Unfolding of band structures

In the subsection, we show how the band structure calculated for a supercell can be unfolded into the Brillouin zone of a reference unit cell that a user specifies. As an example, let us consider again SiC in a two-dimensional honeycomb structure without imperfection. However, the unit cell in this case is extended to the ($2\times 2$) supercell as given by

  Atoms.Number 12
  Atoms.SpeciesAndCoordinates.Unit  FRAC  # Ang|AU
  <Atoms.SpeciesAndCoordinates
  1 C    0.16666666   0.33333333   0.50000000  2 2
  2 C    0.66666666   0.33333333   0.50000000  2 2
  3 C    0.16666666   0.83333333   0.50000000  2 2
  4 C    0.66666666   0.83333333   0.50000000  2 2
  5 Si   0.33333333   0.16666666   0.50000000  2 2
  6 Si   0.83333333   0.16666666   0.50000000  2 2
  7 Si   0.33333333   0.66666666   0.50000000  2 2
  8 Si   0.83333333   0.66666666   0.50000000  2 2
  9 Te   0.00000000   0.00000000   0.50000000  0 0
  10 Te   0.50000000   0.00000000   0.50000000  0 0
  11 Te   0.00000000   0.50000000   0.50000000  0 0
  12 Te   0.50000000   0.50000000   0.50000000  0 0
  Atoms.SpeciesAndCoordinates>

  Atoms.UnitVectors.Unit             Ang   # Ang|AU
  <Atoms.UnitVectors
  6.138  0.0000000000  0.00
  -3.069  5.3156639282  0.00
  0.000  0.0000000000 10.00
  Atoms.UnitVectors>

The SCF calculation for the supercell of the two-dimensional SiC can be performed as

  % mpirun -np 16 openmx SiC_C_NSP_P.dat > sic_c_nsp_p.std &

where the input file 'SiC_C_NSP_P.dat' can be found in the directory 'work/unfolding_example'. After finishing the SCF calculation, you obtain the following files relevant to the unfolding calculation:

  sic_c_nsp_p.unfold_totup
  sic_c_nsp_p.unfold_orbup
  sic_c_nsp_p.unfold_plotexample

By plotting 'sic_c_nsp_p.unfold_totup' with gnuplot together with the band structure of the primitive cell as a reference, you may obtain a figure as shown in Fig. 51(a). It is confirmed that the unfolded bands of the supercell exactly recover those of the primitive cell as expected.

To perform the unfolding of bands in the calculation, the following keywords are specified in addition to the keywords explained in the previous subsection:

  <Unfolding.ReferenceVectors
  3.0690  0.0000000000   0.000
  -1.5345  2.6578319641   0.000
  0.0000  0.0000000000  10.000
  Unfolding.ReferenceVectors>

  <Unfolding.Map
  1 1
  2 1
  3 1
  4 1
  5 2
  6 2
  7 2
  8 2
  9 3
  10 3
  11 3
  12 3
  Unfolding.Map>

The specification of the keywords above are listed below.

• Unfolding.ReferenceVectors

  With the keyword 'Unfolding.ReferenceVectors' one can define a reference unit cell for which the unfolding of bands is performed based on Eq. (24) in Ref. [98]. In the example, the primitive cell was used as the reference cell. The format is the same as that for the keywords 'Atoms.UnitVectors', i.e., the first, second, and third lines correspond to a-, b-, and c-axes, respectively. It is also noted that the unit of the reference unit cell is the same as that used for the supercell, which is specified by the keyword 'Atoms.UnitVectors.Unit'. In the subsection 'Analysis of band structures', the keyword 'Unfolding.ReferenceVectors' was not specified explicitly. In such a case, 'Atoms.UnitVectors' is automatically set for 'Unfolding.ReferenceVectors'.

• Unfolding.Map

  The keyword 'Unfolding.Map' specifies how atoms in the supercell can be mapped to atoms in the reference cell. The first and second columns are the serial number of atom, which corresponds to the number of the first column in the specification of the keyword 'Atoms.SpeciesAndCoordinates', and an identification number representing the group to which the atom belongs. In the example above, atoms 1 to 4 belong to the group 1, and atoms 5 to 6 the group 2, and atoms 9 to 12 the group 3. Based on the information given by the keyword 'Unfolding.Map', the relabelling of the index for each atom is performed as given by Eq. (17) in Ref. [98], which is the the essential part in the unfolding procedure. It is also noted that the identification number should be positive and integer, while the the integer needs not to start from '1', and not to appear in ascending order.

