mirror of https://github.com/abinit/abinit.git
292 lines
12 KiB
Markdown
292 lines
12 KiB
Markdown
---
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authors: SS, XG
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---
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# fold2bloch analysis tool
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## Unfolding the electronic structure of a lattice of Hydrogen atoms.
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Supercells are often used in electronic structure calculations in order to
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model compound alloys, defects, etc. The band structure obtained directly from
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these calculations is hard to interpret due to the Brillouin zone folding as a
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result of its reduced size for the supercell, compared to that for the
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unperturbed host unit cell.
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The unfolding technique used in Abinit is the one presented in [[cite:Rubel2014]].
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This tutorial aims at demonstrating how to unfold the band structure of a
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supercell and present it in the basis of conventional Bloch wave vectors
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inherent to the unperturbed unit cell. We will construct a supercell of 6
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hydrogen atoms, perform self-consistent cycle and plot the corresponding band
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structure. At the end, we will recover the familiar dispersion relation using
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the *fold2Bloch* utility. See also the [[help:fold2bloch]].
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This tutorial should take about 1 hour.
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[TUTORIAL_README]
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## Creating a Hydrogen supercell structure
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Let's begin with the simplest structure: a lattice of hydrogen atoms. It is a
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convenient starting point since the hybridization between $s$-orbitals results
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in a well known dispersion relation for the energy eigenvalues as a function
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of the wave vector $k$:
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$$E(k)=E_0 - 2A \cos(kb)$$
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Here $b$ is the lattice spacing and $A$ reflects the strength of
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hybridization between adjacent $s$-orbitals. For more details please refer to
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the Feynman's Lectures on Physics (vol 3, chapter 13).
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Our model structure will be a cubic lattice of hydrogen atoms spaced 3 Bohr apart.
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The primitive cell contains only one atom. Here we chose to represent
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the same structure with a 6 atom supercell. Such a large cell is redundant in
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this case, since there is nothing that disturbs the original symmetry.
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However, we made this choice on purpose in order to observe the zone folding.
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: Fig. 1: Primitive unit cell (a) and 1x2x3 supercell (b) that represent a cubic
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lattice of hydrogen atoms.
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Since we are interested in the band structure, we need to select a path in the
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reciprocal space which will be used for plotting. The supercell was expanded
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in Y and Z directions. Accordingly, the reciprocal space shrinks along the
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same directions. We select those directions for the band structure plot as
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they will be affected by the zone folding.
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: Fig. 2: Brillouin zone of the supercell.
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*Before beginning, you might consider to work in a different subdirectory as
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for the other tutorials. Why not Work_fold2Bloch?*
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In order to use the *fold2Bloch*, you need to first generate a wave function file (*WFK* file).
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For this purpose, copy the file
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tests/tutorial/Input/tfold2bloch_1.abi in the directory ~abinit/tests/tutorial/Input/Work_fold2Bloch.
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```sh
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cd $ABI_TESTS/tutorial/Input
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mkdir Work_fold2Bloch
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cd Work_fold2Bloch
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cp ../tfold2bloch_1.abi .
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```
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{% dialog tests/tutorial/Input/tfold2bloch_1.abi %}
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The input file has two datasets,
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the first to generate the *WFK* file, and the second to draw the band structure.
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Now you are ready to run Abinit. Issue the following:
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abinit tfold2bloch_1.abi > tfold1bloch_1.log &
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This will generate a self consistent charge density for the 6 Hydrogen atom
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supercell structure, and the wave function file, *tfold2bloch_1o_DS1_WFK*, which is
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needed for unfolding with *fold2Bloch*.
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## Folded band structure
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Before we proceed with the unfolding, let's plot the "standard" band structure
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of a supercell. We will be looking for signatures of the zone folding. In
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order to do this, there are several possibilities. You might e.g. rely on Abipy (see below), or on Matlab.
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This tutorial was initally illustrated using Matlab, for which
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you will need the following scripts
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1. energy_eig-abinit.sh
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2. plot_band.m
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that are located at $ABI_HOME/doc/tutorial/fold2Bloch_assets/ .
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Copy them to your work directory.
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Execute the *energy_eig-abinit.sh* script
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./energy_eig-abinit.sh tfold2bloch_1o_DS2_EIG
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This will generate an output file: *tfold2bloch_1o_DS2_EIG.dat*
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Edit *plot_band.m* file and point to the newly created *tfold2bloch_1o_DS2_EIG.dat* file.
