BandDispersion: Calculation on the band dispersion relation

'kSpin' has four ways to calculate the k-space spin density matrix, which can be specified by a keyword 'Calc.Type'. Here we introduce BandDispersion among the four ways. BandDispersion can calculate the k-space spin density matrix resolved to each atom on the band dispersion relation within a user-specified energy range. First, the calculation of the band dispersion using BandDispersion is illustrated here with a simple model of an Au(111) surface. Then, you can calculate spin textures using 53.6 MulPCalc.

After a calculation of OpenMX with an input file 'Au111Surface_BD.dat' stored in the directory 'work', you may try to run 'kSpin'. The keywords relevant to the executation of kSpin can be found at the bottom of the input file 'Au111Surface_BD.dat' stored in the directory 'work' as shown below:

List of keywords relevant to kSpin

    Filename.scfout      Au111Surface.scfout 
    Filename.outdata     Au111Surface_BD     
    Calc.Type            BandDispersion    # FermiLoop, GridCalc,
                                             BandDispersion, or MulPOnly 
                                             default: MulPOnly
    Energy.Range        -1.0  1.0          # eV; default: 0.0  0.0
    Band.Nkpath                2
    <Band.kpath
      135  0.0  0.500000  0.000000    0.0   0.000000  0.000000     M  G
      135  0.0  0.000000  0.000000    0.0  -0.500000  0.000000     G -M
    Band.kpath>

Specification of keywords

The specification of each keyword is explained below:

Filename.scfout
Specify the name of the scfout file which will be read by 'kSpin'.

Filename.outdata
Specify a name for output files. This keyword corresponds to the keyword 'System.Name' for OpenMX calculations.

Calc.Type
Choose either FermiLoop, GridCalc, BandDispersion, or MulPOnly. The default setting is MulPOnly. Here we choose BandDispersion for the exercise.

Energy.Range
Specify the two different values for an energy range for which BandDispersion should search bands. The unit is in eV. The default is '-0.5 0.5' (i.e. the range is [-0.5, 0.5]).

Band.Nkpath
Specify the number of k-paths along whichBandDispersion should calculate the band dispersion. The notation is the same as that in the conventional calculation.

Band.kpath
Specify k-paths along which BandDispersion should calculate the band dispersion. The notation is the same as that in the conventional calculation.

Calculation

The k-space spin density matrix resolved to each atom is calculated by a post-processing code 'kSpin' in the directory 'work'. Please move to the directory 'work', and perform a calculation as follows:

    % ./kSpin Au111Surface_BD.dat

or for the MPI calculation, for example, the case with 4 MPI processes

    % mpirun -np 4 ./kSpin Au111Surface_BD.dat

As the calculation proceeds, you may see the following standard output:

    ******************************************************************
    ******************************************************************
     kSpin:
     code for evaluating spin related properties
     in momentum space of solid state materials.
     Copyright (C), 2019,
     Hiroki Kotaka, Naoya Yamaguchi and Fumiyuki Ishii.
     This software includes the work that is distributed
     in version 3 of the GPL (GPLv3).
 
     Please cite the following article:
     H. Kotaka, F. Ishii, and M. Saito,
     Jpn. J. Appl. Phys. 52, 035204 (2013).
     DOI: 10.7567/JJAP.52.035204.
    ******************************************************************
    ******************************************************************


    Input filename is "Au111Surface.scfout"  


    Start "BandDispersion" Calculation (3).  
    line_Nk[1]:  0.665829
    ( 0.000000,  0.500000,  0.000000 ) -> ( 0.000000,  0.000000,  0.000000 )
    line_Nk[2]:  1.331658
    ( 0.000000,  0.000000,  0.000000 ) -> ( 0.000000, -0.500000,  0.000000 )
    ########### ORBITAL DATA ##################
    ClaOrb_MAX[0]:   2
    ClaOrb_MAX[1]:   8
    Total Band (2*n): 124
    ###########################################
    Band.Nkpath: 2
    135 ( 0.000000,  0.500000,  0.000000 ) >>> ( 0.000000,  0.000000,  0.000000 ) M G
    135 ( 0.000000,  0.000000,  0.000000 ) >>> ( 0.000000, -0.500000,  0.000000 ) G -M
    l_min:  49   l_max:  56   l_cal:   8 
    Au111Surface_BD.Band49_1
    Au111Surface_BD.Band50_1
    Au111Surface_BD.Band51_1
    Au111Surface_BD.Band52_1
    Au111Surface_BD.Band53_1
    Au111Surface_BD.Band54_1
    Au111Surface_BD.Band55_1
    Au111Surface_BD.Band56_1
    l_min:  49   l_max:  56   l_cal:   8 
    Au111Surface_BD.Band49_2
    Au111Surface_BD.Band50_2
    Au111Surface_BD.Band51_2
    Au111Surface_BD.Band52_2
    Au111Surface_BD.Band53_2
    Au111Surface_BD.Band54_2
    Au111Surface_BD.Band55_2
    Au111Surface_BD.Band56_2
    ###########################################
    Total MulP data:2176
    ############ CALC TIME ####################
      Total Calculation Time:  4.772406 (s)
    ###########################################

When the calculation is completed normally as shown above, you can find the following output files in the directory 'work':