Here we show one more example, i.e., a SiC ($2\times 2$) supercell in a two-dimensional honeycomb structure with a Si vacancy. The SCF calculation can be performed by

  % mpirun -np 16 openmx SiC_C_SP_V.dat > sic_c_sp_v.std &

where the input file 'SiC_C_SP_V.dat' can be found in the directory 'work/unfolding_example'. By removing a Si atom from the ($2\times 2$) supercell cell structure, the honeycomb structure with a Si vacancy was created as

  Atoms.Number 11
  Atoms.SpeciesAndCoordinates.Unit  FRAC  # Ang|AU
  <Atoms.SpeciesAndCoordinates
  1 C    0.16666666   0.33333333   0.50000000  2.5 1.5
  2 C    0.66666666   0.33333333   0.50000000  2.5 1.5
  3 C    0.16666666   0.83333333   0.50000000  2.5 1.5
  4 C    0.66666666   0.83333333   0.50000000  2.5 1.5
  5 Si   0.33333333   0.16666666   0.50000000  2.5 1.5
  6 Si   0.83333333   0.16666666   0.50000000  2.5 1.5
  7 Si   0.33333333   0.66666666   0.50000000  2.5 1.5
  8 Te   0.00000000   0.00000000   0.50000000  0.0 0.0
  9 Te   0.50000000   0.00000000   0.50000000  0.0 0.0
  10 Te   0.00000000   0.50000000   0.50000000  0.0 0.0
  11 Te   0.50000000   0.50000000   0.50000000  0.0 0.0
  Atoms.SpeciesAndCoordinates>

The mapping of atoms in the supercell to those in the reference cell was specified by

  <Unfolding.Map
  1 1
  2 1
  3 1
  4 1
  5 2
  6 2
  7 2
  8 3
  9 3
  10 3
  11 3
  Unfolding.Map>

After finishing the SCF calculation, you obtain the following files relevant to the unfolding calculation:

  sic_c_sp_v.unfold_totup
  sic_c_sp_v.unfold_totdn
  sic_c_sp_v.unfold_orbup
  sic_c_sp_v.unfold_orbdn
  sic_c_sp_v.unfold_plotexample

By plotting 'sic_c_sp_v.unfold_totup' and 'sic_c_sp_v.unfold_totdn' with gnuplot together with the band structure of the primitive cell as a reference, you may obtain a figure as shown in Fig. 51(b). It is found from the unfolded spectral weights that the characteristic feature of the perfect case is still preserved in spite of introduction of the vacancy, and the chemical potential is pushed up largely. We also see that the electronic state is spin-polarized due to dangling bonds of carbon atoms. The analysis shows that the unfolding method would be useful to analyze how the original bands are pertubed by the introduced imperfection.

When you consider an impurity instead of introduction of vacancy, you only have to assign the impurity with an identification number. For example, if you introduce an impurity of atom 13 in the SiC ($2\times 2$) supercell, you can define the mapping rule in relabelling as

  <Unfolding.Map
  1 1
  2 1
  3 1
  4 1
  5 2
  6 2
  7 2
  8 2
  9 3
  10 3
  11 3
  12 3
  13 4
  Unfolding.Map>

In case of multi-impurities and existence of surfaces, you can define the mapping in a similar way above.

Figure 51: (a) Band structure of a SiC ($2\times 2$) supercell in a two-dimensional honeycomb structure without imperfection as shown in the inset, where the red line was obtained by the conventional calculation for the primitive cell, and the green circles represent the total spectral weight obtained by unfolding. The radius reflects the magnitude of the weight. The calculation was performed by using an input file 'SiC_C_NSP_P.dat' in the directory 'work/unfolding_example'. (b) Band structure of a SiC ($2\times 2$) supercell in a two-dimensional honeycomb structure with a Si vacancy as shown in the inset, where the red line was obtained by the conventional calculation for the primitive cell, and the green and blue circles represent the total spectral weight obtained by the unfolding procedure for up- and down-spin states, respectively. The radius reflects the magnitude of the weight. The calculation was performed by using an input file 'SiC_C_SP_V.dat' in the directory 'work/unfolding_example'.