nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

k.p dispersion in bulk unstrained ZnS, CdS and CdSe (wurtzite)

Note: This tutorial's copyright is owned by Stefan Birner, www.nextnano.de.

Author: Stefan Birner
Please send comments to nextnano3 (-at-) wsi.tum.de.

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1Dbulk_6x6kp_dispersion_ZnS.in
-> 1Dbulk_6x6kp_dispersion_CdS.in
-> 1Dbulk_6x6kp_dispersion_CdSe.in

Note: Not all features (e.g. shift-holes-to-zero, sg_bulk_dispersion.dat, bulk-kp-dispersion-3D) are implemented into the nextnano³ version of 2004_08_24. If you are interested in the executable that includes these features, please contact stefan.birner@nextnano.de.

k.p dispersion in bulk unstrained ZnS, CdS and CdSe (wurtzite)

This tutorial is based on

Valence band parameters of wurtzite materials
J.-B. Jeon, Yu.M. Sirenko, K.W. Kim, M.A. Littlejohn, M.A. Stroscio
Solid State Communications 99, 423 (1996)

• We want to calculate the dispersion E(k) from |k|=0 to |k|=0.2 [1/Angstrom] along the following directions in k space:
  - [000] to [0001], i.e. parallel to the c axis (Note: The c axis is parallel to the z axis.)
  - [000] to [110], i.e. perpendicular to the c axis (Note: The (x,y) plane is perpendicular to the c axis.)
  We compare 6x6 k.p theory results vs. single-band (effective-mass) results.
• We calculate E(k) for bulk ZnS, CdS and CdSe (unstrained).

STEP 1: Bulk dispersion along [0001]

• $output-kp-data
 destination-directory = kp_data1/

 bulk-kp-dispersion    = yes
 grid-position         = 5d0              ! in units of [nm]

 !----------------------------------------
 ! STEP 1:
 ! Dispersion along [001] direction, i.e. parallel to c=[0001] axis in wurtzite
 ! maximum |k| vector = 0.2 [1/Angstrom]
 !----------------------------------------
 k-direction           = 0d0  0d0  0.2d0 ! k-direction and range for dispersion plot [2pi/a] [1/Angstrom]

 number-of-k-points    = 99              ! number of k points to be calculated (resolution)
  $end_output-kp-data
• We calculate the pure bulk dispersion at grid-position=5d0, i.e. for the material located at the grid point at 5 nm. In our case this is ZnS but it could be any strained alloy. In the latter case, the k.p Bir-Pikus strain Hamiltonian will be diagonalized.
  The grid point at grid-position must be located inside a quantum cluster.
  shift-holes-to-zero = yes forces the top of the valence band to be located at 0 eV.
  How often the bulk k.p Hamiltonian should be solved can be specified via number-of-k-points. To increase the resolution, just increase this number.
• Start the calculation.
  The results of STEP 1 can be found in:
 kp_data1/kp_bulk_dispersion.dat (6x6 k.p)
   kp_data1/sg_bulk_dispersion.dat (single-band approximation)
  
  kp_bulk_dispersion.dat:
  The first column contains the |k| vector in units of [1/Angstrom], the next six columns the six eigenvalues of the 6x6 k.p Hamiltonian for this k=(k_x,k_y,k_z) point.
  The resulting energy dispersion is usually discussed in terms of a nonparabolic and anisotropric energy dispersion of heavy, light and split-off holes, including valence band mixing.
  
  sg_bulk_dispersion.dat:
  The first column contains the |k| vector in units of [1/Angstrom], the next three columns the energy for heavy (A), light (B) and crystal-field split-off (C) hole  for this k=(k_x,k_y,k_z) point.
  The single-band effective mass dispersion is parabolic and depends on a single parameter: The effective mass m*.
  Note that in wurtzite materials, the mass tensor is usually anisotropic with a mass m_zz parallel to the c axis, and two masses perpendicular to it m_xx=m_yy.

STEP 2: Bulk dispersion along [110]

• $output-kp-data
 ...
 !----------------------------------------
 ! STEP 2:
 ! Dispersion along [110] direction, i.e. perpendicular to c=[0001] axis in wurtzite
 ! maximum |k| vector = 0.2 [1/Angstrom]
 !----------------------------------------
 k-direction           = 0.141421356d0 0.141421356d0 0d0 ! k-direction and range for dispersion plot [2pi/a] [1/Angstrom]
 
• Here we use another direction, i.e. from [000] to [110]. The maximum value of |k| is 0.2 [1/Angstrom].
  Note that for values of |k| larger than 0.2 [1/Angstrom], k.p theory might not be a good approximation any more.
  This depends on the material system, of course.
• Start the calculation.
  The results of STEP 2 can be found in:
 kp_data1/kp_bulk_dispersion.dat (6x6 k.p)
   kp_data1/sg_bulk_dispersion.dat (single-band approximation)

STEP 3: Plotting the energy dispersion E(k)

• Here we will have to visualize the results of both STEP 1 and STEP 2.
  The final figures will look like this (left: dispersion along [0001] (STEP 1), right: dispersion along [110] (STEP 2)):
  
  
  
  
  
  
   
• These three figures are in excellent agreement to figure 1 of the paper by [Jeon].
• The dispersion along the hexagonal c axis is substantially different than the dispersion in the plane perpendicular to the c axis.
  The effective mass approximation is indicated by the dashed, grey lines.
  For the heavy holes (A), the effective mass approximation is very good for the dispersion along the c axis, even at large k vectors.
   
• For comparison, the single-band (effective-mass) dispersion is also shown. For ZnS, it corresponds to the following effective hole masses:
  
 valence-band-masses = 0.35d0  0.35d0  2.23d0 ! [m0] heavy hole A  (2.23 along c axis)
                       0.485d0 0.485d0 0.53d0 ! [m0] light    hole B  (0.53 along c axis)
                       0.75d0  0.75d0  0.32d0 ! [m0] crystal hole C  (0.32 along c axis)

The effective mass approximation is a simple parabolic dispersion which is anisotropic if the mass tensor is anisotropic (i.e. it also depends on the k vector direction).
  
  One can see that for |k| < 0.05 [1/Angstrom] the single-band approximation is in excellent agreement with 6x6 k.p but differs at larger |k| values substantially.
   
• Plotting E(k) in three dimensions
  Alternatively one can print out the 3D data field of the bulk E(k) = E(k_x,k_y,k_z) dispersion.
  
  $output-kp-data
  ...
  bulk-kp-dispersion-3D = yes

!----------------------------------------
! maximum |k| vector = 0.12 [1/Angstrom]
!----------------------------------------
  k-direction           = 0d0  0d0 0.2d0   ! k-direction and range for dispersion plot [2pi/a] [1/Angstrom]
  number-of-k-points    = 41               ! number of k points to calculated (resolution)

  The meaning of number-of-k-points = 41 is the following:
  40 k points from '- maximum |k| vector' to zero and
  40 k points from zero to  '+ maximum |k| vector' along all three directions,
  i.e. the whole 3D volume then contains 81 * 81 * 81 = 531441 k points.

• Please help us to improve our tutorial! Send comments to nextnano3 (-at-) wsi.tum.de.