Theoretical background

For theoretical background, chapter 14 of Solid State Physics by Ashcroft/Mermin is a good reference to understand why oscillations happen.

This repository contains a set of tools I found to calculate quantum oscillation frequencies for magnetic susceptibility (de HAAS-van ALPHEN effect), conductivity (Shubnikov-de Haas effect), and magnetorestriction, etc.

The quantum oscillation frequencies are proportional to extremal Fermi surface areas (Onsage relation, eq 14.15 in Ashcroft/Mermin), so the key is to calculate Fermi surface, and find extremal area perpendicular to applied magnetic field.

I have found a set of useful online tools and developped a working flow. I also made additional changes to adjust for more needs.

prerequesites

pip install scikit-image

Working flow

DFT calculation to obtain Fermi surface

First run a VASP job on a dense K-grid for Fermi surface calculation, and use c2x (https://www.c2x.org.uk/fermi/vasp.html) to get a fermi_surface.bxsf file. The .bxsf file stores the information of Fermi surface by saving band energies at the dense K-point grid.

I have found several other codes, like vaspkit, that can also produce .bxsf file. However, they do not use symmetry of the k grid to reduce resource cost.

File format change

The .bxsf file needs some modifications as the requirement of the next step. Here are a complete set of notes for this step.

(1) Download xcrysden in a linux system (http://www.xcrysden.org/Download.html). Xcrysden is supposed to work for MacOS, but I did not figure it out for Mac.

(2) Output band-specific .bxsf files in xcrysden for bands with Fermi surface (http://www.xcrysden.org/doc/fermi.html). You may also refer to SKEAF README to see why the requirements are necessary.

Quantum oscillation frequencies by SKEAF

The implementation principle is well illustrated in the SKEAF paper (https://arxiv.org/pdf/0803.1895.pdf). In principle, you can download the original version of SKEAF (http://www.democritos.it/pipermail/xcrysden/2012-July/001234.html), but I have made some changes to the source for an easier usage. Please download my updated SKEAF version skeaf_v2_update4vasp.tar.gz.

The original version of SKEAF contains many .F90 files: the format change file ELK_exciting_BXSFconverter_v04.F90, skeaf_v1p3p0_r149.F90, and a bunch example files. I have editted two of them ELK_exciting_BXSFconverter_v04.F90 and skeaf_v1p3p0_r149.F90.

Compile

Compile the SKEAF codes by

gfortran skeaf_v1p3p0_r149.F90 -o skeaf
gfortran ELK_exciting_BXSFconverter_v04.F90 -o bxsfconverter

You will have two binaries compiled named skeaf and bxsfconverter. Move these two binaries to your working directory, or add the their location to $PATH in .bashrc

Format and unit change

Run bxsfconverter. It will give you a set of questions. Make the following choices: band.bxsf(filename from Xcrysden), n(not on a periodic grid), n(not having factor 2*Pi), e(energy unit is eV), n(not divided exponent), n(not switched sign), converted.bxsf(new filename)

As a result, you will see a file with name converted.bxsf, and we will use it for skeaf calculation.

Note if you don't use my updated file ELK_exciting_BXSFconverter_v04.F90, you will not have the choice of energy unit eV.

• The .bxsf file energy is in the unit of eV, but the original converter file only have two choices for energy unit: hartree and rydberg. The edit is made in the updated ELK_exciting_BXSFconverter_v04.F90 file.

• The orginal code has a limitation on a size of k grid in .bxsf file: smaller than 100 k points in a, b, and c directions. I have changed the limitation to a larger k grid: smaller than 150 k points in all directions.

Run skeaf

Run SKEAF to by directly typing skeaf in your terminal. You can play with its input and output to get some idea.

Note, if you don't use my updated package, the length unit of the original code assumes Bohr (1 Bohr=0.53 Angstrom).

• The .bxsf file length is in the unit of angstrom (The VASP output is by default Angstrom), original code assumed Bohr unit. I have updated skeaf_v1p3p0_r149.F90 file to assume angstrom unit.

Automation

Working flow to run massive SKEAF calculations Some automation codes. (Will be uploaded later) The rotational angles is not properly implemented in the original SKEAF code. It changes angles linearly.

Fermi surface plot

Of course, you can use xcrysden to visualize and plot the Fermi surface. I have other files for visualization in python, which is easier for publication-level graphs and further changes.

fermi_surface_plot_VASP.py can visualize the fermi surface by taking VASP inputs (EIGENVAL, KPOINTS, POSCAR). This file is developed based on the fs.py file in QijingZheng/VASP_FermiSurface. I made additional changes to this file because I encountered two practical issues.

The first issue: the symmetry tag should be on for using fs.py, which is not the case for spin-orbit coupling calculations with ISYM=-1.

The second issue: an implementation error which is not working for ISPIN=2 (spin-polarized calculations). It works perfectly with the default VASP tag ISPIN=1.

If you replaced fermi_surface_plot_VASP.py with fs.py, the above two issues are resolved. Please use python *.py --help for usage hints. Or you may go to QijingZheng/VASP_FermiSurface for the tutorial.

Note, fermi_surface_plot_VASP.py is only for VASP inputs. Therefore, the other files fermi_surface_plot_bxsf.py and bxsf_parser.py are designed for more general cases, where you only need to input a .bxsf file. Please run

python fermi_surface_plot_bxsf.py *.bxsf

This function only supports matplotlib, where I have removed other visualization tools like Mayavi in the orginal code.

Extremal orbital visualization

Visualization of extremal Fermi surface orbitals Working on it. It will be on soon.

Fermi surface on a denser k-grid

fermi_5_write_bxsf.py and fermi_6_fourier_interp.py can use Fourier interpolation to generate a .bxsf file with denser k-grid. Or, you can follow the tutorial of wannier90 to get a denser .bxsf file by the wannier interpolation.