Dataset DOI: 10.5061/dryad.t76hdr8f9

Description of the data and file structure

This dataset contains the numerical outputs of a first-principles-based microscopic theory designed to evaluate the electron-phonon coupling in twisted bilayer graphene. The model utilizes a momentum-space continuum approach, bypassing the computational cost of periodic supercells. The results include calculations within the generalized Eliashberg-McMillan framework, providing a comparison between adiabatic approximations and a full non-adiabatic treatment of the moiré flat bands. Here, we include the data necessary to reproduce the key results. 

1. Filename: lambda.zip

Description: lambda.zip contains the calculated electron-phonon coupling (EPC) constant and Eliashberg function. 

Variables in lambda_<<theta_deg>>.mat files: 

• E_list: list of Fermi energies 
• lambda: EPC constant calculated using the Choi Phys. Rev. Lett. 127, 167001 (2021) definition, assuming adiabaticity 
• lambda2: EPC constant calculated using the general expression without adiabaticity 
• prefac: scaling factor for EPC matrix element 
• alpha2F: Eliashberg function (or frequency-dependent EPC) calculated using the Choi Phys. Rev. Lett. 127, 167001 (2021) definition, assuming adiabaticity 
• alpha2F2: Eliashberg function calculated using the general expression without adiabaticity 

Manuscript context (Fig. 4): This data is used to plot the twist-angle dependence of electron-phonon coupling and superconducting critical temperature. More specifically, by plotting lambda2 against the twist angles represented in the filenames, the manuscript shows a sharp peak in coupling strength at 1.1˚ and persists up to 1.4˚. In addition, users can compare the lambda with and without adiabatic approximations. 

2. Filename: g_avg_new.zip

Description: This archive contains the mode-resolved coupling strengths averaged over the moiré Brillouin zone. This data identifies the resonance conditions between the electronic flat bands and specific vibrational modes. It highlights the contribution of Gamma-point optical phonons and their role in the modification of the moiré potential. Users can use these matrix elements to determine which phonon branches drive the enhancement of EPC near the magic angle (1.1˚).

The filename format within the zip file is g_avg_<<theta>>deg.mat. Each file contains a 400x119 variable named g_avg, with the first dimension being the flattened phonon frequencies and the second dimension being the phonon branch index.

Manuscript Context (Fig. 3): This data is used to identify specific phonon branches that contribute most significantly to the EPC.

3. Filename: ph_data.zip

Description: ph_data.zip contains the phonon frequency vs. momentum for various twist angles and cutoff radii. The twist angles and cutoff radii are indicated in the filename of each individual file. 

The zip files contain the following data types: 

• evecs_mode_<<nu>>_theta_<<theta_deg>>_re.txt: Real part of the phonon eigenvectors of mode number nu and twist angle theta_deg. 
• evecs_mode_<<nu>>_theta_<<theta_deg>>_re.txt: Imaginary part of the phonon eigenvectors of mode number nu and twist angle theta_deg. 
• eval_theta_<<theta_deg>>.txt: Phonon eigenvalues for all modes for twist angle theta_deg.
• q_ph_gr_theta_<<theta_deg>>.txt: Phonon momentum-space grid for the data files above 

Note: If a file does not have a _cutoff suffix, the cutoff radius is 2.1 in units of moiré reciprocal lattice vector.  If there is a suffix, the cutoff is the number after the suffix. 

Manuscript Context (Figs. 1 & 2): 

• Fig. 1 (Electronic/Phonon Band Structures): Provides the foundational dispersion relationships for the moiré supercell.

• Fig. 2 (Phonon Comparison): Visualizes the difference between phonons with high versus low EPC strength, identifying how specific modes modify the moiré potential.

Code/software

Codes used to generate the data are available upon request.