron-phonon coupling in twisted bilayer graphene

TL;DR

The paper develops a first-principles–based, twist-angle–dependent EPC framework for twisted bilayer graphene using a momentum-space continuum model and a non-adiabatic Eliashberg-McMillan theory to compute $\lambda$ and $T_c$. It demonstrates a pronounced EPC enhancement near the magic angle ($\theta \approx 1.1^{\circ}$) with a predicted $T_c$ of about $0.9\ \mathrm{K}$ that persists up to $\sim 1.4^{\circ}$, governed by a resonance between the electronic bandwidth $t$ and dominant phonon frequencies, particularly low-energy moiré modes around $10\ \mathrm{meV}$. The work identifies specific $\Gamma$-point moiré phonons (layer-breathing and layer-shearing modes, and a chiral mode) that contribute strongly to EPC and could be probed by Raman spectroscopy, and discusses experimental implications, substrate effects, and generalization to other moiré systems. Overall, it provides a quantitative, angle-dependent, mode-resolved EPC input that can be integrated into advanced theories of superconductivity in moiré materials.

Abstract

The origin of superconductivity in twisted bilayer graphene -- whether phonon-driven or electron-driven -- remains unresolved, in part due to the absence of a quantitative and efficient model for electron-phonon coupling (EPC). In this work, we develop a first-principles-based microscopic theory to calculate EPC in twisted bilayer graphene for arbitrary twist angles without requiring a periodic moiré supercell. Our approach combines a momentum-space continuum model for both electronic and phononic structures with a generalized Eliashberg-McMillan theory beyond the adiabatic approximation. Using this framework, we find that the EPC is strongly enhanced near the magic angle. The superconducting transition temperature induced by low-energy phonons peaks at $1.1^\circ$ around 1 K, and remains finite for a range of angles both below and above the magic angles. We predict that superconductivity persists up to $\sim 1.4^\circ$, where superconductivity has been recently observed despite the dispersive electronic bands. Beyond a large density of states, we identify a key condition for strong EPC: resonance between the electronic bandwidth and the dominant phonon frequencies. We also show that the EPC strength of a specific phonon corresponds to the modification of the moiré potential. In particular, we identify several $Γ$-phonon branches that contribute most significantly to the EPC, which are experimentally detectable via Raman spectroscopy.

Figures (9)

• Figure 1: Electronic and phonon band structures for twisted bilayer graphene. Electronic band structure and DOS for (a)-(b) $\theta=1.1^\circ$ and (c)-(d) $\theta=1.7^\circ$.(e) Electronic DOS as a function of the twist angle. (f) -(j) Same as (a) -(e) but for moiré phonons.
• Figure 2: Comparison between phonons with large and small electron-phonon coupling. Real space $\Gamma$-point phonon displacement pattern, $\delta \bm{u}_{\Gamma\nu} (\bm{r}) = \delta \bm{u}^{(2)}_{\Gamma\nu} (\bm{r}) - \delta \bm{u}^{(1)}_{\Gamma\nu} (\bm{r})$ for (a) $\omega = 5.2$ meV and (b)-(c) degenerate $\omega = 8.9\,$meV at $\theta = 1.0^\circ$. Arrows show in-plane displacement vectors, and colors show out-of-plane displacement. The displacements are normalized such that the maximum magnitude is 1. (d)-(f) Electronic structures with phonons corresponding to (a) -(c), respectively (solid blue lines). Gray dashed lines show the band structure without phonons. (g)-(i) Redistribution of local stacking order due to phonons. The y-axis shows the probability, and the x-axis shows the distance to AA stacking. Scattered points show the stacking distribution after relaxation and without phonons, which is mostly AB stacking. The histogram shows the stacking distribution with phonons corresponding to (a)-(c), respectively. (j) -(l) Redistribution of the interlayer spacing, $\Delta h_\nu = h(\mathbf{b} + \bm{u}(\bm{b})) + \delta u_{\mathrm{z},\nu}(\mathbf{b}) - h(\bm{b} + \bm{u} (\bm{b}) + \delta u_{\tilde{\bm{q}}\nu\parallel}(\mathbf{b}))$, that corresponds to the phonons in(a) -(c) respectively. Scale bars are 30 nm in (a) -(c) and (j) -(l).
• Figure 3: Identification of phonon branches with large electron-phonon coupling. (a) Averaged magnitude of the EPC matrix element, $1/{N_{\tilde{\bm{k}}} \mathcal{N}_b^2}\sum_{nm\tilde{\bm{k}}}|g_{mn\nu} (\tilde{\bm{k}}, \tilde{\bm{q}})|^2$ in arbitrary units, as a function of the phonon frequency for different twist angles. (b) Frequency as a function of the twist angle for $\Gamma$-phonon branches with large EPC identified from(a). Each curve represents a branch labeled by text in the same color: layer breathing (LB), layer shearing (LS), and chiral (C). (c)-(e) Displacement fields, $\delta \bm{u}_{\Gamma\nu} (\bm{r}) = \delta \bm{u}^{(2)}_{\Gamma\nu} (\bm{r}) - \delta \bm{u}^{(1)}_{\Gamma\nu} (\bm{r})$, of selective phonons with large EPC at $\theta =$(c) $1.6^\circ$, (d) $1.3^\circ$, (e) $1.0^\circ$. Branch labels are on the lower left corner of each panel. All scale bars are 30 nm.
• Figure 4: Twist-angle dependence of electron-phonon coupling and superconducting critical temperature. (a) $\alpha^2(\omega) F(\omega)$ as a function of phonon frequency and twist angle, averaged over the energy window $E_\mathrm{F} \in [-20, 20]$ meV. (b) $\lambda$ as a function of the Fermi level, $E_\mathrm{F}$, and the twist angle. (c) Maximum $\lambda$ for each twist angle. (d) Estimated critical temperature as a function of the twist angle using McMillan formula (blue dashed line) and McMillan formula (Eq. \ref{['eqn:tc_anti']}) with the finite bandwidth correction (solid purple line).
• Figure 5: Phonon DOS at $\theta=1.1^\circ$, $F(\omega)$, for different momentum space cutoffs shown in different colors. Note that the curve for the $8^\mathrm{th}$-shell (green) overlaps with that of the $10^\mathrm{th}$-shell, suggesting convergence.