Data for: Restriction of macroscopic structural superlubricity due to structure relaxation by the example of twisted graphene bilayer

Atomistic calculations of the potential energy surface (PES) of interlayer interaction have been performed with account of structure relaxation for a set of twisted graphene bilayers corresponding to commensurate moiré patterns. The calculations are carried out using the registry-dependent Kolmogorov-Crespi potential [1] and the second-generation Brenner potential [2] as implemented in the LAMMPS code [3]. For moiré patterns with small unit cells, structure relaxation is found to have a negligible effect on the PES. For large moiré pattern unit cells,  there are drastic changes in the amplitude of PES corrugations, although the PES shape does not change and can be described by the first Fourier harmonics in the same was as for rigid layers [4].

Two types of constraints are applied to atoms to compute the PESs with account of structure relaxation. In the first case, in-plane positions of all atoms are fixed, which corresponds to the out-of-plane relaxation. In the second case, only one atom per layer is fixed. The calculations with these two types of constraints give qualitatively similar results indicating that out-of-plane relaxation is sufficient to capture the prinicipal relaxation effects. 

The input and output files for LAMMPS calculations are enclosed:

• lammps_relax_bond: graphene bond length optimization.
• lammps_nm: data for the moiré pattern (n,m) including optimization of the interlayer distance for rigid layers (*_dist*), PES for rigid layers at the optimal interlayer distance (*_unrelaxed), PESs with structure relaxation with two types of constraints applied (*_all and *_2atoms*, respectively), full 2D PES with out-of-plane relaxation (*_pes, only for (2,1) and (3,1) moiré patterns).

The folder "data" contains raw data in XYZ format for PES contour plots shown in Figure 2a-b of the article. File "Contour plot.m" in each subfolder contains Mathematica code for reproduction (2,1) and (3,1) PESs. Coordinates in DAT files are given in angstroms and energies are given in microelectronvolts.

Steps to reproduce:
- Open "Contour plot.m" file with Mathematica (ver. 8 or newer).
- Execute code by pressing "Run Package/Run All Code" button or "Shift-Enter" for every cell.
- Contour plots for corresponding PES data should appear below the code.

[1] A. N. Kolmogorov and V. H. Crespi, Phys. Rev. B 71, 235415 (2005).

[2] D. W. Brenner, O. A. Shenderova, J. A. Harrison, S. J. Stuart, B. Ni, and S. B. Sinnott, J. Phys.: Condens. Matter 14, 783 (2002).

[3] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, Comp. Phys. Comm. 271, 108171 (2022).

[4] A. S. Minkin, I. V. Lebedeva, A. M. Popov, S. A. Vyrko, N. A. Poklonski, and Y. E. Lozovik, Phys. Rev. B 108, 085411 (2023).

A.S.M., A.M.P. and Y.E.L. acknowledge the support by the Russian Science Foundation grant No. 23-42-10010, https://rscf.ru/en/project/23-42-10010/. I.V.L. acknowledges the IKUR HPC project ``First-principles simulations of complex condensed matter in exascale computers'' funded by MCIN and by the European Union NextGenerationEU/PRTR-C17.I1, as well as by the Department of Education of the Basque Government through the collaboration agreement with nanoGUNE within the framework of the IKUR Strategy. A.M.P. and Y.E.L. acknowledge the support by project FFUU-2024-0003 of the Institute of Spectroscopy of the Russian Academy of Sciences. S.A.V. and N.A.P. acknowledge the support by the Belarusian Republican Foundation for Fundamental Research (Grant No.~F23RNF-049) and by the Belarusian National Research Program ``Convergence-2025''. This work has been particularly carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC ``Kurchatov Institute'', http://ckp.nrcki.ru