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Example 3: Supercell with non-diagonal transformation matrix

Genral discussion

In the previous examples, our supercells have the same shape as the primitive cell but in real calculation, there are many cases that the SC have different shape with the PC. Mathematically, this means that the transformation matrix \(\bf M\) is non-diagonal. As a result, the K-path before/after unfolding are not collinear.

A simple example of a non-diagonal \(\bf M\) is shown in this figure:

where red and blue indicate SC and PC respectively. In figure (a) above, the transformation matrix between the SC and PC is:

\[ {\bf M} = \begin{pmatrix} 2 & 2 & 0 \\\ 2 & -2 & 0 \\\ 0 & 0 & 1 \end{pmatrix} \]

The point here is that, when \(\bf M\) is non-diagonal, the PC k-path and the corresponding SC K-path are not collinear in the fractional coordinates. As shown in figure (b), in reciprocal space the PC k-path:

\[\Gamma (0, 0, 0) - X (0 ,0.5,0)\]

corresponding to the SC K-path:

\[{\bf -M} (-0.5, -0.5, 0) - {\bf M} (0.5 ,0.5,0)\]

which can be checked by formula: \({\bf K} = {\bf k} \cdot {\bf M}\). Thus, in order to get unfolded band along PC k-path ( \(\Gamma - X\) ), one need to calculate band structure along ( \(\bf -M - M\) ) in the SC.

Silicon conventional cell

In the following example, we will unfold the band structure of a silicon conventional cell to the PC k-path \(\Gamma (0, 0, 0) - M (0 ,0.5,0.5)\).

Geometry

A perfect silicon conventional cell in diamond structure is:

lattice_vector 5.417040018        0.0000000000000000 0.0000000000000000 
lattice_vector 0.0000000000000000 5.417040018        0.0000000000000000 
lattice_vector 0.0000000000000000 0.0000000000000000 5.417040018
atom 0.0000000000000000 0.0000000000000000 0.0000000000000000 Si
atom 0.0000000000000000 2.7085200087860000 2.7085200087860000 Si
atom 2.7085200087860000 0.0000000000000000 2.7085200087860000 Si
atom 2.7085200087860000 2.7085200087860000 0.0000000000000000 Si
atom 1.3542600043930000 1.3542600043930000 1.3542600043930000 Si
atom 1.3542600043930000 4.0627800131790000 4.0627800131790000 Si
atom 4.0627800131790000 1.3542600043930000 4.0627800131790000 Si
atom 4.0627800131790000 4.0627800131790000 1.3542600043930000 Si
In comparison, the primitive cell is:

lattice_vector 2.709        2.709 0.0000000000000000 
lattice_vector 2.709 0.000000000000    2.709
lattice_vector 0.0000000000000000 2.709 2.709
atom 0.0000000000000000 0.0000000000000000 0.0000000000000000 Si
atom 1.3542600043930000 1.3542600043930000 1.3542600043930000 Si
They are connected by the transformation matrix:

\[ {\bf M} = \begin{pmatrix} 1 & 1 & -1 \\\ 1 & -1 & 1 \\\ -1 & 1 & 1 \end{pmatrix} \]

Control tags

Since we want to get the bands along PC k-path \(\Gamma (0, 0, 0) - M (0 ,0.5,0.5)\), by formula \({\bf K} = {\bf k} \cdot {\bf M}\), we need to calculate SC K-path \({\bf -Z} (0, 0, -0.5) - {\bf Z} (0 ,0,0.5)\).

output band 0.0 0.0 -0.5 0.0 0.0 0.5 65
bs_unfolding .true.
The band structure along these two paths in SC and PC are:

Other input files

 1  1 -1
 1 -1  1 
-1  1  1 


1
1
1
1
5
5
5
5
The format are exatlly the same as the previous examples.

Run the calculation

A parallel simulation with N processes can be launced by typing

mpirun -n N your_aims.x > aims.out
The binary name your_aims.x should be replaced by your FHI-aims binary file.

Checking the results

The format of output files are exactly the same as the previous example. After post-processing, we can visualize the unfolded band:

Warning

Currently, the band unfolding implementation in FHI-aims is limited to SC with atomic displacements, meaning the total number and species of atoms remain unchanged, with no vacancies or interstitial atoms in the SC. This constraint ensures that we can easily establish a mapping between each atom in the SC and its counterpart in the PC. In cases involving vacancies or interstitials, the number of atoms in the PC and SC are not divisible, and some atoms in the SC may lack a corresponding atom in the PC, making it difficult to construct the translational operator of basis functions.

However, our band unfolding methodology remains valid for such systems. One potential solution is to introduce ghost atoms or ghost orbitals to ensure that the number of atoms in the SC and PC are divisible. This feature should be implemented in the future.

Solutions

You find all the solution to all the above exercises by clicking on the button below.

Show solutions to Part 3