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data=load('tfold2bloch_1o_DS2_EIG.dat');
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Then, run the *plot_band.m* script in MatLab
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plot_band.m
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This will plot the band structure of the 6 atom Hydrogen supercell created.
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Lastly, compare the image obtained to the band structure image below.
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: Fig. 3: 6 atom hydrogen supercell band structure plot
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Here you can see that the band structure does not look like a cosine function
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along directions Y-Gamma and Z-Gamma. The band structure is folded according
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to the multiplicity along those directions used when constructing the supercell (Fig. 1b).
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!!! tip
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If |AbiPy| is installed on your machine, you can use the |abiopen| script
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with the `--expose` option to visualize the band dispersion:
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abiopen.py tfold2bloch_1o_DS2_GSR.nc --expose
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## Bloch spectral weights with fold2Bloch
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Next step is to execute *fold2Bloch* using the wave function file from the 2nd
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dataset, and multiplicity in the corresponding directions, used when
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constructing the super-cell (x:y:z), included as part of the command line
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arguments. For this example the multiplicity used was (1:2:3)
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Execute the following command:
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fold2Bloch tfold2bloch_1o_DS2_WFK 1:2:3
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You should see the following:
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```
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***********************
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** Fold2Bloch V 1.1 **
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**Build Mar 16, 2015**
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***********************
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Reading eigenvalues from: tfold2bloch_1o_DS2_WFK , with iomode: IO_MODE_MPI
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wfk_read_eigenvalues completed. cpu: 0.03 [s] , wall: 0.04 [s] <<< TIME
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- Creating netcdf file WITHOUT MPI-IO support: tfold2bloch_FOLD2BLOCH.nc
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2% Processing K point: 0.000000 -0.500000 0.000000
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4% Processing K point: 0.000000 -0.450000 0.000000
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7% Processing K point: 0.000000 -0.400000 0.000000
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...
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95% Processing K point: 0.000000 0.000000 0.400000
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97% Processing K point: 0.000000 0.000000 0.450000
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100% Processing K point: 0.000000 0.000000 0.500000
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Number of K points processed: 42
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Data was written to: tfold2bloch.f2b
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Data format: KX, KY, KZ, Eigenvalue(Ha), Weight
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```
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That output tells us which K-point was processed, total number of K-points
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processed, output file, and the format that the data is written in.
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Now take a look at the *tfold2Bloch.f2b*. The first few lines should be as follows:
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```
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0.000000 -0.250000 0.000000 -0.308342 0.589037
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0.000000 -0.250000 0.333333 -0.308342 0.000000
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0.000000 -0.250000 -0.333333 -0.308342 0.000000
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0.000000 0.250000 0.000000 -0.308342 0.410963
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0.000000 0.250000 0.333333 -0.308342 0.000000
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0.000000 0.250000 -0.333333 -0.308342 0.000000
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0.000000 -0.250000 0.000000 -0.308342 0.410963
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0.000000 -0.250000 0.333333 -0.308342 0.000000
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0.000000 -0.250000 -0.333333 -0.308342 0.000000
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0.000000 0.250000 0.000000 -0.308342 0.589037
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0.000000 0.250000 0.333333 -0.308342 0.000000
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0.000000 0.250000 -0.333333 -0.308342 0.000000
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0.000000 -0.250000 0.000000 -0.083709 0.000000
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0.000000 -0.250000 0.333333 -0.083709 0.593925
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0.000000 -0.250000 -0.333333 -0.083709 0.025041
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0.000000 0.250000 0.000000 -0.083709 0.000000
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0.000000 0.250000 0.333333 -0.083709 0.133575
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0.000000 0.250000 -0.333333 -0.083709 0.247460
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0.000000 -0.250000 0.000000 -0.083709 0.000000
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0.000000 -0.250000 0.333333 -0.083709 0.193298
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0.000000 -0.250000 -0.333333 -0.083709 0.218686
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0.000000 0.250000 0.000000 -0.083709 0.000000
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0.000000 0.250000 0.333333 -0.083709 0.564322
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0.000000 0.250000 -0.333333 -0.083709 0.023694
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0.000000 -0.225000 0.000000 -0.332652 1.000000
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0.000000 -0.225000 0.333333 -0.332652 0.000000
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...
