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Energy levels in a pyramidal shaped InAs/GaAs quantum dot including strain
and piezoelectric fields
Author:
Stefan Birner
> 3DInAsGaAsQDPyramid_PryorPRB1998_10nm_nn3.in / *nnp.in 
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Energy levels in a pyramidal shaped InAs/GaAs quantum dot including strain
and piezoelectric fields
> 3DInAsGaAsQDPyramid_PryorPRB1998_10nm.in 
QD with 10 nm base length
This tutorial is based on the following paper:
Eightband calculations of strained InAs/GaAs quantum dots compared with
one, four, and sixband approximations
Craig Pyror
Physical Review B 57 (12), 7190 (1998)
We use identical material parameters with respect to this paper in order to
make it possible to reproduce Pryor's results.
We note that meanwhile more realistic material parameters are available and
that for the simulation of realistic quantum dots the inclusion of the wetting
layer and an appropriate nonlinear InGaAs alloy profile is recommended.
We make the following simplifications in order to be consistent with Pryor:
 The wetting layer is omitted for simplicity.
 The QD material is purely InAs.
 The barrier material is purely GaAs.
 The dielectric constant in the barrier material (GaAs) is the one for
InAs.
 Periodic boundary conditions are assumed in all three directions for the
strain equation.
 The QD shape is a pyramid with a square base (base length = 10 nm) and a
height of 5 nm.
The four side walls of the pyramid are oriented in the (011), (011), (101)
and (101) planes, respectively.
The whole simulation area has the dimensions 44 nm x 44 nm x 40 nm.
Conduction and valence band profiles
The following figures shows the conduction and valence band edges (heavy hole,
light hole and splitoff hole) for a 10 nm pyramidal shaped QD along two
different line scans.
The energies of the bands have been obtained by diagonalizing the 8band k.p
Hamiltonian at k = 0 (including the BirPikus strain Hamiltonian) for
each grid point, taking into account the local strain tensor and deformation
potentials.
Note that piezoelectric effects are not included yet in this band profile.
The left figure shows the band profile along the z axis through the
center of the QD (x = y = 0 nm).
The right figure shows the band profile along the x axis through the base
of the QD (y = z = 0 nm).
The figures compare well with Pryor's Figs. 2(a) and 2(b). However, there are
some differences:
Due to valence band mixing of the states in the k.p Hamiltonian, we do
not have pure heavy and light hole eigenstates any more.
Thus there is some arbitrariness to assign the labels "heavy" and "light" to the
relevant eigenstates h_{1} and h_{2}.
Obviously, when solving the full 6band or 8band k.p Hamiltonian, this
labelling becomes irrelevant because all three hole band edges enter the
Hamiltonian simultaneously (in contrast to a singleband effective mass approach
where only individual "heavy" hole or "light" hole band edges would be
considered).
Electron wave function of the ground state
(singleband effectivemass approximation)
... to be continued.
Preliminary:
10 nm quantum dot (Note: Pryor's Fig. 7 shows the energies
for a 14 nm quantum dot).
cb edge: 0.762074 eV
vb edge:  0.756926 eV
=> band gap: 1.519 eV
Electron energies
(i) effective mass (m_{e} = 0.023 m_{0}) =>
0.7000983 eV (only one confined electron state) + 0.756926 eV = 1.4570243
eV
(ii) effective mass (m_{e} = 0.04 m_{0})
=> eV
(iii) effective mass (m_{e}(r) = ... m_{0})
=> not implemented in nextnano³
(iv) 8band k.p
=> eV
Hole energies
( ) effective mass (m_{hh} = 0.41 m_{0}) =>
hh1 = 0.585198481 eV + 0.756926 eV = 0.171727 eV
=> hh1 = 0.61776 eV + 0.756926 eV = 0.139166 eV
=> hh1 = 0.62275 eV + 0.756926 eV = 0.134172 eV
(i) 6band k.p
=> 1.0081402 eV (?) (bad eigenvalues using 6band k.p with
finitedifferences )
(ii) 8band k.p
=> eV
Transition energy electron  hole:
 (i)  ( ): exciton correction 2.9 meV (Pryor: 27 meV)
E_ex [eV] E_el  E_hl E_el0  E_hl0
Delta_Ex REAL(inter_matV(1))
1.28238 1.27958
1.28530 0.00291947
0.428169
14 nm quantum dot (Pryor's Fig. 7).
Electron energies
(i) effective mass (m_{e} = 0.023 m_{0}) =>
0.6458949 eV (only one confined
electron state) + (1.519  0.752916) eV = 1.412 eV (in substrate layer below QD)
(i) effective mass (m_{e} = 0.023 m_{0}) =>
0.6458949 eV (only one confined
electron state) + (1.519  0.765522) eV = 1.399 eV (in substrate layer at
corner)
(i) effective mass (m_{e} = 0.04 m_{0})
=> 0.6248762 eV (only one confined electron
state) + (1.519  0.765522) eV = 1.378 eV (in substrate layer at corner)
14 nm, 6x6k.p, box, nonsym:
0.56607270
0.58734305
0.59621434
0.60757551
0.62802221
0.63650764
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