Band structure of nanocrystals

The 'fuzzy band structure' of an nanocrystal (i.e. cluster without periodic boundary condition) can be obtained by Fourier transform of it's eigenstates. See Hapala et.al,Phys.Rev.B87,195420. In fireball there are two methods to do this projection.

1. 3D Fourier transform of molecular orbitals - in this method we choose several molecular orbitals, project them on grid and let fireball execute Fast fourier transform on this grid. The problem is that the result is complex ( real part correspond to symmetric components of the real wavefunction and imaginary part to antisymetric componenets ). For easier visualization we compute square of absolute value, which corresponds to density of the state in k-space. This property is exported as .xsf file. There also other possible formats of outputs - as described further.
2. projection of Molecular orbitals on just selected planewaves sampled along some 1D lines in k-space - again, molecular orbitals are first projected onto real space grid, then on this gird are multiplied by planewave and product is integrated.

this is implemented only in Prokop's personal versions of Fireball located in

/data/home/hapala/Fireball_dev/src_1.0-BSfinal
/data/home/hapala/Fireball_dev/progs_Jellium_mod

Interface for outputing 3D fourier transform of MOs is modified form standard interface for outputting real-space projection of MOs. You have to switch on iwrtewf = 1 and set iewform = 5. Than you choose particular set of MO to plot by npbands and pbands. The example fireball.in looks like this:

fireball.in

&OPTION
basisfile  = answer.bas 
nstepf     = 1
icluster   = 1
ifixcharge = 1
dt         = 0.5
&END
&OUTPUT
iwrtewf     = 1
&END
&MESH
iewform = 5
npbands = 5
pbands = 159,160,161,162,163
&END

You have to provide also file called kmap.optional where are written more speccific stettings. In this file you can choose byEnerg option to automatically evaluate all orbitals in particular energy window Emin and Emax instead of listing all orbitals in pbands variable. Then there are options of different output format - either 3D grid in .xsf or 2D image .ppn. Both options can be switched on simultaneously. In case you choose the .ppn image the 3D grid is collapsed into 2D by evaluationg maximum along z-axis. You can also plot independently real and imaginary part of projection ( instead of just square of absolute value) by option PlotImag .

Then there are some very specific options GetMaxPos to output position of global maximum projection for each orbital, and ValIn to output value of the projection in particular set of points in k-space. This options are here mostly for historical reasons, and I don't expect they will be much used in general. They are shortly described later, for completeness.

The example of kmap.optional looks like:

1                  # byEnerg   - if 1 choose orbitals by energy range else choose by pbands list
-8.59705   1.09186 # Emin Emax - energy range
0                  # PlotXSF?  - save .xsf files
0                  # PlotPPN?  - save 2D projections (max along z-axis) into .ppn image file for quick overview 
0                  # PlotImag? - save real and imaginery part to independent files  fft_Im_XXXX.xsf and fft_Re_XXXX.xsf
1                  # GetMaxPos - if 1 for each MO writes out position of global maximum of k-space density int kmaxs.dat
5.50               # alat      - lattice constant in angstroem
0                  # ValIn     - write out value in different points defined in ksamples.optional

If you choose ValIn option you have to specify points in k-space where the value of projection should be taken. This is done in ksample.optional file. Example of ksample.optional file is there:

ksample.optional

  26                  nksamples
  2.0  0.0  0.0   # HOMOs Gamma 100
  0.0  2.0  0.0   
  0.0  0.0  2.0    
 -2.0  0.0  0.0 
  0.0 -2.0  0.0   
  0.0  0.0 -2.0   
  1.0  1.0  1.0   # HOMOs Gamma 111
 -1.0  1.0  1.0   
  1.0 -1.0  1.0   
 -1.0 -1.0  1.0   
  1.0  1.0 -1.0   
 -1.0  1.0 -1.0   
  1.0 -1.0 -1.0   
 -1.0 -1.0 -1.0   
  0.0  1.0  1.0   # LUMOs   X
  0.0 -1.0  1.0   
  0.0  1.0 -1.0   
  0.0 -1.0 -1.0   
  1.0  0.0  1.0  
 -1.0  0.0  1.0   
  1.0  0.0 -1.0   
 -1.0  0.0 -1.0   
  1.0  1.0  0.0   
 -1.0  1.0  0.0 
  1.0 -1.0  0.0 
 -1.0 -1.0  0.0 

The code outputs fourier_XXXX.xsf if you choose writeXSF and orbfft_XXXX.ppm if you choose writePPN option. If you choose PlotImag option you in addition get separate real and imaginary part of projection ( corresponds to symmetric and antisymmetric component ) in files fft_Im_XXXX.xsf and fft_Re_XXXX.xsf.

