Electronic Structure and Topology in Gulf-edged Zigzag Graphene Nanoribbons

Electronic Structure, Topology and Spin-Polarization in Gulf-edged Zigzag Graphene Nanoribbons

This repository collects the necessary calculation files to reproduce the results shown in our manuscript (arXiv). It includes the following parts:* A) Structure files* B) TB calculations* C) DFT calculations* D) GW calculations* E) Parametrization of TB with Hubbard-U (TB+U)* F) TB+U calculations* G) ZGNR systems * H) Calculations for different $U$ values

(A) Structure files

ZGNR-G structures, created with a C-C (C-H) bond length of 1.4 Ang (1.1 Ang) and bond angles of 120°. Structures are given with and without saturation of dangling bonds by hydrogen atoms. The unit cells are rectangular. The GNR is periodic in the x-direction, and a vacuum gap of 20 Ang between the carbon atoms is added in the y- and z-direction. We did not perform geometry optimization. Files are given in XYZ, XSF, and CIF format. The structural parameters are varied in the following range:* $N$=4...11* $a$=3...10* $M$=2...9 (depending on $a$)* $b$=0...a/2 (depending on $a$ and whether $N$ is odd or even)* S and L inversion center

The files are named in the following way: * Carbon only (used in TB calculations): $N$-ZGNR-G$M$_$a$_$b$_.* Saturated systems (used in DFT calculations): $N$-ZGNR-G$M$_$a$_$b$__saturated.

(B) TB calculations

Minimal calculation files for the complete set of structures:* Structure file in CIF format* PythTB input file (onsite energy $\alpha$=0, 1st-NN hopping element $t_1$=-1)* Calculation results in JSON format

The data files contain the calculated band gaps and Z2 topological invariants, sorted into tables by structural parameters. A table is given for each combination of $N$, $M$, and inversion center. Each table varies the parameter $a$ in the rows (value of $a$ given first in each line) and the parameter $b$ in the columns (values not explicitly given, varied from 0 (0.5) to $a$/2 for even (odd) $N$). The band gaps are given in units of the 1st-NN hopping element $t_1$. The Z2 topological invariant is calculated using the Zak phase. A value is given for metallic systems even though the equations are not applicable to these systems. Additionally, the results are given as a simple list.

(C) DFT calculations

Calculation files for the subset of systems studied on the DFT/HSE06 level with a tight tier 1 basis and a k-grid of 18x3x3 using FHI-aims. ZGNR-G systems for this subset are selected to have a maximum of 100 carbon atoms in the primitive unit cell. Calculations are performed both without and with spin polarization. Spin-polarized systems are run with both an antiferromagnetic (AFM) and ferromagnetic (FM) initial guess (by placing an initial spin moment on the zigzag edge atoms), resulting in an AFM or FM magnetic state, respectively. 

For each calculation, the following files are stored:* geometry.in: input geometry* control.in: input file for FHI-aims* aims.out: output file for FHI-aims * band1001.out: band structure file for the first spin channel* band2001.out: band structure file for the second spin channel (only for spin-polarized calculations)* cube_001_spin_density.cube: converged spin density in CUBE format (compressed in ZIP format to save storage space after extracting calculation files)* spin-polarization.png: plot of spin moments for carbon atoms (only for spin-polarized calculations)* gap.dat: band gap, extracted from the band structure file* spin_max.dat: maximum absolute spin moment, extracted from the Mulliken projection resultsAdditionally, for each structure, a plot comparing the band structure without spin polarization, the band structure of the AFM state, and the band structure of the FM state are stored. The DAT files are not stored for the FM state as those are never the magnetic ground state and thus were not further analyzed.

In addition to the calculation files, the main results are collected in DAT files: the total energy (without spin polarization, AFM state, FM state), the band gap (without spin polarization and AFM state), and the maximum spin moment (only for the AFM state).

(D) GW calculations

Calculation files for the GW calculations, performed for 4-ZGNR, 5-ZGNR, and 6-ZGNR. They are calculated at the GW@PBE level and compared against calculations on the DFT/PBE and DFT/HSE06 level. Calculations are performed using FHI-aims with a tier 1 or tier 2 basis set and varying k-grids as visible from the file names. For each calculation, the input and output files are stored. They are sorted into subdirectories by their properties in the following order:* Studied system,* Method (DFT or GW),* Functional, and* Basis set and k-grid.

(E) Parametrization of TB with Hubbard-U (TB+U)

The parametrization of TB+U was done in two steps: (1) parametrization of the 1st NN hopping element $t_1$ and (2) the subsequent parametrization of the Hubbard-U, using the previously parametrized $t_1$.