    Au111Surface_BD.BAND
    Au111Surface_BD.Band49_1
    Au111Surface_BD.Band50_1
    Au111Surface_BD.Band51_1
    Au111Surface_BD.Band52_1
    Au111Surface_BD.Band53_1
    Au111Surface_BD.Band54_1
    Au111Surface_BD.Band55_1
    Au111Surface_BD.Band56_1
    Au111Surface_BD.Band49_2
    Au111Surface_BD.Band50_2
    Au111Surface_BD.Band51_2
    Au111Surface_BD.Band52_2
    Au111Surface_BD.Band53_2
    Au111Surface_BD.Band54_2
    Au111Surface_BD.Band55_2
    Au111Surface_BD.Band56_2
    Au111Surface_BD.AMulPBand
    Au111Surface_BD.AMulPBand_s
    Au111Surface_BD.AMulPBand_p
    Au111Surface_BD.AMulPBand_p1
    Au111Surface_BD.AMulPBand_p2
    Au111Surface_BD.AMulPBand_p3
    Au111Surface_BD.AMulPBand_d
    Au111Surface_BD.AMulPBand_d1
    Au111Surface_BD.AMulPBand_d2
    Au111Surface_BD.AMulPBand_d3
    Au111Surface_BD.AMulPBand_d4
    Au111Surface_BD.AMulPBand_d5
    Au111Surface_BD.plotexample
    Au111Surface_BD.atominfo
    temporal_12345.input

As an example, by executing the following command, you can obtain a figure of the band dispersion for the Rashba spin splitting in the Au(111) surface as shown in Fig. 67(a).

    % gnuplot Au111Surface_BD.plotexample

In order to obtain information of spin textures on the band dispersion, you may try to analyze by 53.6 MulPCalc.

**Figure 67:** (a) Band dispersion for the Rashba spin splitting at the Au(111) surface stored in a BAND file 'Au111Surface_BD.BAND'. You can get this figure by a plotexample file 'Au111Surface_BD.plotexample'. (b) k-space spin density matrix resolved to each atom for the Rashba spin splitting at the Au(111) surface. The circles represent the difference between the number of electrons with $\alpha$ - and $\beta$ -spin at each k-point, and the radius and color reflect its magnitude and sign, respectively, stored in a MulPop file 'Au111Surface_BD_MC.MulPop'. You can get this figure by a plotexample file 'Au111Surface_BD_MC.plotexample'.
$\includegraphics[width=16.0cm]{Rashba-Fig4.eps}$

Output files

The content of each output file is explained below:

BAND file
This file stores data for the band dispersion. The first and second columns correspond to the distance for the k-points along each k-path (in units of Bohr $^{-1}$ ) and the energy for each of them (in units of eV), respectively.

Band_YY_Z file
This file stores data for the band dispersion of each branch with the band index YY and k-path index Z. The notation of contents of this file is the same as the BAND file.

AMulPBand file
This file stores data of the k-space spin density matrix resolved to each atom, and can be analyzed by 53.6 MulPCalc.

AMulPBand_xx file
This file stores data of the xx-component of the k-space spin density matrix resolved to each atom, and can be analyzed by 53.6 MulPCalc.

plotexample file
This file supplies an example of gnuplot scripts.

atominfo file
This file supplies information of lattice vectors and PAOs.

temporal_12345.input
This file is a copy of the input file stored in the scfout file.

Analysis of the k-space spin density matrix resolved to each atom

MulPCalc can extract data to analyze the k-space spin density matrix resolved to each atom from AMulPBand files or AMulPBand_xx files. The executable file can be obtained by compilation in the directory 'source' as follows:

    % make MulPCalc

After the successful compilation, you can find the executable file 'MulPCalc' in the directory 'work'. Let us analyze the k-space spin density matrix resolved to the atomic contribution on the band dispersion for the Rashba spin splitting in the Au(111) surface. First, add the following keywords and values into the input file 'Au111Surface_BD.dat', for example:

    Filename.atomMulP Au111Surface_BD.AMulPBand # default: default
    Filename.xyzdata   Au111Surface_BD_MC       # default: default
    Num.of.Extract.Atom        3                # default: 1
    Extract.Atom               1 2 3            # default: 1 2 ... (Num.of.Extract.Atom)

After the above calculation, you can analyze the k-space spin density matrix resolved to the atomic contribution as follows:

    % ./MulPCalc Au111Surface_BD.dat

In addtion, you can adjust data for making better figures by the following keywords and values:

    MulP.Vec.Scale       0.1  0.1  0.1          # default: 1.0  1.0  1.0
    Data.Reduction             1                # default: 1

After executing MulPCalc, you can find the following output files in the directory 'work'.

    Au111Surface_BD_MC.MulPop
    Au111Surface_BD_MC.MulPop49
    Au111Surface_BD_MC.MulPop50
    Au111Surface_BD_MC.MulPop51
    Au111Surface_BD_MC.MulPop52
    Au111Surface_BD_MC.MulPop53
    Au111Surface_BD_MC.MulPop54
    Au111Surface_BD_MC.MulPop55
    Au111Surface_BD_MC.MulPop56
    Au111Surface_BD_MC.plotexample

As an example, by executing the following command, you can obtain a figure of the band dispersion for the Rashba spin splitting in the Au(111) surface with the k-space spin density matrix resolved to the atom contribution as shown in Fig. 67(b).

    % gnuplot Au111Surface_BD_MC.plotexample

For more information of MulPCalc, see also the subsection of 53.6 MulPCalc.