If you choose option GetMaxPos you obtaion file kmaxs.dat with following structure:

kmaxs.dat

 iband,    Ei               imax,    jmax,     kmax, ValMax,              rimax,          rjmax,        rkmax
 107     -7.40951516        -3        -3        -3   2968.71621697     -0.70854607     -0.70769619     -0.70733483
 108     -7.38731466        -4        -3        -2   3133.46757266     -0.94472810     -0.70769619     -0.47155655
 109     -7.38677828        -4        -2        -3   3128.34104148     -0.94472810     -0.47179746     -0.70733483
 110     -7.26838823        -4        -4         0   1878.06049291     -0.94472810     -0.94359492      0.00000000
 ...

where Ei is eigenvalue of the MO, (imax,jmax,kmax) is index of position of maximum inside the grid, ValMax is value of projection in the maximum, and (rimax,rjmax,rkmax) are coordinates of the maximum in multiples of reciprocal lattice vector.

ksamples.dat

  Ei         rho(k1)          rho(k2)        rho(k3)         rho(k4)          rho(k5)        rho(k6)
-2.08815     16.68657904      3.16266264     10.09450414     16.68657904      3.16266264     10.09450414     ...
-2.08681      5.87885299     17.02497532     10.22090870      5.87885299     17.02497532     10.22090870     ... 
-2.06449     28.89941918     27.76432636     29.01794378     28.89941918     27.76432636     29.01794378     ...
...

To evaluate band structure projections densly sampled along particular 1D hi-symmetry lines in reciprocal space, you have to again set 'iwrtewf = 1' as in previous case but with option 'iewform = 6'. So the 'fireball.in' file looks eg. like this:

fireball.in

&OPTION
basisfile  = answer.bas 
nstepf     = 1
icluster   = 1
ifixcharge = 1
dt         = 0.5
&END
&OUTPUT
iwrtewf     = 1
&END
&MESH
iewform = 6
&END

In addition you have to provide file kscan.optional which contains information about the 1D cut directions. In principle it is possible take 1D cuts in arbitrary direction in k-space independent of the crystal symmetry. However, usually it make sense to respect crystal symmetry, and sample all the equivalent lines. In this particular example we choose all equivalent lines which conect the point X and Gamma in 2'nd reciprocal unit cell of silicon lattice. There is 48 such directions. Each of this lines is sampled by 100 equidistant k-points. And we choose to project all molecular orbitals in energy range from -8.000 to 1.000 eV. The example kscan.optional then look like this:

kscan.optional

1                                # byEnerg
-8.0000  1.0000                  # Emin Emax
5.50                             # alat
48   100                         # nlines nkpoints
2.0  0.0  0.0   1.0  1.0  0.0
2.0  0.0  0.0   1.0 -1.0  0.0 
2.0  0.0  0.0   1.0  0.0  1.0 
2.0  0.0  0.0   1.0  0.0 -1.0
0.0  2.0  0.0   1.0  1.0  0.0
0.0  2.0  0.0  -1.0  1.0  0.0
0.0  2.0  0.0   0.0  1.0  1.0
0.0  2.0  0.0   0.0  1.0 -1.0
0.0  0.0  2.0  -1.0  0.0  1.0    
0.0  0.0  2.0   1.0  0.0  1.0  
0.0  0.0  2.0   0.0  1.0  1.0  
0.0  0.0  2.0   0.0 -1.0  1.0  
-2.0  0.0  0.0  -1.0  1.0  0.0
-2.0  0.0  0.0  -1.0 -1.0  0.0 
-2.0  0.0  0.0  -1.0  0.0  1.0 
-2.0  0.0  0.0  -1.0  0.0 -1.0 
0.0 -2.0  0.0   1.0 -1.0  0.0
0.0 -2.0  0.0  -1.0 -1.0  0.0
0.0 -2.0  0.0   0.0 -1.0  1.0
0.0 -2.0  0.0   0.0 -1.0 -1.0
0.0  0.0 -2.0   1.0  0.0 -1.0    
0.0  0.0 -2.0  -1.0  0.0 -1.0  
0.0  0.0 -2.0   0.0  1.0 -1.0  
0.0  0.0 -2.0   0.0 -1.0 -1.0 
1.0  1.0  1.0    0.0  1.0  1.0
1.0  1.0  1.0    1.0  0.0  1.0
1.0  1.0  1.0    1.0  1.0  0.0
-1.0  1.0  1.0    0.0  1.0  1.0
-1.0  1.0  1.0   -1.0  0.0  1.0
-1.0  1.0  1.0   -1.0  1.0  0.0
1.0 -1.0  1.0    0.0 -1.0  1.0 
1.0 -1.0  1.0    1.0  0.0  1.0 
1.0 -1.0  1.0    1.0 -1.0  0.0 
-1.0 -1.0  1.0    0.0 -1.0  1.0
-1.0 -1.0  1.0   -1.0  0.0  1.0
-1.0 -1.0  1.0   -1.0 -1.0  0.0
1.0  1.0 -1.0    0.0  1.0 -1.0 
1.0  1.0 -1.0    1.0  0.0 -1.0 
1.0  1.0 -1.0    1.0  1.0  0.0 
-1.0  1.0 -1.0    0.0  1.0 -1.0 
-1.0  1.0 -1.0   -1.0  0.0 -1.0 
-1.0  1.0 -1.0   -1.0  1.0  0.0 
1.0 -1.0 -1.0    0.0 -1.0 -1.0
1.0 -1.0 -1.0    1.0  0.0 -1.0
1.0 -1.0 -1.0    1.0 -1.0  0.0
-1.0 -1.0 -1.0    0.0 -1.0 -1.0 
-1.0 -1.0 -1.0   -1.0  0.0 -1.0 
-1.0 -1.0 -1.0   -1.0 -1.0  0.0 