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```
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Let's take a moment to analyse the output. Columns 1-3 correspond to kx, ky
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and kz of the unfolded bands; the 4th column is the energy eigenvalue in [Ha]
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and the 5th column corresponds to a spectral weight of the k-point after
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unfolding. Do not confuse it with k-point weight, which represents its
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multiplicity in the Brillouin zone. Since our supercell contains 6 unit cells
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(1 x 2 x 3), each eigenvalue in the solution contains information about 6 Bloch
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wave vectors, i.e., unfolded into 6 k-points. The relative contribution of
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these k-points is determined by the spectral weight.
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Lines 1-6 represent unfolding of the 1st eigenvalue of -0.31796 Ha. This
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eigenvalue corresponds to the Bloch wave vectors of (0 ±1/2 0) as they have a
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non-zero weight. The weights total 1 for normalization. This eigenvalue is
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degenerate, so lines 7-12 look very similar. When a disorder is present (in a
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form of defects, distortions, impurities) individual eigenstates may not
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express an exclusive Bloch character any longer. This can have some
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interesting consequences for transport or optical properties, which are not
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apparent from the folded band structure.
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!!! note
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the whole output is much bigger than the sample provided above. For the
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band structure visualization proceed to the next step.
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## Unfolded band structure
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Lets visualize the unfolded band structure. It is different from a regular
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band structure plot, though. Now we have one additional dimension -- the Bloch spectral weight.
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There are several alternative visualization strategies. Here
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we use the scatter plot with the point size proportional to the spectral weight.
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The following MatLab script will help you build a graph for any
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*fold2Bloch* output: [ubs_dots.m](fold2bloch_assets/ubs_dots.m)
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Make sure the following parameters in *ubs_dots.m* are set as follows:
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```matlab
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KPATH = [0 1/2 0; ...
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0 0 0; ...
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0 0 1/2];
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finpt='tfold2bloch_1o.f2b';
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```
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and:
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```matlab
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G = [0.3333333 0.0000000 0.0000000;
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0.000000 0.1666667 0.0000000;
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0.000000 0.000000 0.1111111]; % Reciprocal latt. vect. [Bohr^-1] from *.out
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```
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Reciprocal lattice vector information must match that in *tfold2bloch_1.abo*:
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Real(R)+Recip(G) space primitive vectors, cartesian coordinates (Bohr,Bohr^-1):
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R(1)= 3.0000000 0.0000000 0.0000000 G(1)= 0.3333333 0.0000000 0.0000000
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R(2)= 0.0000000 6.0000000 0.0000000 G(2)= 0.0000000 0.1666667 0.0000000
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R(3)= 0.0000000 0.0000000 9.0000000 G(3)= 0.0000000 0.0000000 0.1111111
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Unit cell volume ucvol= 1.6200000E+02 bohr^3
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Angles (23,13,12)= 9.00000000E+01 9.00000000E+01 9.00000000E+01 degrees
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For graphing any other *fold2Bloch* output, make sure that the "%%Init Parameters"
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are set accordingly to size of the supercell constructed.
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After running the script you should see the following graph:
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As you can see the unfolded band structure perfectly reproduces the
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anticipated dispersion relation $E(k)=E_0 - 2A \cos(kb)$. We can even
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estimate the magnitude of the hopping matrix element between adjacent atoms
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$V_{ssG} = -A$. The band width is $4A = (-1) - (-12) = 11$ eV which results in $V_{ssG} = -2.75$ eV.
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To analyze the results with AbiPy use:
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abiopen.py tfold2bloch_FOLD2BLOCH.nc
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to open the file inside |ipython| and then type, at the request of ipython:
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```ipython
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# Plot unfolded bands along the path defined by kbounds.
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# 'In [x]' are ipython prompts, do no type them, type only 'kbounds ...' then 'klabels ...' then 'abifile...'
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In [1] kbounds = [0, 1/2, 0, 0, 0, 0, 0, 0, 1/2]
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In [2] klabels = ["Y", r"$\Gamma$", "X"]
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In [3] abifile.plot_unfolded(kbounds, klabels, title="Unfolded bands")
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```
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See also
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[this example](http://abinit.github.io/abipy/gallery/plot_fold2bloch.html#sphx-glr-gallery-plot-fold2bloch-py).
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