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  1D bulk k.p dispersion in II-VI




nextnano3 - Tutorial

next generation 3D nano device simulator

1D Tutorial

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

Author: Stefan Birner

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k.p dispersion in bulk unstrained ZnS, CdS, CdSe and ZnO (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 [1/nm] to |k|=2.0 [1/nm] 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 6-band k.p theory results vs. single-band (effective-mass) results.
  • We calculate E(k) for bulk ZnS, CdS and CdSe (unstrained).


Bulk dispersion along [0001] and [110]

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

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

     ! Dispersion along [001] direction, i.e. parallel      to c=[0001] axis in wurtzite
     ! Dispersion along [110] direction, i.e. perpendicular to c=[0001] axis in wurtzite
     ! maximum |k| vector = 2.0 [1/nm]
     k-direction-from-k-point = 0d0          0d0          2.0d0 !
    k-direction and range for dispersion plot [1/nm]
     k-direction-to-k-point   = 1.41421356d0 1.41421356d0 0d0   !
    k-direction and range for dispersion plot [1/nm]

    The dispersion is calculated from the k point 'k-direction-from-k-point' to Gamma, and then from the Gamma point to 'k-direction-to-k-point'.

     number-of-k-points       = 100             !
    number of k points to be calculated (resolution)
  • 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.
  • The maximum value of |k| is 2.0 [1/nm].
    Note that for values of |k| larger than 2.0 [1/nm], k.p theory might not be a good approximation any more.
    This depends on the material system, of course.
  • Start the calculation.
    The results can be found in:
    (6-band k.p)
     kp_bulk/bulk_sg_dispersion.dat                                      (single-band approximation)

    The first column contains the |k| vector in units of [1/nm], the next six columns the six eigenvalues of the 6-band k.p Hamiltonian for this k=(kx,ky,kz) 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.

    The first column contains the |k| vector in units of [1/nm], the next three columns the energy for heavy (A), light (B) and crystal-field split-off (C) hole  for this k=(kx,ky,kz) 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 mzz parallel to the c axis, and two masses perpendicular to it mxx=myy.




  • Here we visualize the results.
    The final figures will look like this (left: dispersion along [0001], right: dispersion along [110]):

  • 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| < 0.5 [1/nm] the single-band approximation is in excellent agreement with 6-band 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(kx,ky,kz) dispersion.

      bulk-kp-dispersion-3D  = yes

    ! maximum |k| vector = 2.0 [1/nm]
      k-direction-to-k-point = 0d0  0d0 2.0d0   !
    k-direction and range for dispersion plot [1/nm]
      number-of-k-points     = 40               !
    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 (plus the Gamma point) and
    40 k points from zero to  '+ maximum |k| vector' (plus the Gamma point) along all three directions,
    i.e. the whole 3D volume then contains 81 * 81 * 81 = 531441 k points.


k.p dispersion in bulk unstrained ZnO

The following figure shows the bulk 6-band k.p energy dispersion for ZnO.
The gray lines are the dispersions assuming a parabolic effective mass.

The following files are plotted:
- kp_bulk/bulk_6x6kp_dispersion_as_in_inputfile_kxkykz_000_kxkykz.dat
- kp_bulk/bulk_sg_dispersion.dat
The files
- bulk_6x6kp_dispersion_axis_-100_000_100.dat
- bulk_6x6kp_dispersion_diagonal_-110_000_1-10.dat
contain the same data because for a wurtzite crystal, due to symmetry, the dispersion in the plane perpendicular to the kz direction (corresponding to [0001]) is isotropic.

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