Two kinds of outputfiles are produced. klines_XXXXX.dat is producet for each molecular orbital, and in addition klines_TOT.dat summarize results for all orbitals together. In the klines_XXXXX.dat files the raw projection along each lines are written separately in colmuns (each row is different k-sample along the line, each column is different line). The format looks like this:

klines_XXXXX.dat

       1    391.30476603    391.30476603    391.30476603    391.30476603    287.45162513  ...  
       2    391.20959758    391.21921233    390.08793464    391.61800721    295.99428664  ...  
       3    389.17880205    389.77130848    387.81208053    388.10778349    306.53352273  ...  
       ...

Contrarly, in klines_TOT.dat each row correspond to one molecular orbital (ordered by energy), the first column is the orbital energy eigenvalue, and in columns are projections corresponding to each k-sample along the line, aggreated (reduced) over all lines. In current implementation maximum over each line is used for aggregation (but it would make sense also to use summation). Naturally, this aggregated post processed information make sense only if all k-lines are equivalent.

klines_TOT.dat

Ei          rho(k1)         rho(k2)         rho(k3)         rho(k4)         rho(k5)       ...    
-7.99755    413.91048549    428.76133304    440.87314746    450.14882297    456.52837031  ...
-7.99140    394.18797633    403.02567283    409.86012222    414.38007661    416.29758353  ... 
-7.97472    418.51477832    433.83793597    448.50901654    462.18377663    474.53904963  ...
-7.96729    262.72449682    269.69434762    274.26597685    275.89514172    285.23659456  ... 
...

The file klines_TOT.dat is supposed to be used as a quick overview of the whole bandstructure, but detailed information should be read directly from klines_XXXXX.dat. Note that even though in bulk material (e.g. bulk silicon) all k-lines can be equivalent due to the symmetry of the lattice, this symmetry is usually not conserved in case of finite nanocrystal with some surface passivation. For this reason the value of projection may be quite different for different k-lines, which would be equivalent from lattice symmetry point of view. An other reason why the symmetric k-lines may not be equivalenet is that there are several symmetry degenerated states, and eingenvalue solver choose some arbitrary linear combination of these eingenvectors diregarding the symmetry.

For easy and quick plotting of results from 'klines_TOT.dat' file you may use this simple python. To make it work you will need installed pylab

#!/usr/bin/python
from pylab import *
import numpy
dE    = 0.02             # [ eV ]choose width of energy-bins for plotting
F = transpose(genfromtxt( 'klines_TOT.dat' ))
Es = F[0]
Ps = transpose(F[1:])
Emin =Es.min(); Emax =Es.max()
Pgrid = zeros( ( int( (Emax-Emin)/dE )+1 ,  shape(Ps)[1] ) )
n = len(Es)
for i in range(n):
	iE = int((Es[i]-Emin)/dE)
	#Pgrid[iE] += Ps[i] 					      # overlaping (=degenerated) states are summed up
	Pgrid[iE] = numpy.vstack([Pgrid[iE],Ps[i] ]).max(axis=0)      # for overlaping (=degenerated) states is taken maximum
# choose color map reference here http://wiki.scipy.org/Cookbook/Matplotlib/Show_colormaps
cmap='spectral'
figure( figsize=(5,10) )
extent = ( 0,2, Emin, Emax ) 
imshow( Pgrid, origin='image', extent=extent, cmap = cmap )
xticks([0,2] , ['$\Gamma$','X'], fontsize=16)        # x-axis ticks Gamma and X vector
colorbar()
savefig('FuzzyBand.png',   bbox_inches='tight' )
show()