DOI: 10.5061/dryad.t76hdr8f9
title: Data from: Microscopic theory for electron-phonon coupling in twisted bilayer graphene

description: This dataset provides the underlying numerical results from a
 first-principles-based microscopic theory investigating electron-phonon
 coupling in twisted bilayer graphene. This data package specifically
 catalogs the electron-phonon interactions, Eliashberg functions, and
 mode-specific coupling strengths generated via a momentum-space continuum
 model. Dataset structure and contents: The data is organized into three
 primary compressed directories representing the evolution of EPC from the
 atomic to the macroscopic scale: EPC Constants and Eliashberg Functions
 (lambda.zip): Contains .mat files indexed by twist angle (\theta). Key
 variables include the Eliashberg function (\alpha^2F) and EPC constants
 (\lambda) calculated under both adiabatic and non-adiabatic frameworks,
 allowing for comparison of dynamical effects near the magic angle.
 Mode-Averaged Matrix Elements (g_avg_new.zip): Data identifying the
 contribution of specific phonon branches to the total coupling strength.
 Phonon Dispersion Profiles (ph_data.zip): Comprehensive phonon frequency
 vs. momentum data for various twist angles and cutoff radii, defining the
 phononic band structure of the moiré supercell. Reuse potential: This
 dataset is intended for condensed matter physicists and materials
 scientists seeking to validate many-body theories or compare experimental
 transport data with microscopic EPC calculations. By providing both
 adiabatic and general Eliashberg functions, the data enables further
 sensitivity analysis of superconducting transition temperatures and
 building an effective theory in twisted bilayer graphene.
description: # Data from: Microscopic theory for electron-phonon coupling in twisted
 bilayer graphene Dataset DOI:
 [10.5061/dryad.t76hdr8f9](https://doi.org/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_&lt;&gt;.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_&lt;&gt;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_&lt;&gt;_theta_&lt;&gt;_re.txt`:
 Real part of the phonon eigenvectors of mode number `nu` and twist angle
 `theta_deg`.  *
 `evecs_mode_&lt;&gt;_theta_&lt;&gt;_re.txt`: Imaginary part of the phonon eigenvectors of mode number `nu` and twist angle `theta_deg`.  * `eval_theta_&lt;&gt;.txt`: Phonon eigenvalues for all modes for twist angle `theta_deg.` * `q_ph_gr_theta_&lt;&gt;.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.