(1) Parametrization of $t_1$

The parametrization of $t_1$ was done using the ZGNR-G systems available for DFT calculations. The TB and DFT calculations without spin polarization were used from steps B and C. Systems were excluded from the data set if the position of the DFT band gap was not reproduced in TB, leaving 372 ZGNR-G systems in the data set. The parametrization itself was done by linear regression of the DFT band gap in eV as a function of the TB band gap in units of $t_1$, resulting in y=3.328x-0.072 (R^2=0.951), giving $t_1$=3.328 eV. The TB and DFT band gaps are stored in the file "step1_parametrize_t1.dat"; the calculation files are taken directly from steps B and C.

(2) Parameterization of $U$

The parameterization of the Hubbard-U was done by first running TB+U calculations with different $U$ values. For this purpose, we varied $U$ from 0 to 5 in intervals of 0.2, using $t_1$=-1 to keep this step independent of the parametrization of $t_1$. The resulting band gaps are stored in DAT files in the subdirectory "calc_step2_variation_U" with a single file per ZGNR-G system. To save storage space, we did not upload further calculation files - the input files are equivalent to those uploaded in step F, just with different values for $t_1$ and $U$. 

Afterward, we used these results to obtain the optimal $U$ value for each system, focusing on systems that show a band gap opening in the AFM state on the DFT/HSE06 level. We performed the parametrization by identifying which value of $U$ in each system gives the best agreement of the TB+U band gap with the DFT band gap in the AFM state, using the calculations from step C and interpolating linearly between the $U$ values of the scan described above. We then ran a TB+U calculation with the obtained $U$ value to check the agreement with the DFT calculations. We generally obtained good agreement with a few exceptions that were filtered out: systems that resulted in a $U$ of zero and those without a band gap opening in the AFM state of the TB+U calculation. The results of the remaining 414 ZGNR-G systems are summarized in "step2_parametrize_U.dat". The final value of $U$ was obtained by averaging over those systems, yielding an average of 1.720 $t_1$, equivalent to 5.723 eV. The calculation files used for parametrization, including those filtered out, are stored in "calc_step2_TB+U_calculations". Plots comparing the band structures on DFT/HSE06 level, TB, and TB+U are in "plots_step2_fit_agreement".

(F) TB+U calculations

Calculation files for the complete set of structures:* Structure file in XYZ format* PythTB input file (onsite energy $\alpha$=0, 1st-NN hopping element $t_1$=-1, $U$=1.72 $t_1$)* Calculation results in JSON format* Plot of the band structure from TB vs. TB+U (AFM state)* Plot of spin moments as an overlay over the atomic structure

The data files contain the band gaps on TB and TB+U level ("results_band_gaps.dat"), the position of VBM and CBM on TB and TB+U level ("results_band_edge_positions.dat"), and spin momentum quantities ("results_spin_moments.dat"). Please note that, compared to the JSON files, a factor of 2 is applied to obtain the spin moment; this corrects the PythTB calculations, which multiply the final spin-polarization by a factor of 1/2 to account for electrons being particles with spin 1/2.

(G) ZGNR systems

ZGNR systems without gulf edges are included in the data set as a reference system. Structures are included in the subdirectory "structures" with widths $N$ from 2 to 50, analogously to part A. For all ZGNRs, DFT and TB+U calculations were performed. The provided files are equivalent to parts C and F. Additionally, for the DFT calculations with spin polarization, files for the maximum, minimum, and average (over the carbon atoms) spin moments are provided, distinguished by results from Mulliken and Hirshfeld analysis. The plots of the spin moments are also given for both the Mulliken and Hirshfeld analysis results.

The data files contain the band gaps of the AFM state on DFT and TB+U level ("results_band_gaps_AFM_state.dat"), the total energy of the DFT calculations without and with spin-polarization in the AFM and FM state ("results_total_energies_DFT.dat"), as well as the maximum, minimum, and average (averaged over the C atoms) spin moment of the TB+U and DFT calculations, distinguished by Mulliken and Hirshfeld analysis ("results_spin-moments_maximum.dat," "results_spin-moments_minimum.dat," "results_spin-moments_average.dat"). Please note that, compared to the JSON files, a factor of 2 is applied to obtain the spin moment for the TB+U calculations.

(H) Calculations for different $U$ values

TB+U calculations similar to part F were performed for ZGNR and ZGNR-G systems. The main difference is that different values of $U$ were used: 1.20, 1.50, 1.72, and 2.00 in units of $t_1$. Please note that, compared to the JSON files, a factor of 2 is applied to obtain the spin moment for the TB+U calculations.