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

Input files:
• bulk_6x6kp_dispersion_ZnS_nnp.in

• bulk_6x6kp_dispersion_CdS_nnp.in

• bulk_6x6kp_dispersion_CdSe_nnp.in

• bulk_6x6kp_dispersion_ZnO_nn3.in

Scope:

We calculate $$E(k)$$ for bulk $$ZnS$$, $$CdS$$, $$CdSe$$ and $$Zn0$$ (unstrained). In this tutorial we aim to reproduce results of [Jeon1996].

Introduction¶

We want to calculate the dispersion $$E(k)$$ from $$|k|$$ = 0 [1/nm] to $$|k|$$ = 1.0 [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.

Bulk dispersion along [0001] and [110]¶

quantum{
region{
...
bulk_dispersion{
path{ # dispersion along arbitrary path in k-space
name = "user_defined_path"
position{ x = 5.0 }
point{ k = [0.7071, 0.7071, 0.0] }
point{ k = [0.0, 0.0, 1.0] }
spacing  = 0.01                # [1/nm]
shift_holes_to_zero = yes
}
}
}
}


We calculate the pure bulk dispersion at grid position x = 5.0, 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 inside position{} must be located inside a quantum region. 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 spacing. To increase the resolution, just increase this number. The maximum value of $$|k|$$ is 1.0 [1/nm]. Note that for values of $$|k|$$ larger than 1.0 [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 the folder bias_00000\Quantum\Bulk_dispersions.

The files bulk_6x6kp_dispersion_as_in_inputfile_kxkykz_000_kxkykz.dat for instance contain 6-band k.p dispersions: The first column contains the $$|k|$$ vector in unitsHere we visualize the results. The final figures will look like this (left: dispersion along [0001], right: dispersion along [110]): of [1/nm], the next six columns the six eigenvalues of the 6-band k.p Hamiltonian for this $$k$$ = ($$k_x$$, $$k_y$$, $$k_z$$) point.

The resulting energy dispersion in 6-band k.p theory is usually discussed in terms of a nonparabolic and anisotropic energy dispersion of heavy, light and split-off holes, including valence band mixing.

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 $$m_{xx}$$ = $$m_{yy}$$.

Results¶

We visualize now the results in Figure 2.4.10.1, Figure 2.4.10.2 and Figure 2.4.10.3. The final figures will look like this (left: dispersion along [0001], right: dispersion along [110]):

These three figures are in excellent agreement to Fig. 1 of the paper by [Jeon1996]. The dispersion along the hexagonal c axis is substantially different from the dispersion in the plane perpendicular to the c axis. The effective mass approximation is indicated by the dashed, gray 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_bands{
HH{ mass_l = 2.23  mass_t = 0.35}   # [m0] heavy hole A  (2.23 along c axis)
LH{ mass_l = 0.53  mass_t = 0.485}  # [m0] light hole B  (0.53 along c axis)
SO{ mass_l = 0.32  mass_t = 0.75}   # [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 $$E(k)$$ = $$E(k_x, k_y,k_z)$$ dispersion.

full{ # 3D dispersion on rectilinear grid in k-space
name = "3D"
position{ x = 5.0 }
kxgrid {
line{ pos = -1  spacing = 0.04 }
line{ pos =  1  spacing = 0.04 }
}
kygrid {
line{ pos = -1  spacing = 0.04 }
line{ pos =  1  spacing = 0.04 }
}
kzgrid {
line{ pos = -1  spacing = 0.04 }
line{ pos =  1  spacing = 0.04 }
}
shift_holes_to_zero = yes
}
}


k.p dispersion in bulk unstrained ZnO¶

Figure 2.4.10.4 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:

• bulk_6x6kp_dispersion_as_in_inputfile_kxkykz_000_kxkykz.dat

• bulk_sg_dispersion.dat

The files

• bulk_6x6kp_dispersion_axis_-100_000_100.dat and

• 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 $$k_z$$ direction (corresponding to [0001]) is isotropic.